\section{Introduction} The energy momentum tensor should be symmetric due to the conservation of angular momentum \autocite{landau2013classical}. For a given Lagrangian \(\mathscr{L}\) of a field, there are two major ways to obtain the energy momentum tensor: from general relativity or from the canonical way. The energy momentum tensor obtained from general relativity is automatically symmetric. However, the canonical energy momentum is not symmetric, in general, if the field is not a scalar field. This situation occurs in an electromagnetic field. This asymmetric issue also causes the canonical energy-momentum tensor not to be gauge-invariant. Several ways are used to fix this asymmetric and resolve the gauge invariant, but all seem unnatural \autocite{freese2022noether}. \section{Abelian Gauge Theory (Electromagnetic Field)} \label{sec:Ab} \subsection{Derivation} We denote \(p\) is a point in spacetime manifold \(M\), \(A(p)\) as gauge potential 1-form of the electromagnetic field, and the field strength is \(F = dA\). Let \(G\) be a Lie group acting smoothly on a spacetime manifold \(M\) from the left, and let \(\mathfrak{g}\) denote the Lie algebra of \(G\). For a Lie algebra element \(X_{\mathfrak{g}} \in \mathfrak{g}\), when applying an infinitesimal group action \(g = \exp(t X_{\mathfrak{g}})\) to a point \(p \in M\), the point is actively moved to \(g * p\). The induced vector field \({\delta X}(p) \in \mathfrak{X}(M)\) is defined via the infinitesimal action: \[\begin{align*} {\delta X}(p) := \left. \frac{d}{dt} \right|_{t=0} \left( \exp(-t X_{\mathfrak{g}}) * p \right) = - T_e p (X_{\mathfrak{g}}). \end{align*}\] In Noether's variational framework, one compares physical fields—such as the gauge potential 1-form \(A\)—at the same spacetime point before and after the transformation. This requires pulling back the transformed field to the original point \(p\) for comparison. In this context, the Lie derivative reflects the infinitesimal pullback of the field to the original point, which inherently introduces a negative direction in the group flow. This geometric interpretation explains the frequently observed identity: \[\begin{align*} \delta A_\nu = - \left(\hat{\mathcal{L}}_{\delta X} A\right)_\nu=-\left(A_{\nu,\gamma}\delta X^\gamma+A_\gamma \delta {X^\gamma}_{,\nu}\right), \end{align*}\] which expresses that the variation of \(A\) under spacetime symmetry is equivalent to its Lie derivative along the \emph{backward} flow generated by \(X\). This convention aligns with the principle that physical variations are evaluated at fixed spacetime points in the coordinate chart. The Lagrangian \(\mathscr{L}_{EM}\) and the action \(S\) of electromagnetic field are \[\begin{align*} \mathscr{L}_{EM} = -\frac{1}{16\pi c} g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}F_{\mu\nu}\sqrt{-g} \end{align*}\] and \[ S = \int d^4x \, \mathscr{L}_{EM}[A_\mu(x^\gamma), A_{\nu,\mu}(x^\gamma), x^\gamma] \] We derive the equation of motion (EoM) and Noether theorem follow standard procedure. The variation of action \(\Delta S\) divides into two terms: \[ \delta S = \int \delta d^4x \cdot \mathscr{L}_{EM} + \int d^4x \cdot \delta \mathscr{L}_{EM} \] The first term is the variation of volume form, which is \autocite{ryder1996quantum} \[ \delta d^4x = {\delta{X}^\gamma }_{,\gamma} \cdot d^4x \] The second term is the variation of \(\mathscr{L}_{EM}\) \[\begin{align} \label{eq:ab_lag} \Delta \mathscr{L}_{EM} &= \mathscr{L}_{EM}[\tilde{A}_\nu(\tilde{x}^\gamma), \tilde{A}_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] - \mathscr{L}_{EM}[A_\nu(x^\gamma), A_{\nu,\mu}(x^\gamma), x^\gamma] \\ & = \left[ \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu} \delta A_\nu + \frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)} \delta (\partial_\mu A_\nu) \right] (x^\gamma) + \left[ {\mathscr{L}_{EM}}_{, \gamma} \delta x^\gamma \right] (x^\gamma) + O(\delta^2) \nonumber \end{align}\] Hence, \(\Delta S\) is \[\begin{align} \label{eq:Delta S_ab} \Delta S = \int \left[ \frac{\partial \mathscr{L}}{\partial A_\nu} - \partial_\mu\left(\frac{\partial\mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}\right) \right] \delta A_\nu d^4 x + \int \left[ \partial_\mu\left(\frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)}\delta A_\nu\right) + (\mathscr{L}_{EM} \delta x^\gamma)_{,\gamma}\right] d^4 x \end{align}\] The EoM is \[ \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu} - \partial_\mu \left( \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right) = 0 \] Now we introduce the total variation \(\Delta A = \delta A + \hat{\mathcal{L}}_{\delta x} A \), instead of the traditional \(\Delta A_\nu=\delta A_\nu +\delta x^\mu \partial_\mu A_\nu\). Using \(\delta A_\nu = \Delta A_\nu - A_{\nu,\gamma} \delta x^\gamma - A_\gamma \delta x^\gamma_{,\nu}\), \[\begin{align*} \Delta S &= \int \{EoM\} \delta A_\nu d^4 x + \int \partial_\mu\left[\frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} (\Delta A_\nu - A_{\nu,\gamma}\delta x^\gamma - A_\gamma \delta x_{,\nu}^\gamma) + \delta^\mu_\gamma\mathscr{L}_{EM} \delta x^\gamma\right]d^4x\\ &= \int \{EoM\} \delta A_\nu d^4 x + \int \partial_\mu \left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}\Delta A_\nu - \left( \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}A_{\nu,\gamma} \delta x^\gamma + \underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}A_\gamma \delta x^\gamma_{,\nu}}_{(*)} - \delta^\mu_\gamma \mathscr{L}_{EM}\delta x^\gamma\right)\right]d^4x \end{align*}\] Evaluate the \((*)\) term: \[\begin{align*} \underbracket[0.4pt][0pt]{\partial_\mu \left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} A_{\gamma} \delta x^\gamma_{,\nu} \right]}_{(*)} &= \underbracket[0.4pt][0pt]{\left[ \frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)} A_\gamma \delta x^\gamma \right]_{,\nu \mu}}_{(*1)} -\partial_\mu \left[ \underbracket[0.4pt][0pt]{ \left( \frac{\partial \mathscr{L}_{EM}}{ \partial (\partial_\mu A_\nu)}\right)_{,\nu} A_\gamma \delta x^\gamma}_{(*2)}\right] -\partial_\mu \left[ \underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}A_{\gamma,\nu} \delta x^\gamma}_{(*3)} \right] \end{align*}\] Note that \(\frac{\partial\mathscr{L}_{EM}}{\partial(\partial_\mu A_{\nu})}\) is \[\begin{align} \label{eq:ab_partial} \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} &= -\frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta} \end{align}\] The \((*1)\) term \[\begin{align} \label{eq:anti_ab} \underbracket[0.4pt][0pt]{\left[ \frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)} A_\gamma \delta x^\gamma\right]_{,\nu \mu}}_{(*1)} = \left[ -\frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta} A_\gamma \delta x^\gamma \right]_{,\nu \mu} = 0 \end{align}\] due to the antisymmetric of \(F_{\mu\nu}\) and the symmetric of second order derivative \( \{_{,\nu\mu}\}\).\\ The \((*2)\) term: \[\begin{align} \label{eq:*2_ab} \underbracket[0.4pt][0pt]{ \left( \frac{\partial \mathscr{L}_{EM}}{ \partial (\partial_\mu A_\nu)}\right)_{,\nu} A_\gamma \delta x^\gamma}_{(*2)}=0 \end{align}\] Hence \((*)\) only left \((*3)\) term: \[ \underbracket[0.4pt][0pt]{\partial_\mu\left[\frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} A_\gamma \delta x^\gamma \right]_{,\nu}}_{(*)} =-\partial_\mu\left[ \underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} A_{\gamma, \nu} \delta x^\gamma }_{(3)} \right] \] The final result, \[\begin{align*} \Delta S &= \int \{EoM\}\delta A_\nu d^4 x + \int\partial_\mu \left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}\Delta A_\nu - \left( \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} A_{\nu,\gamma} \delta x^\gamma - \underbracket[0.4pt][0pt]{ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} A_{\gamma, \nu} \delta x^\gamma }_{(*)=(3)} \right) - \delta^\mu_\gamma \mathscr{L}_{EM}\delta x^\gamma \right]d^4x \\ &= \int \{EoM\}\delta A_\nu d^4 x + \int \partial_\mu \left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}\Delta A_\nu - \left( \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} F_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{EM} \right) \delta x^\gamma \right]d^4x \end{align*}\] Hence we have \[\begin{align*} \left(t_{EM}\right)^\mu_\gamma &= \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} F_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{EM} \\ &=-\frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta} F_{\alpha\beta} F_{\gamma\nu}\sqrt{-g} + \delta^\mu_\gamma \frac{1}{16\pi c} g^{\varepsilon\alpha}g^{\nu\beta}F_{\alpha\beta}F_{\varepsilon\nu}\sqrt{-g}\\ &= -\frac{1}{4\pi c}F^{\mu\nu}F_{\gamma\nu}\sqrt{-g} + \delta^\mu_\gamma \frac{1}{16\pi c} F^{\alpha\beta}F_{\alpha\beta}\sqrt{-g}\\ &=\underbracket[0.4pt][0pt]{\left( -\frac{1}{4\pi c}F^{\mu\nu}F_{\gamma\nu} + \delta^\mu_\gamma \frac{1}{16\pi c} F^{\alpha\beta}F_{\alpha\beta}\right)}_{\left({T_{EM}}\right)^\mu_\gamma}\sqrt{-g} \\ &=-\frac{1}{4\pi c}\delta^\mu_\varepsilon g^{\varepsilon\alpha} g^{\nu\beta} F_{\alpha\beta} F_{\gamma\nu}\sqrt{-g} + \delta^\mu_\gamma \frac{1}{16\pi c} g^{\varepsilon\alpha}g^{\nu\beta}F_{\alpha\beta}F_{\varepsilon\nu}\sqrt{-g}\\ &=\frac{1}{4\pi c}\left(-\delta^\mu_\varepsilon F_{\alpha\beta} F_{\gamma\nu} + \delta^\mu_\gamma \frac{1}{4} F_{\alpha\beta}F_{\varepsilon\nu}\right)g^{\varepsilon\alpha} g^{\nu\beta} \sqrt{-g} \end{align*}\] . This symmetric and gauge invariant. \section{Non-Abelian Gauge Theory (Yang-Mills Theory)} \subsection{Introduction} In Sec.\ref{sec:Ab} we derive the symmetrical, gauge invariant canonical energy-momentum tensor for abelian gauge field. In this section we generalize to non-abelian gauge field. A more detailed derivation can be found in the non-Abelian parts in supplementary derivation. \subsection{Derivation} We denote \(p\) is a point in spacetime, \(\mathbf{B}(p)\) as Lie algebra valued gauge potential 1-form of the Yang-Mills field, and the field strength is \(\mathbf{F} \equiv d\mathbf{B} + [\mathbf{B} \wedge \mathbf{B}]\). We will discuss the effect of the variation on gauge potential and the spacetime variation. We denote the variation on gauge potential, \(\mathbf{B} \rightarrow \mathbf{\tilde{B}}= \mathbf{B} + \delta \mathbf{B}\), and the spacetime variation drag by a vector field \(\delta x\) denote as: \[ p \rightarrow \tilde{p} = f_{\delta x}(p) \] The total variation of gauge 1-form is \[ \Delta \mathbf{B} = \mathbf{\tilde{B}}(\tilde{p}) - \textbf{B}(p) = \delta \mathbf{B} + \hat{\mathcal{L}}_{\delta x} \mathbf{B} \] In local coordinate \(p \rightarrow \{x^\mu\}\), the expressions are \[ \mathbf{B}(p) \rightarrow B^a_\mu(x^\gamma) \hat{T}_a \] where \(\hat{T}_a\) is the generator of Lie algebra. The field strength in local coordinate \[\begin{align} \label{eq:Guv} \mathbf{F} \rightarrow \mathbf{F}_{\mu\nu} &= \hat{T}_a \partial_\mu B_\nu^a - \hat{T}_a \partial_\nu B_\mu^a + \lambda [B_\mu^a \hat{T}_a, B_\nu^b \hat{T}_b] \nonumber \\ &= \hat{T}_a \left( \partial_\mu B_\nu^a - \partial_\nu B_\mu^a+ \lambda f_{bc}^a B_\mu^b B_\nu^c \right) \end{align}\] where \[ [\hat{T}_a, \hat{T}_b] = f_{ab}^c \hat{T}_c \] and \[ F_{\mu\nu}^a = \partial_\mu B_\nu^a - \partial_\nu B_\mu^a + \lambda f^a_{bc}B^b_\mu B^c_\nu \] . \(f^c_{ab}\) is the structure constant of Lie algebra. The local coordinate representation of variations are \[p \rightarrow x^\nu = x^\nu + \delta x^\nu\] \[\Delta B^a_\mu = \delta B^a_\mu + \partial_\nu B^a_\mu \delta x^\nu + B^a_\nu \partial_\mu \delta x^\nu\] The Lagrangian \(\mathscr{L}_{YM}\) of Yang-Mills field is \[\begin{align} \label{eq:Lagrangian} \mathscr{L}_{YM} = Tr\left( -\frac{1}{16\pi c} g^{\mu\alpha} g^{\nu\beta} \mathbf{F}_{\mu\nu} \mathbf{F} _{\alpha\beta} \sqrt{-g} \right) = -\frac{1}{16\pi c} K_{ab} g^{\mu\alpha} g^{\nu\beta} F^a_{\mu\nu} F^b_{\alpha\beta} \sqrt{-g} \end{align}\] Here, we denote the Killing form/metric as \[ K_{ab} = Tr(f_{ad}^c f_{be}^d) = f^c_{ad} f^d_{bc} \] The action \(S\) \[ S = \int d^4x \, \mathscr{L}_{YM}[B^a_\nu(x^\gamma), B^a_{\nu,\mu}(x^\gamma), x^\gamma] \] We derive the equation of motion (EoM) and the Noether theorem follow the standard procedure [1]. The variation of action \(\Delta S\) divides into two terms: \[ \Delta S = \int \Delta d^4 x * \mathscr{L}_{YM} + \int d^4 x * \Delta \mathscr{L}_{YM} \] The first term is the variation of the volume form, which is \autocite{ryder1996quantum} \[ \Delta d^4 x = \partial_\gamma \delta x^\gamma \cdot d^4x \] The second term is the variation of \(\mathscr{L}_{YM}\) \[\begin{align} \label{eq:Delta_L} \Delta \mathscr{L}_{YM} &= \mathscr{L}_{YM}[\tilde{B}^a_\nu(\tilde{x}^\gamma), \tilde{B}^a_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] - \mathscr{L}_{YM}[B^a_\nu(x^\gamma), B^a_{\nu,\mu}(x^\gamma), x^\gamma] \nonumber \\ & = \left[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_\nu} \delta B^a_\nu + \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)} \delta (\partial_\mu B^a_\nu) \right] (x^\gamma) + \left[ {\mathscr{L}_{YM}}_{, \gamma} \delta x^\gamma \right] (x^\gamma) + O(\delta^2) \end{align}\] Hence the \(\Delta S\) is \[\begin{align} \label{eq:Delta_S} \Delta S = \int \left[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_\nu} - \partial_\mu\left(\frac{\partial\mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}\right) \right] \delta B^a_\nu d^4 x + \int \left[ \partial_\mu\left(\frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)}\delta B^a_\nu\right) + (\mathscr{L}_{YM} \delta x^\gamma)_{,\gamma}\right] d^4 x \end{align}\] The EoM is \[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_\nu} - \partial_\mu \left( \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}\right) = 0 \] Using \(\delta B_\nu^a = \Delta B_\nu^a - B_{\nu,\gamma}^a \delta x^\gamma - B_\gamma^a \delta x_{,\nu}^\gamma\): \[\begin{align*} \Delta S &= \int \{EoM\} \delta B^a_\nu d^4 x + \int \partial_\mu\left[\frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} (\Delta B^a_\nu - B^a_{\nu,\gamma}\delta x^\gamma - B^a_\gamma \delta x_{,\nu}^\gamma) + \delta^\mu_\gamma\mathscr{L}_{YM} \delta x^\gamma\right]d^4x\\ &= \int \{EoM\} \delta B^a_\nu d^4 x + \int \underbracket[0.4pt][0pt]{\partial_\mu}_{(*)} \left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}\Delta B^a_\nu - \left( \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}B^a_{\nu,\gamma} \delta x^\gamma + \underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}B^a_\gamma \delta x^\gamma_{,\nu}}_{(*)} - \delta^\mu_\gamma \mathscr{L}_{YM}\delta x^\gamma\right)\right]d^4x \end{align*}\] Evaluate the \((*)\) term: \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align} \label{eq:(*)} \underbracket[0.4pt][0pt]{\partial_\mu \left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} B^a_{\gamma} \delta x^\gamma_{,\nu} \right]}_{(*)} &= \underbracket[0.4pt][0pt]{\left[ \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)} B^a_\gamma \delta x^\gamma \right]_{,\nu \mu}}_{(*1)} -\partial_\mu \left[\underbracket[0.4pt][0pt]{ \left( \frac{\partial \mathscr{L}_{YM}}{ \partial (\partial_\mu B^a_\nu)}\right)_{,\nu} B^a_\gamma \delta x^\gamma }_{(*2)} \right] - \partial_\mu \left[\underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}B^a_{\gamma,\nu} \delta x^\gamma }_{(*3)} \right] \\ &=\underbracket[0.4pt][0pt]{ 0}_{(*1)} \textcolor{red}{+}\partial_\mu \left[ \underbracket[0.4pt][0pt]{ \left( \frac{\partial \mathscr{L}_{YM}}{ \partial \left( \partial_{\textcolor{red}{\nu}} B^a_{\textcolor{red}{\mu}}\right) } \right)_{,\nu} B^a_\gamma \delta x^\gamma }_{(*2)} \right] \textcolor{blue}{ -\partial_\mu\left[\left( \frac{\partial \mathscr{L}_{YM}}{ \partial B^a_\mu} \right) B^a_\gamma \delta x^\gamma \right]} \textcolor{blue}{ +\partial_\mu\left[\left( \frac{\partial \mathscr{L}_{YM}}{ \partial B^a_\mu} \right) B^a_\gamma \delta x^\gamma \right]} -\partial_\mu \left[\underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}B^a_{\gamma,\nu} \delta x^\gamma }_{(*3)} \right] \\ &=\underbracket[0.4pt][0pt]{ 0}_{(*1)} +\partial_\mu \left[ \underbracket[0.4pt][0pt]{ \left[ \left( \frac{\partial \mathscr{L}_{YM}}{ \partial \left( \partial_{\nu} B^a_{\mu}\right) } \right)_{,\nu} -\frac{\partial \mathscr{L}_{YM}}{ \partial B^a_\mu} \right] B^a_\gamma \delta x^\gamma }_{(*2)} \right] +\partial_\mu \left[\frac{\partial\mathscr{L}_{YM}}{\partial(\partial_{\mu}B_{\nu}^{a})} \lambda f_{bn}^{a} B^{b}_\gamma B_{\nu}^{n}\delta x^\gamma \right] -\partial_\mu \left[\underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}B^a_{\gamma,\nu} \delta x^\gamma }_{(*3)} \right] \end{align}\] \end{adjustwidth} We first calculate \(\frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B_\nu^a)}\) for later use: \[\begin{align} \label{eq:Non-ab_partial} \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B_\nu^a)} = -\frac{1}{4\pi c} K_{ab} g^{\alpha\mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} \end{align}\] The \((*1)\) term term: \[\begin{align} \label{eq:(*1)} \left( \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B_\nu^a)} B_\gamma^a \delta x^\gamma \right)_{,\nu\mu} =\left( -\frac{1}{4\pi c} K_{ab} g^{\alpha\mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} B_\gamma^a \delta x^\gamma \right)_{,\nu\mu} = 0 \end{align}\] due to the antisymmetric of \(F_{\mu\nu}\) and the symmetric of second order derivative \( \{_{,\nu\mu}\}\), similar with Eq.\eqref{eq:anti_ab} in Abelian case.\\ The \((*2)\) term is: \[\begin{align} \label{eq:(*2)} \underbracket[0.4pt][0pt]{ \left( \frac{\partial \mathscr{L}_{YM}}{ \partial (\partial_\mu B^a_\nu)}\right)_{,\nu} B^a_\gamma \delta x^\gamma}_{(*2)} = \frac{\partial\mathscr{L}_{YM}}{\partial(\partial_{\mu}B_{\nu}^{{a}})}\left( \lambda f_{{b}n}^{{a}}B_{\nu}^{n} B^{{b}}_\gamma \delta x^\gamma \right) \end{align}\] We now have: Hence \((*)\) becomes: \[ \underbracket[0.4pt][0pt]{\partial_\mu \left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} B^a_{\gamma} \delta x^\gamma_{,\nu} \right]}_{(*)} =-\partial_\mu \left[\underbracket[0.4pt][0pt]{-\frac{\partial\mathscr{L}_{YM}}{\partial(\partial_{\mu}B_{\nu}^{a})} \lambda f_{bn}^{a} B^{b}_\gamma B_{\nu}^{n}\delta x^\gamma }_{(*2)} \right]-\partial_\mu \left[\underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}B^a_{\gamma,\nu} \delta x^\gamma }_{(*3)}\right] \] \[\begin{align*} \Delta S &= \int \{EoM\} \delta B_\nu^a \, d^4x + \int \partial_\mu \left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B_\nu^a)} \Delta B_\nu^a - \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B_\nu^a)} \left( B^a_{\nu, \gamma} - \underbracket[0.4pt][0pt]{B^a_{\gamma,\nu} }_{(*3)} +\underbracket[0.4pt][0pt]{ \lambda f_{bn}^{a} B^{b}_\gamma B_{\nu}^{n} }_{(*2)}\right) \delta x^\gamma + \delta^\mu_\gamma \mathscr{L}_{YM} \delta x^\gamma \right] \, d^4x \\ &=\int \{EoM\} \delta B_\nu^a \, d^4x + \int \partial_\mu \left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B_\nu^a)} \Delta B_\nu^a - \left( \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B_\nu^a)} F^a_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{YM}\right) \delta x^\gamma \right] \, d^4x \end{align*}\] Hence we have \[\begin{align*} \left(t_{YM}\right)^\mu_\gamma &= \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B_\nu^a)} F^a_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{YM}\\ &= \left( -\frac{1}{4\pi c} K_{ab} g^{\alpha\mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} \right)F^a_{\gamma\nu} - \delta^\mu_\gamma \left( -\frac{1}{16\pi c} K_{ab} g^{\mu\alpha} g^{\nu\beta} F^a_{\mu\nu} F^b_{\alpha\beta} \sqrt{-g} \right)\\ &= -\frac{1}{4\pi c} F^{\mu\nu}_a F^a_{\gamma\nu} \sqrt{-g} + \delta^\mu_\gamma \frac{1}{16\pi c} F^{\alpha\beta}_a F^a_{\alpha\beta} \sqrt{-g} \end{align*}\] \[\begin{align*} g_{\psi\mu}g^{\phi\gamma}\left(t_{YM}\right)^\mu_\gamma &= g_{\psi\mu}g^{\phi\gamma}\left( -\frac{1}{4\pi c} F^{\mu\nu}_a F^a_{\gamma\nu} \sqrt{-g} + \delta^\mu_\gamma \frac{1}{16\pi c} F^{\alpha\beta}_a F^a_{\alpha\beta} \sqrt{-g} \right)\\ &= -\frac{1}{4\pi c} g_{\psi\mu}g^{\phi\gamma} F^{\mu\nu}_a F^a_{\gamma\nu} \sqrt{-g} +g_{\psi\mu}g^{\psi\gamma} \delta^\mu_\gamma \frac{1}{16\pi c} F^{\alpha\beta}_a F^a_{\alpha\beta} \sqrt{-g} \\ &= -\frac{1}{4\pi c} g_{\psi\mu}g^{\phi\gamma} g^{\mu\zeta}g^{\nu\xi}F^a_{\zeta\xi} F^a_{\gamma\nu} \sqrt{-g} +\delta^\phi_\psi \frac{1}{16\pi c} F^{\alpha\beta}_a F^a_{\alpha\beta} \sqrt{-g} \\ &= -\frac{1}{4\pi c} g^{\phi\gamma} g^{\nu\xi}F^a_{\psi\xi} F^a_{\gamma\nu} \sqrt{-g} +\delta^\phi_\psi \frac{1}{16\pi c} F^{\alpha\beta}_a F^a_{\alpha\beta} \sqrt{-g} \\ &= -\frac{1}{4\pi c} g_{\psi\mu}g^{\phi\gamma} g^{\mu\zeta}g^{\nu\xi}F^a_{\zeta\xi} F^a_{\gamma\nu} \sqrt{-g} +\delta^\phi_\psi \frac{1}{16\pi c} F^{\alpha\beta}_a F^a_{\alpha\beta} \sqrt{-g} \\ &= -\frac{1}{4\pi c} F^a_{\psi\xi} F_a^{\phi\xi} \sqrt{-g} +\delta^\phi_\psi \frac{1}{16\pi c} F^{\alpha\beta}_a F^a_{\alpha\beta} \sqrt{-g} \\ &=\left(t_{YM}\right)^\phi_\psi \end{align*}\] \section{1-form nature of gauge potential} For \textbf{non-abelian} gauge theories, the definition of the curvature must account for the non-commuting nature of the gauge group generators. The field strength is given by: \[ F = dA + A \wedge A \] In terms of components \(A = A^a_\mu T^a dx^\mu\) and \(F = \frac{1}{2} F^a_{\mu\nu} T^a dx^\mu \wedge dx^\nu\), the \(A \wedge A\) term generates the commutator term involving the Lie algebra structure constants \(f^{abc}\): \[ F^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g f^a_{bc} A^b_\mu A^c_\nu \] where \(g\) is the coupling constant. This non-linear term reflects the self-interaction of the non-abelian gauge bosons (e.g., gluons interacting with gluons). In both cases, the 2-form \(F\) (or its components \(F_{\mu\nu}\) or \(F^a_{\mu\nu}\)) represents the physically measurable field strength or curvature associated with the gauge connection \(A\). \subsection{1-form Nature of Gauge Potential} Mathematically, the gauge connection is a 1-form field. However, in physics, normally treat gauge potential as a vector field \(A^\mu \) or as a 1-form field \(A_\mu \) equivalently. These two point of view are related by metric tensor field \(g\): \[\begin{align*} A^\mu&=g^{\mu\nu}A_\nu\\ A_\mu&=g_{\mu\nu}A^\nu\\ \delta^\mu_\nu&=g^{\mu\gamma}g_{\gamma\nu} \end{align*}\] Furthermore, the concept of differentiation in differential geometry is intimately linked with forms. The \textbf{exterior derivative}, denoted by \(d\), is a fundamental operator acting on differential forms, increasing their degree by one (e.g., acting on a 0-form function \(f\) gives the 1-form differential \(df = \frac{\partial f}{\partial x^\mu} dx^\mu\)). A key property is that applying it twice yields zero: \(d^2 = 0\). The physical \textbf{field strength} (or \textbf{curvature} in geometric terms), denoted by the 2-form \(F\), is defined via the exterior derivative of the connection 1-form \(A\). For an \textbf{abelian} gauge theory (like electromagnetism), the relationship is simply: \[ F = dA \] To see how this relates to the familiar component form, we write \(A = A_\mu dx^\mu\) (summation implied) and apply the exterior derivative using its properties (linearity, product rule \(d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^p \alpha \wedge d\beta\) where \(p\) is the degree of \(\alpha\), and \(d^2=0\)): \[\begin{align*} F = d(A_\mu dx^\mu) &= (dA_\mu) \wedge dx^\mu + A_\mu \wedge (d(dx^\mu)) \\ &= \left(\frac{\partial A_\mu}{\partial x^\nu} dx^\nu\right) \wedge dx^\mu + A_\mu \wedge (0) \quad (\text{since } d(dx^\mu) = d^2x^\mu = 0) \\ &= \frac{\partial A_\mu}{\partial x^\nu} dx^\nu \wedge dx^\mu \end{align*}\] The standard way to write a 2-form \(F\) in terms of its components \(F_{\mu\nu}\) is \(F = \frac{1}{2} F_{\mu\nu} dx^\mu \wedge dx^\nu\). Comparing the two expressions and using the anti-symmetric property of the wedge product (\(dx^\nu \wedge dx^\mu = - dx^\mu \wedge dx^\nu\)): \[ \frac{\partial A_\mu}{\partial x^\nu} dx^\nu \wedge dx^\mu = \frac{1}{2} \left( \frac{\partial A_\nu}{\partial x^\mu} - \frac{\partial A_\mu}{\partial x^\nu} \right) dx^\mu \wedge dx^\nu \] Comparing the coefficients of \(dx^\mu \wedge dx^\nu\) (e.g., by setting specific indices), we identify the components: \[ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \] This recovers the familiar definition of the electromagnetic field strength tensor. In physics, the field strength can also be written in 2-vector: \[\begin{align*} F^{\mu\nu} &=\partial^\mu A^\nu-\partial^\nu A^\mu\\ &=g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta} \end{align*}\] The Lagrangian has two ways to express: \[\begin{align*} \mathscr{L}^{A_\mu}&= -\frac{1}{16\pi c} g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}F_{\mu\nu}\\ \mathscr{L}^{A^\mu}&= -\frac{1}{16\pi c} g_{\mu\alpha}g_{\nu\beta}F^{\alpha\beta}F^{\mu\nu}\\ \end{align*}\] These two Lagrangian are equivalent when derived the EoM, either using \(A_\mu\to A_\mu+\delta A_\mu \) or \(A^\mu\to A^\mu+\delta A^\mu \). However, when apply Noether's theorem, only 1-form potential can derive correct canonical energy momentum tensor. The difference comes from the Lie derivative in total variation: \[\begin{align*} \Delta A_\nu &= \delta A_\nu + A_{\nu,\gamma} \delta x^\gamma + A_\gamma {\delta x^\gamma}_{,\nu}\\ \Delta A^\nu &= \delta A^\nu + {A^{\nu}}_{,\gamma} \delta x^\gamma\underbracket[0.4pt][0pt]{- A^\gamma {\delta x^\nu}_{,\gamma}}_ \end{align*}\] Previously we demonstrates the 1-form potential \(A_\mu\) can derive desired canonical energy momentum tensor. \subsection{The 1-form Nature of the Gauge Potential} From a rigorous mathematical perspective, particularly within the framework of differential geometry used to describe gauge theories, the fundamental object representing the gauge field is a \textbf{connection 1-form}, often denoted simply as \(A\). This connection lives on a principal bundle (or an associated vector bundle) over spacetime and provides a way to relate, or "connect," the internal gauge degrees of freedom at infinitesimally separated spacetime points. It defines the concept of parallel transport for fields charged under the gauge group. For non-abelian gauge theories (like Yang-Mills theory), this 1-form \(A\) takes values in the Lie algebra associated with the gauge group. Its components in a local coordinate basis \(dx^\mu\) can be written as \(A = A^a_\mu T^a dx^\mu\), where \(T^a\) are the generators of the Lie algebra. \section{General Relativity} \subsection{Introduction} The stress-energy-momentum pseudotensor is traditional method to describe the concept of energy-momentum of gravitational field. The Landau-Lifshitz pseudotensor \(t^{\mu\nu}_{LL}\) is derived from the Einstein field equation \(\mathbb{G}^{\mu\nu} =\kappa T^{\mu\nu}\) to satisfied the conservation law \((T^{\mu\nu} + t^{\mu\nu}_{LL})_{,\nu} = 0\), where \(\mathbb{G}^{\mu\nu}\) is Einstein field tensor, \(\kappa=\frac{8\pi G}{c^4}\), and \(T^{\mu\nu}\) is the energy-momentum tensor of source. \[\begin{align*} t^{\mu\nu}_{LL} = -\frac{1}{2\kappa}\mathbb{G}^{\mu\nu} + \frac{1}{2\kappa(-g)}[(-g)(g^{\mu\nu}g^{\alpha\beta}-g^{\mu\alpha}g^{\nu\beta} )]_{,\alpha\beta} \end{align*}\] \(t^{\mu\nu}_{LL}\) is symmetrical but depend explicit in the Christoffel symbols (i.e., coordinate dependent and vanish in specific coordinate). The Dirac pseudotensor \(t^{\mu\nu}_D\) starting from the equivalent action \(\mathscr{L}_{GR}^*\) and derived using standard Noether's theorem derivation. The original Einstein-Hilbert action \(\mathscr{L}_{GR} = R\sqrt{-g}=\mathscr{L}_{GR} [g^{\mu\nu},{g^{\mu\nu}}_{,\gamma}, {g^{\mu\nu}}_{,\gamma\eta}]\) depend on the second derivative of metric tensor, where \(R\) and \(g\) are Ricci scalar and \(g = \det(g_{\mu\nu})\), respectively. The Ostrogradsky instability indicate the Lagrangian should not depend on higher order derivative more than 1st order. The equivalent action \[\begin{align*} \mathscr{L}_{GR}^* &= \mathscr{L}_{GR} - \partial_\mu(\sqrt{-g}g^{\mu\nu}\Gamma^\sigma_{\nu\sigma} - \sqrt{-g}g^{\sigma\nu}\Gamma^\mu_{\nu\sigma})\\ &= \sqrt{-g}g^{\mu\nu}\left(\Gamma^\tau_{\mu\nu}\Gamma^\sigma_{\tau\sigma} - \Gamma^\tau_{\mu\sigma}\Gamma^\sigma_{\tau\nu}\right)\\ &=\mathscr{L}_{GR}^* [g^{\mu\nu},{g^{\mu\nu}}_{,\gamma}] \end{align*}\] have same EoM with advantage only depend on the 1st order derivative of metric tensor, which can apply standard Noether's theorem derivation. However, the equivalent action \(\mathscr{L}_{GR}^*\) lost the scalar property. Also, the Dirac pseudotensor \(t^{\mu\nu}_D\): \[\begin{align*} t^{\mu\nu}_D = \frac{1}{2\kappa(-g)}\left[g^{\mu\gamma}\left(g^{\alpha\beta}\sqrt{-g} \right)_{,\gamma}\left( \Gamma^\nu_{\alpha\beta} -\delta^\nu_\beta \Gamma^\sigma_{\alpha\sigma} \right) - g^{\mu\nu}g^{\alpha\beta} \left( \Gamma^\rho_{\alpha\beta}\Gamma^\sigma_{\rho\sigma}-\Gamma^\rho_{\alpha\sigma}\Gamma^\sigma_{\beta\rho}\right) \right] \end{align*}\] lost the symmetric property, and is coordinate dependent and vanishes in specific coordinate as \(t^{\mu\nu}_{LL}\).\\ As we previously derive the symmetrical, gauge invariant canonical energy-momentum tensor for abelian and non-abelian gauge field, in this article we generalize to general relativity. To reserve the scalar property of the Lagrangian and avoid the higher-order derivatives, we will use Palatini variation for derivation the EoM. To further apply Noether's theorem, we will use the vielbeins technique for deriving the conservation law. A more detailed derivation can be found in the supplementary derivation. \subsection{Derivation of Palatini variation} The Palatini variation treat metric tensor \(g^{\omega\sigma}\) and the connection \(\Gamma^\varepsilon_{\kappa\gamma}\) as independent field. The curvature tensor \(R^\varepsilon_{\kappa\omega\sigma}\): \[\begin{align} \label{eq:R_curvature} R^\varepsilon_{\kappa\omega\sigma} = \Gamma^\varepsilon_{\kappa\sigma,\omega} - \Gamma^\varepsilon_{\kappa\omega,\sigma} + \Gamma^\varepsilon_{\gamma\omega}\Gamma^\gamma_{\kappa\sigma} - \Gamma^\varepsilon_{\gamma\sigma}\Gamma^\gamma_{\kappa\omega} \end{align}\] and the torsion tensor \(T^\alpha_{\beta\gamma}\): \[\begin{align} \label{eq:torsion} T^\alpha_{\beta\gamma} = \Gamma^\alpha_{\beta\gamma} - \Gamma^\alpha_{\gamma\beta}=- T^\alpha_{\gamma\beta} \end{align}\] In the following, we will NOT assume the connection to be torsion-free, i.e., \(T^\alpha_{\beta\gamma} \neq 0 \leftrightarrow \Gamma^\alpha_{\beta\gamma} \neq \Gamma^\alpha_{\gamma\beta}\) in general. The Einstein-Hilbert action: \[\begin{align*} \mathscr{L}_{GR} = \frac{1}{2\kappa}\sqrt{-g}g^{\mu\nu}\delta^\omega_\varepsilon R_{\kappa\omega\sigma}^\varepsilon = \mathscr{L}_{GR}[g^{\mu\nu}, \Gamma^\kappa_{\mu\nu}, \Gamma^\kappa_{\mu\nu,\gamma}] \end{align*}\] \[\begin{align*} S = \frac{1}{2\kappa}\int d^4x \sqrt{-g} g^{\kappa\sigma}\delta^\omega_\varepsilon R^\varepsilon_{\kappa\omega\sigma} \end{align*}\] Here we note that the Ricci tensor \(R_{\kappa\sigma}\) and Ricci scalar \(R\) are: \[\begin{align*} R_{\kappa\sigma} = \delta^\omega_\varepsilon R_{\kappa\omega\sigma}^\varepsilon \end{align*}\] and \[\begin{align*} R = g^{\kappa\sigma}R_{\kappa\sigma} \end{align*}\] , so \(\mathscr{L}_{GR} = \frac{1}{2\kappa}\sqrt{-g}R\). Since the metric tensor \(g^{\mu\nu}\) and the connection \(\Gamma^\alpha_{\mu\nu}\) are independent field, the Lagrangian only depend on 1st order derivative of the connection. The Palatini variation: \[\begin{align} \label{eq:Action_g} \delta S &= \int \left(\frac{\partial \mathscr{L}_{GR}}{\partial g^{\mu\nu}}\delta g^{\mu\nu} + \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\mu\nu}} \delta \Gamma^\alpha_{\mu\nu} + \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\mu\nu,\gamma}} \delta \Gamma^\alpha_{\mu\nu,\gamma} \right) d^4x \nonumber \\ &= \int \underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{GR}}{\partial g^{\mu\nu}}}_{EoM\#1}\delta g^{\mu\nu} d^4x + \int \underbracket[0.4pt][0pt]{\left[ \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\mu\nu}} - \left(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\mu\nu,\gamma}}\right)_{,\gamma}\right]}_{EoM\#2} \delta \Gamma^\alpha_{\mu\nu} d^4 x + \int \left[ \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\mu\nu,\gamma}}\delta\Gamma^\alpha_{\mu\nu}\right]_{,\gamma} d^4x \end{align}\] Two equations of motion (EoM) are \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial g^{\mu\nu}} &= 0 \\ \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\mu\nu}} &= \left(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\mu\nu,\gamma}}\right)_{,\gamma} \end{align*}\] \subsection{EoM\#1 - Einstein Field Equation} Since the curvature tensor do not depend on metric tensor, the first EoM: \[\begin{align} \label{eq:Einstein} \frac{\partial \mathscr{L}_{GR}}{\partial g^{\mu\nu}} = \frac{1}{2\kappa} \sqrt{-g}\left( R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \right)=0 \end{align}\] Define the Einstein field tensor \(G_{\mu\nu}\): \[\begin{align*} \mathbb{G}_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R \end{align*}\] For vacuum, \(\mathbb{G}_{\mu\nu}=0\) is the Einstein field equation in vacuum. If the matter presents, for example, photon field, \(\mathscr{L}_{GR}=\mathscr{L}_{GR}+\mathscr{L}_M\), where \(\mathscr{L}_{GR}=\frac{1}{2\kappa}\sqrt{-g}R\) and \(\mathscr{L}_M\) is matters. The variation gives: \[\begin{align*} \frac{\partial \left( \mathscr{L}_{GR}+\mathscr{L}_M \right) }{\partial g^{\mu\nu}} &=\frac{\partial \mathscr{L}_{GR} }{\partial g^{\mu\nu}}+\frac{\partial \mathscr{L}_M }{\partial g^{\mu\nu}}= \frac{1}{2\kappa} \sqrt{-g}\, \mathbb{G}_{\mu\nu} +\frac{\partial \mathscr{L}_M }{\partial g^{\mu\nu}}=0\\ &\to \mathbb{G}_{\mu\nu}=\kappa \left( \frac{-2}{\sqrt{-g}} \frac{\partial \mathscr{L}_M }{\partial g^{\mu\nu}} \right)\equiv \kappa \left(T_M\right)_{\mu\nu} \end{align*}\] We take EM field as an example. \[\begin{align*} \left(T_{EM}\right)_{\mu\nu}= \frac{-2}{\sqrt{-g}} \frac{\partial \mathscr{L}_{EM} }{\partial g^{\mu\nu}} = \frac{-2}{\sqrt{-g}} \frac{\partial }{\partial g^{\mu\nu}}\left(-\frac{1}{16\pi c} g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}F_{\mu\nu}\sqrt{-g}\right) \end{align*}\] \subsection{EoM\#2 - Relate to The Metric Compatible Condition} To show the second EoM relates to the metric compatible condition, we will use the following relations: \[\begin{align} \label{eq:Fact_1} \frac{1}{\sqrt{-g}}\left(\sqrt{-g}g^{\mu\nu}\right)_{;\gamma} = \frac{1}{\sqrt{-g}}\left(\sqrt{-g}g^{\mu\nu}\right)_{,\gamma} -g^{\mu\nu}\Gamma_{\eta\gamma}^{\eta} +g^{\eta\nu}\Gamma_{\eta\gamma}^{\mu} +g^{\mu\eta}\Gamma_{\eta\gamma}^{\nu} \end{align}\] We first calculate \(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu}^\alpha}\) and \(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha}\) for later use: \[\begin{align} \label{eq:EoM2_R} \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu}^\alpha} = \sqrt{-g} \left( g^{\kappa\sigma} \delta_\alpha^\nu \Gamma_{\kappa\sigma}^\mu + g^{\mu\nu} \Gamma_{\alpha\omega}^\omega - g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu - g^{\mu\sigma} \Gamma_{\alpha\sigma}^\nu\right) \end{align}\] \[\begin{align} \label{eq:EoM2_D} \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha} = \sqrt{-g} \left( g^{\mu\nu} \delta_\alpha^\gamma - g^{\mu\gamma} \delta_\alpha^\nu \right) \end{align}\] \[\begin{align} \label{eq:EoM2_DD} \left(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha}\right)_{,\gamma} = (\sqrt{-g} \, g^{\mu\nu})_{,\alpha} - (\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} \delta_\alpha^\nu \end{align}\] Substituting Eq.\eqref{eq:EoM2_R} and Eq.\eqref{eq:EoM2_DD} into EoM(more detail derivation, see supplement Eq.\eqref{eq:EoM2_cal} below): \[\begin{align} \label{eq:EoM2_cal} \underbracket[0.4pt][0pt]{{(\sqrt{-g} \, g^{\mu\nu})_{,\alpha}} -{(\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} \delta_\alpha^\nu}}_{\left( \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha} \right)_{,\gamma} } &=\underbracket[0.4pt][0pt]{{\sqrt{-g} \, g^{\kappa\sigma} \delta_\alpha^\nu \Gamma_{\kappa\sigma}^\mu} + {\sqrt{-g}\, g^{\mu\nu} \Gamma_{\alpha\omega}^\omega} -{\sqrt{-g}\, g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu}-{\sqrt{-g}\, g^{\mu\sigma} \Gamma_{\alpha\sigma}^\nu}}_{\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu}^\alpha}} \nonumber\\ {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha} &=\left[ {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\gamma}\right)}_{;\gamma}+g^{\mu\gamma}\, T^\eta_{\eta\gamma}\right] \delta_\alpha^\nu + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\kappa}T^\nu_{\kappa\alpha} \end{align}\] After some calculation (more detail derivation, see supplement Eq.\eqref{eq:EoM2_result} below), we arrive \[\begin{align} \label{eq:EoM2_result} {g^{\mu\nu}}_{;\alpha} &= \frac{1}{3}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu +\frac{1}{3}g^{\mu\nu}T^\eta_{\eta\alpha} + g^{\mu\gamma}T^\nu_{\gamma\alpha}\\ \end{align}\] Eq.\eqref{eq:EoM2_result} indicate that: \begin{table}[h] \centering \begin{tabular}{|>{\Centering}p{0.45\textwidth}|>{\Centering}p{0.45\textwidth}|} % Use 'p' and \Centering \hline Torsion-free then metric compatible & Metric compatible then torsion-free \\ \hline % First row of cells (using minipages for vertical centering) \begin{minipage}[t]{0.45\textwidth} \centering If torsion-free:\, \(T^\alpha_{\beta\gamma}=0\), then \({g^{\mu\nu}}_{;\alpha}=0\) \end{minipage} & \begin{minipage}[t]{0.45\textwidth} \centering If Metric compatible:\, \({g^{\mu\nu}}_{;\alpha}=0\), then \(T^\alpha_{\beta\gamma} = 0\) \end{minipage} \\ \hline \end{tabular} \caption{Comparison of Torsion-free and Metric Compatibility Conditions} \label{tab:torsion_metric} \end{table} If torsion-free, then \[\begin{align*} {g^{\mu\nu}}_{;\alpha}={g^{\mu\nu}}_{,\alpha} + g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu + { g^{\mu\kappa}\Gamma^\nu_{\kappa\alpha}} =0 \end{align*}\] is the metric compatible condition. \subsection{Vielbeins Formalism} In previous tutorials, the variation of the Lie-algebra value gauge 1-form (gauge connection) \(\mathbf{B}\) on principal bundle is : \[\begin{align} \label{eq:B_mu} \Delta \mathbf{B}&=\delta \mathbf{B} + \hat{\mathcal{L}}_{\delta x} \mathbf{B} \end{align}\] In local coordinate \(p \rightarrow \{x^\mu\}\), the expressions of \(\mathbf{B}\) is \[\begin{align*} \mathbf{B}(p) \rightarrow B^a_\mu(x^\gamma) \hat{T}_a \end{align*}\] , where \(\hat{T}_a\) is the generator of Lie algebra. For a given Lie algebra representation: \[\begin{align*} \hat{T}_a \to \left(\hat{T}_a\right)^b_c \end{align*}\] We define the notation: \[\begin{align*} \hat{B}^b_{c\mu} \equiv B^a_\mu \left(\hat{T}_a\right)^b_c \end{align*}\] . The curvature tensor: \[\begin{align} \label{eq:P_curvature} \mathbf{F}\equiv d\mathbf{B} + [\mathbf{B} \wedge \mathbf{B}] \rightarrow \hat{G}^a_{b\mu\nu} &= \hat{B}_{b\nu,\mu}^a - \hat{B}_{b\mu,\nu}^a + \hat{B}_{c\mu}^a\hat{B}_{b\nu}^c-\hat{B}_{c\nu}^a\hat{B}_{b\mu}^c \end{align}\] Eq.\eqref{eq:B_mu} in local coordinate with Lie algebra representation is: \[\begin{align} \label{eq:varaition_gauge} \Delta \hat{B}^b_{c\mu} &= \delta \hat{B}^b_{c\mu} + \underbrace{\delta x^\nu \partial_\nu \hat{B}^b_{c\mu} + \hat{B}^b_{c\mu}\partial_\mu \delta x^\nu}_{\hat{\mathcal{L}}_{\delta x} \mathbf{B}} \end{align}\] However, the connection on tangent bunble is more subtle (see footnote 9. in \autocite{yang1996symmetry}). The curvature on tangent bundle is Eq.\eqref{eq:R_curvature}: \[\begin{align} \tag{\ref{eq:R_curvature}} R^\varepsilon_{\kappa\omega\sigma} = \Gamma^\varepsilon_{\kappa\sigma,\omega} - \Gamma^\varepsilon_{\kappa\omega,\sigma} + \Gamma^\varepsilon_{\gamma\omega}\Gamma^\gamma_{\kappa\sigma} - \Gamma^\varepsilon_{\gamma\sigma}\Gamma^\gamma_{\kappa\omega} \end{align}\] The principal curvature Eq.\eqref{eq:P_curvature} is Lie-algebra value 2-form. In contrast, the tangent bundle curvature takes value in tangent vector. If we directly apply Lie derivative on tangent connection: \[\begin{align*} \left(\hat{\mathcal{L}}_{\delta x} \Gamma\right)^\alpha_{\mu\nu} = {\delta x}^\varepsilon \Gamma^\alpha_{\mu\nu,\varepsilon} + \Gamma^\alpha_{\mu\varepsilon}{\delta x}^\varepsilon_{,\nu} + \underbracket[0.4pt][0pt]{{{\delta x}^\alpha_{,\mu\nu}} - \Gamma^\varepsilon_{\mu\nu}\delta x^\alpha_{,\varepsilon} + \Gamma^\alpha_{\varepsilon\nu}{\delta x}^\varepsilon_{,\mu}}_{\text{due to tangent vector value}} \end{align*}\] We can use Vielbeins formalism. In the vielbeins formalism, the tangent connection \(\Gamma^\varepsilon_{\kappa\sigma}\) becomes the spin connection \(\omega ^a_{b\sigma} \) in the frame bundle, which is \(\frak{gl}\)-value 1-form. From the tangent formalism \(\{\partial_\mu \}\) to the vielbeins formalism \(\{\hat{e}_a\}\), define the transformation \(e^\mu_a\) such that(more detail can be found in Supplementary: Vielbeins): \[\begin{align*} \partial_\mu &=e_\mu^a \hat{e}_a\\ g_{\mu\nu}&= e_\mu^a e_\nu^b\eta_{ab}\\ e^\mu_a e^a_\nu=\delta^\mu_\nu\,\,\, &and \,\,\,e^\mu_a e^b_\mu=\delta^a_b \end{align*}\] The spin conncection: \[\begin{align*} \omega ^a_{b\sigma}=e_\varepsilon^a e_b^\kappa\Gamma^\varepsilon_{\kappa\sigma}+e^a_\mu e^\mu_{b,\sigma} \end{align*}\] \[\begin{align*} e^\alpha_a e^b_\beta \omega ^a_{b\sigma} &=e^\alpha_a e^b_\beta e_\varepsilon^a e_b^\kappa\Gamma^\varepsilon_{\kappa\sigma} +e^\alpha_a e^b_\beta e^a_\mu e^\mu_{b,\sigma}\\ e^\alpha_a e^b_\beta \omega ^a_{b\sigma} &=\delta^\alpha_\varepsilon \delta^\kappa_\beta\Gamma^\varepsilon_{\kappa\sigma} +\delta^\alpha_\mu e^b_\beta e^\mu_{b,\sigma}\\ e^\alpha_a e^b_\beta \omega ^a_{b\sigma} &=\Gamma^\alpha_{\beta\sigma} +e^b_\beta e^\alpha_{b,\sigma}\\ \Gamma^\alpha_{\beta\sigma} &=e^\alpha_a e^b_\beta \omega ^a_{b\sigma} -e^b_\beta e^\alpha_{b,\sigma}\\ \end{align*}\] The frame bundle curvature \(\mathscr{R}^a_{b\omega\sigma}\) is \[\begin{align*} \mathscr{R}^a_{b\omega\sigma}&=\omega ^a_{b\sigma,\omega}-\omega ^a_{b\omega,\sigma}+\omega ^a_{c\omega}\omega ^c_{b\sigma}-\omega ^a_{c\sigma}\omega ^c_{b\omega}\\ &=e^a_\varepsilon e_b^\kappa R^\varepsilon_{\kappa\omega\sigma} \end{align*}\] With this preparation, the Hilbert-Einstein action becomes as follows: \[\begin{align} \label{eq:action_G} S = \frac{1}{2\kappa}\int d^4x \sqrt{-g} g^{\kappa\sigma}\delta^\omega_\varepsilon R^\varepsilon_{\kappa\omega\sigma}= \frac{1}{2\kappa}\int d^4x\,e\, \eta^{ae}e^\sigma_e e^\omega_c \mathscr{R}^c_{a\omega\sigma} \end{align}\] , where \(e=\det(e^a_\mu)=\sqrt{-g}\). \subsection{EoM in Vielbeins } The variation is similar to Eq.\eqref{eq:Action_g}: \[\begin{align} \label{eq:Action_e} \delta S &= \int \left(\frac{\partial \mathscr{L}_{GR}}{\partial e^{\mu}_a}\delta e^{\mu}_a + \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu}} \delta \omega^b_{c\mu} + \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \delta \omega^b_{c\mu,\gamma} \right) d^4x \nonumber \\ &= \int \underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{GR}}{\partial e^{\mu}_a }}_{EoM\#3}\delta e^{\mu}_a d^4x + \int \underbracket[0.4pt][0pt]{\left[ \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu}} - \left(\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\right)_{,\gamma}\right]}_{EoM\#4} \delta \omega^b_{c\mu} d^4 x + \int \left[ \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\delta\omega^b_{c\mu}\right]_{,\gamma} d^4x \end{align}\] Since \(g_{\mu\nu}=\eta_{ab} e_\mu^a e_\nu^b\), the variation \(\delta g_{\mu\nu}=2\,\eta_{ab} e_\mu^a \delta e_\nu^b\), the \(EoM\#3\) is similar to \(EoM\#1\). \(EoM\#4\) is also similar to \(EoM\#2\). \(EoM\#3\) \[\begin{align*} \frac{\partial e}{\partial e^{\mu}_a}&=e\, e^\nu_b\frac{\partial e^b_\nu }{\partial e^{\mu}_a}\\ &=-e\, \frac{\partial e^\nu_b }{\partial e^{\mu}_a}e^b_\nu\\ &=-e\, \delta^\nu_\mu \delta^a_be^b_\nu\\ &=-e\, e^a_\mu\\ \\ e^\nu_b e^c_\nu&=\delta^c_b\\ \frac{\partial }{\partial e^{\mu}_a}\left(e^\nu_b e^c_\nu \right) &=\frac{\partial e^\nu_b}{\partial e^{\mu}_a} e^c_\nu +e^\nu_b\frac{\partial e^c_\nu}{\partial e^{\mu}_a}=0\\ \delta^\nu_\mu\delta^a_b e^c_\nu +e^\nu_b\frac{\partial e^c_\nu}{\partial e^{\mu}_a}&=0\\ e^\nu_b\frac{\partial e^c_\nu}{\partial e^{\mu}_a}&=-\delta^a_b e^c_\mu \\ \frac{\partial e^c_\nu}{\partial e^{\mu}_a}&=-e^b_\nu\delta^a_b e^c_\mu =-e^a_\nu e^c_\mu \\ \end{align*}\] \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial e^{\mu}_a } &=\frac{\partial}{\partial e^{\mu}_a } \left( \frac{1}{2\kappa}e\, \eta^{de}e^\sigma_e e^\omega_c \mathscr{R}^c_{d\omega\sigma} \right)\\ &=\frac{1}{2\kappa}\eta^{de}\mathscr{R}^c_{d\omega\sigma} \frac{\partial}{\partial e^{\mu}_a }\left( e\, e^\sigma_e e^\omega_c \right)\\ &=\frac{1}{2\kappa}\eta^{de}\mathscr{R}^c_{d\omega\sigma} \left( \frac{\partial e}{\partial e^{\mu}_a }\, e^\sigma_e e^\omega_c +e\, \frac{\partial e^\sigma_e}{\partial e^{\mu}_a } e^\omega_c +e\, e^\sigma_e \frac{\partial e^\omega_c }{\partial e^{\mu}_a } \right)\\ &=\frac{1}{2\kappa}\eta^{de}\mathscr{R}^c_{d\omega\sigma} \left( -e\, e^a_\mu e^\sigma_e e^\omega_c +e\, \delta^\sigma_\mu \delta^a_e e^\omega_c +e\, e^\sigma_e \delta^\sigma_\mu \delta^a_c \right)\\ &=\frac{1}{2\kappa} \left( -e\, e^a_\mu e^\sigma_e e^\omega_c \eta^{de}\mathscr{R}^c_{d\omega\sigma} +e\, \delta^\sigma_\mu \delta^a_e e^\omega_c \eta^{de}\mathscr{R}^c_{d\omega\sigma} +e\, e^\sigma_e \delta^\omega_\mu \delta^a_c \eta^{de}\mathscr{R}^c_{d\omega\sigma} \right)\\ &=\frac{1}{2\kappa} \left( -e\, e^a_\mu \eta^{de} e^\sigma_e e^\omega_c \mathscr{R}^c_{d\omega\sigma} +e\, \eta^{da} e^\omega_c \mathscr{R}^c_{d\omega\mu} +e\, \eta^{de} e^\sigma_e \mathscr{R}^a_{d\mu\sigma} \right)\\ &=\frac{1}{2\kappa}\sqrt{-g} \left( -\, e^a_\mu R +\, \eta^{da} e^\kappa_d R_{\kappa\mu} -\, \eta^{de} e^\sigma_e \mathscr{R}^a_{d\sigma\mu} \right)\\ &=\frac{1}{2\kappa}\sqrt{-g} \left( -\, e^a_\mu R +\, \eta^{da} e^\kappa_d R_{\kappa\mu} -\, \eta^{de} e^\sigma_e e^a_\alpha e^\beta_d R^\alpha_{\beta\sigma\mu} \right)\\ &=\frac{1}{2\kappa}\sqrt{-g} \left( -\, e^a_\mu R +\,e^a_\sigma g^{\kappa\sigma} R_{\kappa\mu} -\, e^a_\alpha g^{\beta\sigma} R^\alpha_{\beta\sigma\mu} \right)\\ \\ &=\frac{1}{2\kappa}\sqrt{-g} \left( -\, e^a_\mu R +\,e^a_\sigma g^{\kappa\sigma} R_{\kappa\mu} -\, e^a_\alpha g^{\beta\sigma} g^{\alpha\kappa} R_{\kappa\beta\sigma\mu} \right)\\ &=\frac{1}{2\kappa}\sqrt{-g} \left( -\, e^a_\mu R +\,e^a_\sigma g^{\kappa\sigma} R_{\kappa\mu} +\, e^a_\alpha g^{\beta\sigma} g^{\alpha\kappa} R_{\beta\kappa\sigma\mu} \right)\\ &=\frac{1}{2\kappa}\sqrt{-g} \left( -\, e^a_\mu R +\,e^a_\sigma g^{\kappa\sigma} R_{\kappa\mu} +\, e^a_\alpha g^{\alpha\kappa} R^\sigma_{\kappa\sigma\mu} \right)\\ &=\frac{1}{2\kappa}\sqrt{-g} \left( -\, e^a_\mu R +\,e^a_\sigma g^{\kappa\sigma} R_{\kappa\mu} +\, e^a_\alpha g^{\alpha\kappa} R_{\kappa\mu} \right)\\ &=\frac{1}{2\kappa}\sqrt{-g} \left( -\, e^a_\mu R +2\,e^a_\sigma g^{\kappa\sigma} R_{\kappa\mu} \right) \end{align*}\] \[\begin{align*} \eta^{da} e^\omega_c \mathscr{R}^c_{d\omega\mu} +\eta^{de} e^\sigma_e \mathscr{R}^a_{d\mu\sigma} &=\eta^{de}\delta^a_e \delta^a_e e^\omega_c \mathscr{R}^c_{d\omega\mu} +\eta^{de} e^\omega_e \delta^a_c \mathscr{R}^c_{d\mu\omega}\\ &=\eta^{de} \left( \delta^a_e e^\omega_c -e^\omega_e \delta^a_c \right) \mathscr{R}^c_{d\omega\mu}\\ &=\eta^{de}e^\omega_f \left( \delta^a_e \delta^f_c -\delta^f_e \delta^a_c \right) \mathscr{R}^c_{d\omega\mu}\\ \end{align*}\] \[\begin{align*} R&=\eta^{ae}e^\sigma_e e^\omega_c \mathscr{R}^c_{a\omega\sigma}\\ R_{\kappa\sigma} &= \delta^\omega_\varepsilon R_{\kappa\omega\sigma}^\varepsilon = \delta^\omega_\varepsilon e^a_\kappa e^\varepsilon_c \mathscr{R}_{a\omega\sigma}^c \\ &= e^a_\kappa e^\omega_c\mathscr{R}_{a\omega\sigma}^c\\ e^\kappa_a R_{\kappa\sigma} &= e^\omega_c\mathscr{R}_{a\omega\sigma}^c\\ \end{align*}\] \[\begin{align*} \frac{\partial g^{\omega\sigma}}{\partial e^{\mu}_a} &=\frac{\partial }{\partial e^{\mu}_a} \left( \eta^{ce} e^\omega_c e^\sigma_e \right)\\ &=\eta^{ce} \delta^\omega_\mu \delta^a_c e^\sigma_e +\eta^{ce} e^\omega_c \delta^\sigma_\mu \delta^a_e\\ &=\eta^{ae} \delta^\omega_\mu e^\sigma_e +\eta^{ca} e^\omega_c \delta^\sigma_\mu \\ &=\eta^{ae}\left( \delta^\omega_\mu e^\sigma_e +e^\omega_e \delta^\sigma_\mu \right)\\ \\ \to \delta g^{\omega\sigma}&=\frac{\partial g^{\omega\sigma}}{\partial e^{\mu}_a} \delta e^{\mu}_a\\ &=\eta^{ae}\left( \delta^\omega_\mu e^\sigma_e +e^\omega_e \delta^\sigma_\mu \right) \delta e^{\mu}_a \end{align*}\] \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial g^{\omega\sigma} } \delta g^{\omega\sigma} &=\frac{\partial \mathscr{L}_{GR}}{\partial g^{\omega\sigma} } \frac{\partial g^{\omega\sigma}}{\partial e^{\mu}_a}\delta e^{\mu}_a \\ &=\frac{1}{2\kappa} \sqrt{-g}\left( R_{\omega\sigma} - \frac{1}{2} g_{\omega\sigma} R \right)\eta^{ae}\left( \delta^\omega_\mu e^\sigma_e +e^\omega_e \delta^\sigma_\mu \right) \delta e^{\mu}_a\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\eta^{ae}\left( R_{\omega\sigma}\left( \delta^\omega_\mu e^\sigma_e +e^\omega_e \delta^\sigma_\mu \right) - \frac{1}{2} g_{\omega\sigma} R \left( \delta^\omega_\mu e^\sigma_e +e^\omega_e \delta^\sigma_\mu \right)\right) \delta e^{\mu}_a\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\eta^{ae}\left( R_{\omega\sigma} \delta^\omega_\mu e^\sigma_e +R_{\omega\sigma} e^\omega_e \delta^\sigma_\mu -\frac{1}{2} g_{\omega\sigma} R \delta^\omega_\mu e^\sigma_e -\frac{1}{2} g_{\omega\sigma} R e^\omega_e \delta^\sigma_\mu \right) \delta e^{\mu}_a\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\eta^{ae}\left( R_{\mu\sigma}e^\sigma_e +R_{\omega\mu} e^\omega_e -\frac{1}{2} g_{\mu\sigma} R\, e^\sigma_e -\frac{1}{2} g_{\omega\mu} R\, e^\omega_e \right) \delta e^{\mu}_a\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\eta^{ae}\left( R_{\mu\sigma}e^\sigma_e +R_{\omega\mu} e^\omega_e -g_{\mu\sigma} R\, e^\sigma_e \right) \delta e^{\mu}_a\\ &=\frac{1}{2\kappa} \sqrt{-g} \left( 2R_{\mu\sigma} \eta^{ae}e^\sigma_e -g_{\mu\sigma} R\, \eta^{ae}e^\sigma_e \right) \delta e^{\mu}_a\\ &=\frac{1}{2\kappa} \sqrt{-g}\left( 2R_{\mu\sigma} g^{\sigma\kappa}e^a_\kappa -g_{\mu\sigma} R\, g^{\sigma\kappa}e^a_\kappa \right) \delta e^{\mu}_a\\ &=\frac{1}{2\kappa} \sqrt{-g} \left( 2R_{\mu\sigma} g^{\sigma\kappa}e^a_\kappa - R\,e^a_\mu \right) \delta e^{\mu}_a\\ &=\frac{\partial \mathscr{L}_{GR}}{\partial e^{\mu}_a } \delta e^{\mu}_a \end{align*}\] \(EoM\#4\) \textbf{Eq.\eqref{eq:/omega,}} \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} &= \frac{1}{2\kappa} e\, \eta^{ce} \left(e^\mu_e e^\gamma_b - e^\gamma_e e^\mu_b \right) \end{align*}\] \[\begin{align*} \left(\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \right)_{,\gamma} &= \frac{1}{2\kappa} \eta^{ce} \left(e\ e^\mu_e e^\gamma_b -e\ e^\gamma_e e^\mu_b \right)_{{,\gamma}} \end{align*}\] \textbf{Eq.\eqref{eq:/omega}} \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma}} &= \frac{1}{2\kappa} \eta^{ae} \left(e\ e^{{\mu}}_e e^\gamma_b -e\ e^\gamma_e e^{{\mu}}_b \right)\omega ^c_{a{{\mu}}} -\frac{1}{2\kappa}\eta^{ce}\left(e\ e^{{\mu}}_e e^\gamma_f -e \ e^\gamma_e e^{{\mu}}_f \right)\omega ^f_{b{{\mu}}} \end{align*}\] \[\begin{align*} e_{,\gamma}=\left( \sqrt{-g} \right)_{,\gamma} &= -\frac{1}{2} \sqrt{-g} \, g_{\phi\psi} {g^{\phi\psi}}_{, \gamma} \\ &=-\frac{1}{2} \sqrt{-g} \, g_{\phi\psi} (- g^{\eta\psi}\Gamma_{\eta\gamma}^{\phi} -g^{\phi\eta}\Gamma_{\eta\gamma}^{\psi})\\ &=\frac{1}{2} \sqrt{-g}( g_{\phi\psi} g^{\eta\psi}\Gamma_{\eta\gamma}^{\phi} + g_{\phi\psi} g^{\phi\eta}\Gamma_{\eta\gamma}^{\psi})\\ &=\frac{1}{2} \sqrt{-g}( \delta ^\eta _\phi \Gamma_{\eta\gamma}^{\phi} + \delta ^\eta _\psi \Gamma_{\eta\gamma}^{\psi})\\ &=\frac{1}{2} \sqrt{-g}( \Gamma_{\eta\gamma}^{\eta} + \Gamma_{\eta\gamma}^{\eta})\Gamma_{\eta\gamma}^{\eta})\\ &= \sqrt{-g}\Gamma_{\eta\gamma}^{\eta}\\ &=e\ e^\eta_a e_\eta^b \omega_{b\gamma}^{a}\\ &=e\ \delta^b_a \omega_{b\gamma}^{a}\\ &=e\ \omega_{a\gamma}^{a} \end{align*}\] \subsection{Canonical Energy-Momentum Tensor Derived by Noether's Theorem} Now, we apply the Noether variation: \[\begin{align*} \Delta S = \int \left[EoM \right] d^4 x + \int \left[ \partial_\gamma\left(\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\delta\omega^b_{c\mu}\right) + (\mathscr{L}_{GR} \delta x^\gamma)_{,\gamma}\right] d^4 x \end{align*}\] The spin connection \(\omega^b_{c\mu}\) is \(\frak{gl}\)-value 1-form, we apply similar variation as Eq.\eqref{eq:varaition_gauge}: \[\begin{align} \Delta \omega^b_{c\mu} &= \delta \omega^b_{c\mu} + \underbrace{\delta x^\varepsilon \partial_\varepsilon \omega^b_{c\mu} + \omega^b_{c\varepsilon}\partial_\mu \delta x^\varepsilon}_{\hat{\mathcal{L}}_{\delta x} \mathbf{\omega}} \end{align}\] We have: \[\begin{align*} \Delta S &= \int \left[EoM \right] d^4 x + \int \left[ \partial_\gamma\left(\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\left(\Delta \omega^b_{c\mu} - \delta x^\varepsilon \partial_\varepsilon \omega^b_{c\mu} - \omega^b_{c\varepsilon}\partial_\mu \delta x^\varepsilon \right)\right) + (\mathscr{L}_{GR} \delta x^\gamma)_{,\gamma}\right] d^4 x \\ &= \int \left[EoM \right] d^4 x + \int \underbracket[0.4pt][0pt]{\partial_\gamma}_{(*)}\left[ \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\Delta \omega^b_{c\mu} - \left(\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\mu,\varepsilon}\delta x^\varepsilon + \underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\varepsilon}\delta x^\varepsilon_{,\mu}}_{(*)} - \delta^\gamma_\varepsilon \mathscr{L}_{GR} \delta x^\varepsilon \right)\right] d^4 x \end{align*}\] Evaluate the \((*)\) term: \[\begin{align*} \label{eq:(*)} \underbracket[0.4pt][0pt]{\partial_\gamma \left[\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\varepsilon}\delta x^\varepsilon_{,\mu}\right]}_{(*)}&= \underbracket[0.4pt][0pt]{\partial_\gamma \left[\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\varepsilon}\delta x^\varepsilon\right]_{,\mu}}_{(*1)} -\underbracket[0.4pt][0pt]{\partial_\gamma \left[ \left(\frac{ \partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\right)_{,\mu}\omega^b_{c\varepsilon}\delta x^\varepsilon\right]}_{(*2)} -\underbracket[0.4pt][0pt]{\partial_\gamma \left[\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\varepsilon,\mu}\delta x^\varepsilon\right]}_{(*3)} \end{align*}\] We first calculate \(\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\) for later use: \[\begin{align} \label{eq:/omega,} \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} &= \frac{1}{2\kappa} e\, \eta^{ce} \left(e^\mu_e e^\gamma_b - e^\gamma_e e^\mu_b \right) \left(=- \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma,\mu}}\right) \end{align}\] The \((*1)\) term: \[\begin{align*} \underbracket[0.4pt][0pt]{\partial_\gamma \left[\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\varepsilon}\delta x^\varepsilon\right]_{,\mu}}_{(*1)}=\left[ \underbracket[0.4pt][0pt]{\frac{1}{2\kappa} e\, \eta^{ce} \left(e^{\textcolor{red}{\mu}}_e e^{\textcolor{red}{\gamma}}_b - e^{\textcolor{red}{\gamma}}_e e^{\textcolor{red}{\mu}}_b \right)}_{\eqref{eq:/omega,}} \omega^b_{c\varepsilon}\delta x^\varepsilon\right]_{,\textcolor{red}{\mu\gamma}}=0 \end{align*}\] The \((*2)\) term rely on \(EoM\#4\), using \[\begin{align*} -\underbracket[0.4pt][0pt]{\partial_\gamma \left[ \left(\frac{ \partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\right)_{,\mu}\omega^b_{c\varepsilon}\delta x^\varepsilon\right]}_{(*2)} \underbrace{=}_{\eqref{eq:/omega,}} -\partial_\gamma \left[ \left( \textcolor{red}{-} \frac{ \partial \mathscr{L}_{GR}}{\partial \omega^b_{c\textcolor{red}{\gamma,\mu}}}\right)_{,\mu}\omega^b_{c\varepsilon}\delta x^\varepsilon\right] \underbrace{=}_{EoM\#4} \partial_\gamma \left( \frac{ \partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma}}\omega^b_{c\varepsilon}\delta x^\varepsilon\right) \end{align*}\] and \[\begin{align}\label{eq:/omega} \frac{ \partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma}}=\frac{1}{2\kappa} e\, \eta^{ae}\left(e^\gamma_e e^\mu_b- e^\mu_e e^\gamma_b \right)\omega^c_{a\mu} +\frac{1}{2\kappa} e\, \eta^{ce} \left(e^\mu_e e^\gamma_f -e^\gamma_e e^\mu_f \right)\omega^f_{b\mu} \end{align}\] We can calulate \[\begin{align} \label{eq:/omega*/omega} \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma}}\omega^b_{c\varepsilon} &=\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \left( \omega^b_{{{d}}\varepsilon}\omega^{{d}}_{{{c}}\mu}- \omega^{{b}}_{{{d}}\mu}\omega^{{d}}_{c\varepsilon} \right) \end{align}\] The \((*)\) term than become: \[\begin{align*} \underbracket[0.4pt][0pt]{\partial_\gamma \left[\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\varepsilon}\delta x^\varepsilon_{,\mu}\right]}_{(*)} &=\underbracket[0.4pt][0pt]{0}_{(*1)} -\underbracket[0.4pt][0pt]{\partial_\gamma \left[\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\varepsilon,\mu}\delta x^\varepsilon\right]}_{(*3)} +\underbracket[0.4pt][0pt]{\partial_\gamma \left[ \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \left( \omega^b_{{{d}}\varepsilon}\omega^{{d}}_{{{c}}\mu}- \omega^{{b}}_{{{d}}\mu}\omega^{{d}}_{c\varepsilon} \right)\delta x^\varepsilon \right]}_{(*2)} \end{align*}\] We have: \begin{adjustwidth}{-1.5cm}{-1cm} \[\begin{align*} \Delta S &=\int \left[EoM \right] d^4 x + \int {\partial_\gamma}\left[ \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\Delta \omega^b_{c\mu} - \left(\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\mu,\varepsilon}\delta x^\varepsilon -\underbracket[0.4pt][0pt]{\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\omega^b_{c\varepsilon,\mu}\delta x^\varepsilon}_{(*3)} +\underbracket[0.4pt][0pt]{ \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\left(\omega^b_{d\varepsilon}\omega^d_{c\mu}-\omega^b_{d\mu}\omega^d_{c\varepsilon} \right)\delta x^\varepsilon}_{(*2)} - \delta^\gamma_\varepsilon \mathscr{L}_{GR} \delta x^\varepsilon \right)\right] d^4 x \\ &=\int \left[EoM \right] d^4 x + \int {\partial_\gamma}\left[ \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\Delta \omega^b_{c\mu} - \left(\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\left(\omega^b_{c\mu,\varepsilon} -\omega^b_{c\varepsilon,\mu} + \omega^b_{d\varepsilon}\omega^d_{c\mu}-\omega^b_{d\mu}\omega^d_{c\varepsilon} \right) \delta x^\varepsilon - \delta^\gamma_\varepsilon \mathscr{L}_{GR} \delta x^\varepsilon \right)\right] d^4 x \\ &=\int \left[EoM \right] d^4 x + \int {\partial_\gamma}\left[ \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}}\Delta \omega^b_{c\mu} - \left(\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \mathscr{R}^b_{c\varepsilon\mu} - \delta^\gamma_\varepsilon \mathscr{L}_{GR} \right)\delta x^\varepsilon \right] d^4 x \end{align*}\] \end{adjustwidth} The canonical energy-momentum tensor: \[\begin{align} \left(t_{GR}\right)^\gamma_\varepsilon&=\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \mathscr{R}^b_{c\varepsilon\mu} - \delta^\gamma_\varepsilon \mathscr{L}_{GR} \nonumber\\ &=\frac{1}{2\kappa} e\,\eta^{ce}\left(e^\gamma_e e^\mu_b-e^\mu_e e^\gamma_b \right)\mathscr{R}^b_{c\varepsilon\mu}-\delta^\gamma_\varepsilon \mathscr{L}_{GR} \nonumber\\ &=\frac{1}{2\kappa} \left(g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\sqrt{-g} \label{eq:Noether_t} \end{align}\] \[\begin{align} \left(t_{GR}\right)^\gamma_\varepsilon&=\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \mathscr{R}^b_{c\varepsilon\mu} - \delta^\gamma_\varepsilon \mathscr{L}_{GR} \nonumber\\ &=\frac{1}{2\kappa} e\,\eta^{ce}\left(e^\gamma_e e^\mu_b-e^\mu_e e^\gamma_b \right)\mathscr{R}^b_{c\varepsilon\mu}-\delta^\gamma_\varepsilon e\, \eta^{ae}e^\sigma_e e^\omega_c \mathscr{R}^c_{a\omega\sigma} \nonumber\\ &=\frac{1}{2\kappa} \left(g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\sqrt{-g} \label{eq:Noether_t} \end{align}\] \[\begin{align*} g^{\psi\varepsilon}g_{\phi\gamma}\left(t_{GR}\right)^\gamma_\varepsilon &=\frac{1}{2\kappa} g^{\psi\varepsilon}g_{\phi\gamma}\left(g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left( g^{\psi\varepsilon}g_{\phi\gamma}g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu} +g^{\psi\varepsilon}g_{\phi\gamma}g^{\beta\gamma}R_{\beta\varepsilon} -g^{\psi\varepsilon}g_{\phi\gamma}\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left( g^{\psi\varepsilon}g_{\phi\gamma}g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu} +g^{\psi\varepsilon}R_{\phi\varepsilon} -\delta^\psi_\phi R \right)\sqrt{-g} \\ \end{align*}\] The canonical energy-momentum 2-form: \[\begin{align*} \left(t_{GR}\right)_{\alpha\varepsilon}&=g_{\alpha\gamma}t^\gamma_\varepsilon =\frac{1}{2\kappa} \left(g_{\alpha\gamma}g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu} +R_{\alpha\varepsilon} -g_{\alpha\varepsilon} R \right)\sqrt{-g} \end{align*}\] If metric compatible, we have: \[\begin{align} \label{eq:t_gravity} \left(t_{GR}\right)_{\alpha\varepsilon}&=\frac{1}{\kappa} \mathbb{G}_{{\alpha}{\varepsilon}}\sqrt{-g} \end{align}\] This canonical energy-momentum 2-form has following properties:\\ 1. symmetric\\ 2. coordinate independent\\ 3. vanish when vacuum, which does not depend on the choice of the connection. \subsection{Alternative Derivation of Einstein Field Equation} \[\begin{align*} S = \int d^4x \, \mathscr{L}_{GR}+ \mathscr{L}_{EM} \end{align*}\] \[\begin{align*} \mathscr{L}_{GR}=\frac{1}{2\kappa} \sqrt{-g} g^{\kappa\sigma}\delta^\omega_\varepsilon R^\varepsilon_{\kappa\omega\sigma} = \frac{1}{2\kappa}\, \eta^{ae}e^\sigma_e e^\omega_c \mathscr{R}^c_{a\omega\sigma} \end{align*}\] \[\begin{align*} \mathscr{L}_{EM}= -\frac{1}{16\pi c} g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}F_{\mu\nu}\sqrt{-g} \end{align*}\] \[\begin{align*} x^\gamma &\to x^\gamma+ \delta x^\gamma\\ A_\nu &\to A_\nu+\delta A_\nu \\ \omega^a_{b\nu} &\to \omega^a_{b\nu}+\delta \omega^a_{b\nu} \end{align*}\] \begin{adjustwidth}{-2.3cm}{-1cm} \[\begin{align*} \Delta S = \int \left[ \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\nu}} - \left(\frac{\partial \mathscr{L}_{GR} }{\partial \omega^b_{c\nu,\mu}}\right)_{,\mu}\right]\delta \omega^b_{c\nu} + \left[ \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu} - \left(\frac{\partial\mathscr{L}_{EM}} {\partial A_{\nu,\mu}}\right)_{,\mu} \right] \delta A_\nu +\partial_\mu \left\{\frac{\partial \mathscr{L}_{GR} }{\partial \omega^b_{c\nu,\mu}}\Delta \omega^b_{c\nu} + \frac{\partial \mathscr{L}_{EM}}{\partial A_{\nu,\mu} }\Delta A_\nu - \left[\left(t_{GR} \right)^\mu_\gamma +\left(t_{EM} \right)^\mu_\gamma \right]\delta x^\gamma \right\} d^4x \end{align*}\] \end{adjustwidth} \section{Equivalence of Einstein-Hilbert EMT and Noether canonical EMT} \[\begin{align*} \left(\hat{\mathcal{L}}_{\delta X}\, g\right)_{\mu\nu} &=g_{\mu\nu,\gamma} \delta X^\gamma+g_{\gamma\nu} {\delta X^\gamma}_{,\mu}+g_{\mu\gamma} {\delta X^\gamma}_{,\nu}\\ &=\left(g_{\mu\nu,\gamma}-g_{\eta\nu}\Gamma^\eta_{\mu\gamma}-g_{\mu\eta}\Gamma^\eta_{\nu\gamma}\right) \delta X^\gamma +g_{\eta\nu}\Gamma^\eta_{\mu\gamma}\delta X^\gamma +g_{\mu\eta}\Gamma^\eta_{\nu\gamma}\delta X^\gamma +g_{\gamma\nu} {\delta X^\gamma}_{,\mu}+g_{\mu\gamma} {\delta X^\gamma}_{,\nu}\\ &=g_{\mu\nu;\gamma} \delta X^\gamma +g_{\textcolor{red}{\gamma}\nu}\Gamma^{\textcolor{red}{\gamma}}_{\mu\textcolor{blue}{\eta}}\delta X^{\textcolor{blue}{\eta}} +g_{\mu{\textcolor{red}{\gamma}}}\Gamma^{\textcolor{red}{\gamma}}_{\nu{\textcolor{blue}{\eta}}}\delta X^{\textcolor{blue}{\eta}} +g_{\gamma\nu} {\delta X^\gamma}_{,\mu}+g_{\mu\gamma} {\delta X^\gamma}_{,\nu}\\ &=g_{\mu\nu;\gamma} \delta X^\gamma +g_{\gamma\nu} {\delta X^\gamma}_{;\mu}+g_{\mu\gamma} {\delta X^\gamma}_{;\nu}\\ &= g_{\gamma\nu} {\delta X^\gamma}_{;\mu}+g_{\mu\gamma} {\delta X^\gamma}_{;\nu} \end{align*}\] \[\begin{align*} {g_{\mu\nu,\gamma}} &=\left(g_{\mu\alpha} g_{\nu\beta} g^{\alpha\beta} \right)_{,\gamma}\\ &=g_{\mu\alpha,\gamma} g_{\nu\beta} g^{\alpha\beta} +g_{\mu\alpha} g_{\nu\beta,\gamma} g^{\alpha\beta} +g_{\mu\alpha} g_{\nu\beta} {g^{\alpha\beta}}_{,\gamma} \\ &=g_{\mu\alpha,\gamma}\delta_\nu^\alpha +g_{\nu\beta,\gamma} \delta_\mu^\beta +g_{\mu\alpha} g_{\nu\beta} {g^{\alpha\beta}}_{,\gamma} \\ &=g_{\mu\nu,\gamma} +g_{\nu\mu,\gamma} +g_{\mu\alpha} g_{\nu\beta} {g^{\alpha\beta}}_{,\gamma} \\ \to g_{\mu\nu,\gamma} &=- g_{\mu\alpha} g_{\nu\beta} {g^{\alpha\beta}}_{,\gamma}\\ &=- g_{\mu\alpha} g_{\nu\beta} \left( -g^{\sigma\beta}\Gamma^\alpha_{\sigma\gamma} -g^{\alpha\sigma}\Gamma^\beta_{\sigma\gamma} +\frac{2}{\mathbb{D}-1}g^{\alpha\beta}T_\gamma \right)\\ &=g_{\mu\alpha}\Gamma^\alpha_{\nu\gamma} +g_{\nu\alpha}\Gamma^\alpha_{\mu\gamma} -\frac{2}{\mathbb{D}-1}g_{\mu\nu}T_\gamma \end{align*}\] \[\begin{align*} \frac{\partial\mathscr{L}}{\partial g_{\mu\nu}}\delta g_{\mu\nu} &=\frac{\partial\mathscr{L}}{\partial g_{\mu\nu}}\left(\hat{\mathcal{L}}_{\delta X}\, g\right)_{\mu\nu}\\ &=\frac{1}{2}t^{\mu\nu}\left(g_{\gamma\nu} {\delta X^\gamma}_{;\mu}+g_{\mu\gamma} {\delta X^\gamma}_{;\nu}\right)\\ &=t^{\mu\nu}g_{\gamma\nu} {\delta X^\gamma}_{;\mu}\\ &={t^\mu}_{\gamma} {\delta X^\gamma}_{;\mu}\\ &={T^\mu}_{\gamma}\sqrt{-g}\, {\delta X^\gamma}_{;\mu}\\ &=\left({T^\mu}_{\gamma}{\delta X^\gamma}\right)_{;\mu}\sqrt{-g} \, \, -{T^\mu}_{\gamma;\mu}\sqrt{-g}\,\,{\delta X^\gamma}\\ &=\left({T^\mu}_{\gamma}{\delta X^\gamma}\sqrt{-g}\right)_{,\mu} \, \, -\left({T^\mu}_{\gamma}\sqrt{-g}\right)_{;\mu}\,\,{\delta X^\gamma}\\ &=\left({t^\mu}_{\gamma}{\delta X^\gamma}\right)_{,\mu} \, \, -{t^\mu}_{\gamma;\mu}{\delta X^\gamma}\\ \end{align*}\] \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align*} \left(\hat{\mathcal{L}}_{\delta X}\, g\right)_{\mu\nu} &=g_{\mu\nu,\gamma} \delta X^\gamma+g_{\gamma\nu} {\delta X^\gamma}_{,\mu}+g_{\mu\gamma} {\delta X^\gamma}_{,\nu}\\ &=\left( g_{\mu\nu,\gamma}-g_{\eta\nu}\Gamma^\eta_{\mu\gamma}-g_{\mu\eta}\Gamma^\eta_{\nu\gamma} +\frac{2}{\mathbb{D}-1}g_{\mu\nu}T_\gamma \right) \delta X^\gamma +g_{\eta\nu}\Gamma^\eta_{\mu\gamma}\delta X^\gamma +g_{\mu\eta}\Gamma^\eta_{\nu\gamma}\delta X^\gamma -\frac{2}{\mathbb{D}-1}g_{\mu\nu}T_\gamma \delta X^\gamma +g_{\gamma\nu} {\delta X^\gamma}_{,\mu}+g_{\mu\gamma} {\delta X^\gamma}_{,\nu}\\ &=\underbracket{g_{\mu\nu;\gamma}}_{=0} \delta X^\gamma +g_{\textcolor{red}{\gamma}\nu}\Gamma^{\textcolor{red}{\gamma}}_{\mu\textcolor{blue}{\eta}}\delta X^{\textcolor{blue}{\eta}} +g_{\mu{\textcolor{red}{\gamma}}}\Gamma^{\textcolor{red}{\gamma}}_{\nu{\textcolor{blue}{\eta}}}\delta X^{\textcolor{blue}{\eta}} +g_{\gamma\nu} {\delta X^\gamma}_{,\mu}+g_{\mu\gamma} {\delta X^\gamma}_{,\nu} -\frac{2}{\mathbb{D}-1}g_{\mu\nu}T_\gamma \delta X^\gamma\\ &=g_{{\gamma}\nu}\Gamma^{{\gamma}}_{{\eta}\mu}\delta X^{{\eta}} +g_{{\gamma}\nu}T^{{\gamma}}_{\mu{\eta}}\delta X^{{\eta}} +g_{\mu{{\gamma}}}\Gamma^{{\gamma}}_{{{\eta}\nu}}\delta X^{{\eta}} +g_{\mu{{\gamma}}}T^{{\gamma}}_{\nu{{\eta}}}\delta X^{{\eta}} +g_{\gamma\nu} {\delta X^\gamma}_{,\mu}+g_{\mu\gamma} {\delta X^\gamma}_{,\nu} -\frac{2}{\mathbb{D}-1}g_{\mu\nu}T_\gamma \delta X^\gamma\\ &=g_{\gamma\nu} {\delta X^\gamma}_{;\mu}+g_{\mu\gamma} {\delta X^\gamma}_{;\nu} +\frac{1}{\mathbb{D}-1}g_{{\gamma}\nu}\left( \delta^{{\gamma}}_{\mu} T_{\eta} -\delta^{{\gamma}}_{\eta} T_{\mu} \right)\delta X^{{\eta}} +\frac{1}{\mathbb{D}-1}g_{\mu{{\gamma}}}\left( \delta^{{\gamma}}_{\nu} T_{\eta} -\delta^{{\gamma}}_{\eta} T_{\nu} \right)\delta X^{{\eta}} + -\frac{2}{\mathbb{D}-1}g_{\mu\nu}T_\gamma \delta X^\gamma\\ &=g_{\gamma\nu} {\delta X^\gamma}_{;\mu}+g_{\mu\gamma} {\delta X^\gamma}_{;\nu} +\frac{1}{\mathbb{D}-1}\left( g_{\mu\nu} T_{\eta} -g_{\eta\nu} T_{\mu} \right)\delta X^{{\eta}} +\frac{1}{\mathbb{D}-1}\left( g_{\mu{\nu}} T_{\eta} -g_{\mu{\eta}} T_{\nu} \right)\delta X^{{\eta}} -\frac{2}{\mathbb{D}-1}g_{\mu\nu}T_\gamma \delta X^\gamma\\ &=g_{\gamma\nu} {\delta X^\gamma}_{;\mu}+g_{\mu\gamma} {\delta X^\gamma}_{;\nu} +\frac{1}{\mathbb{D}-1}\left( \cancel{g_{\mu\nu} T_{\gamma}\delta X^{{\gamma}}} -g_{\gamma\nu} T_{\mu}\delta X^{{\gamma}} \right) +\frac{1}{\mathbb{D}-1}\left( \cancel{g_{\mu{\nu}} T_{\gamma}\delta X^{{\gamma}}} -g_{\mu{\gamma}} T_{\nu}\delta X^{{\gamma}} \right) -\cancel{\frac{2}{\mathbb{D}-1}g_{\mu\nu}T_\gamma \delta X^\gamma}\\ &=g_{\gamma\nu} {\delta X^\gamma}_{;\mu}+g_{\mu\gamma} {\delta X^\gamma}_{;\nu} -\frac{1}{\mathbb{D}-1} g_{\gamma\nu} T_{\mu}\delta X^{{\gamma}} -\frac{1}{\mathbb{D}-1} g_{\mu{\gamma}} T_{\nu}\delta X^{{\gamma}} \end{align*}\] \end{adjustwidth} \[\begin{align*} \left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu} &={g^{\mu\nu}}_{,\gamma} \delta X^\gamma-g^{\gamma\nu} {\delta X^\mu}_{,\gamma}-g^{\mu\gamma} {\delta X^\nu}_{,\gamma}\\ &=\left({g^{\mu\nu}}_{,\gamma} +g^{\eta\nu}\Gamma^\mu_{\eta\gamma} +g^{\mu\eta}\Gamma^\nu_{\eta\gamma}\right) \delta X^\gamma -g^{\eta\nu}\Gamma^\mu_{\eta\gamma}\delta X^\gamma -g^{\mu\eta}\Gamma^\nu_{\eta\gamma}\delta X^\gamma -g^{\gamma\nu} {\delta X^\mu}_{,\gamma}-g^{\mu\gamma} {\delta X^\nu}_{,\gamma}\\ &={g^{\mu\nu}}_{;\gamma} \delta X^\gamma -g^{\textcolor{red}{\gamma}\nu}\Gamma^\mu_{\textcolor{red}{\gamma}\textcolor{blue}{\eta}}\delta X^{\textcolor{blue}{\eta}} -g^{\mu{\textcolor{red}{\gamma}}}\Gamma^\nu_{{\textcolor{red}{\gamma}}{\textcolor{blue}{\eta}}}\delta X^{\textcolor{blue}{\eta}} -g^{\gamma\nu} {\delta X^\mu}_{,\gamma}-g^{\mu\gamma} {\delta X^\nu}_{,\gamma}\\ &={g^{\mu\nu}}_{;\gamma} \delta X^\gamma -g^{\gamma\nu} {\delta X^\mu}_{;\gamma}-g^{\mu\gamma} {\delta X^\nu}_{;\gamma}\\ &=-g^{\gamma\nu} {\delta X^\mu}_{;\gamma}-g^{\mu\gamma} {\delta X^\nu}_{;\gamma} \end{align*}\] \[\begin{align*} \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}\delta g^{\mu\nu} &=\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}\left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu}\\ &=\frac{1}{2}t_{\mu\nu}\left( -g^{\gamma\nu} {\delta X^\mu}_{;\gamma}-g^{\mu\gamma} {\delta X^\nu}_{;\gamma} \right)\\ &=-t_{\mu\nu}g^{\mu\gamma} {\delta X^\nu}_{;\gamma}\\ &=-t^\gamma_\nu\, {\delta X^\nu}_{;\gamma}\\ &=-T^\gamma_\nu\sqrt{-g}\, {\delta X^\nu}_{;\gamma}\\ &=-\left(T_\nu^\gamma{\delta X^\nu}\right)_{;\gamma}\sqrt{-g} \, \, +T_{\nu;\gamma}^{\gamma}\sqrt{-g}\,\,{\delta X^\nu}\\ &=-\left(T_\nu^\gamma{\delta X^\nu}\sqrt{-g}\right)_{,\gamma} \, \, +\left(T_{\nu}^{\gamma}\sqrt{-g}\right)_{;\gamma}\,\,{\delta X^\nu}\\ &=-\left(t_\nu^\gamma{\delta X^\nu}\right)_{,\gamma} \, \, +\left(t_{\nu}^{\gamma}\right)_{;\gamma}\,{\delta X^\nu} \end{align*}\] \[\begin{align*} \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}\delta g^{\mu\nu} &=\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}\left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu}\\ &=\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} \left( {g^{\mu\nu}}_{,\gamma} \delta X^\gamma-g^{\gamma\nu} {\delta X^\mu}_{,\gamma}-g^{\mu\gamma} {\delta X^\nu}_{,\gamma} \right)\\ &= \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma -\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} -\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\mu\gamma} {\delta X^\nu}_{,\gamma}\\ &= \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} \end{align*}\] \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align*} \left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu} &={g^{\mu\nu}}_{,\gamma} \delta X^\gamma-g^{\gamma\nu} {\delta X^\mu}_{,\gamma}-g^{\mu\gamma} {\delta X^\nu}_{,\gamma}\\ &=\left({g^{\mu\nu}}_{,\gamma} +g^{\eta\nu}\Gamma^\mu_{\eta\gamma} +g^{\mu\eta}\Gamma^\nu_{\eta\gamma} -\frac{2}{\mathbb{D}-1}g^{\mu\nu}T_\gamma\right) \delta X^\gamma -g^{\eta\nu}\Gamma^\mu_{\eta\gamma}\delta X^\gamma -g^{\mu\eta}\Gamma^\nu_{\eta\gamma}\delta X^\gamma +\frac{2}{\mathbb{D}-1}g^{\mu\nu}T_\gamma \delta X^\gamma -g^{\gamma\nu} {\delta X^\mu}_{,\gamma}-g^{\mu\gamma} {\delta X^\nu}_{,\gamma}\\ &=\underbracket{{g^{\mu\nu}}_{;\gamma}}_{=0} \delta X^\gamma -g^{\textcolor{red}{\gamma}\nu}\Gamma^\mu_{\textcolor{red}{\gamma}\textcolor{blue}{\eta}}\delta X^{\textcolor{blue}{\eta}} -g^{\mu{\textcolor{red}{\gamma}}}\Gamma^\nu_{{\textcolor{red}{\gamma}}{\textcolor{blue}{\eta}}}\delta X^{\textcolor{blue}{\eta}} -g^{\gamma\nu} {\delta X^\mu}_{,\gamma}-g^{\mu\gamma} {\delta X^\nu}_{,\gamma} +\frac{2}{\mathbb{D}-1}g^{\mu\nu}T_\gamma \delta X^\gamma\\ &=-g^{{\gamma}\nu}\Gamma^\mu_{{\eta}{\gamma}}\delta X^{{\eta}} -g^{{\gamma}\nu}T^\mu_{{\gamma}{\eta}}\delta X^{{\eta}} -g^{\mu{{\gamma}}}\Gamma^\nu_{{{\eta}{\gamma}}}\delta X^{{\eta}} -g^{\mu{{\gamma}}}T^\nu_{{{\gamma}}{{\eta}}}\delta X^{{\eta}} -g^{\gamma\nu} {\delta X^\mu}_{,\gamma}-g^{\mu\gamma} {\delta X^\nu}_{,\gamma} +\frac{2}{\mathbb{D}-1}g^{\mu\nu}T_\gamma \delta X^\gamma\\ &=-g^{\gamma\nu} {\delta X^\mu}_{;\gamma}-g^{\mu\gamma} {\delta X^\nu}_{;\gamma} -\frac{1}{\mathbb{D}-1}g^{\gamma\nu} \left( \delta^\mu_{\gamma} T_\eta -\delta^\mu_{\eta} T_\gamma \right) \delta X^{{\eta}} -\frac{1}{\mathbb{D}-1}g^{\mu\gamma} \left( \delta^\nu_{\gamma} T_\eta -\delta^\nu_{\eta} T_\gamma \right) \delta X^{{\eta}} +\frac{2}{\mathbb{D}-1}g^{\mu\nu}T_\gamma \delta X^\gamma \\ &=-g^{\gamma\nu} {\delta X^\mu}_{;\gamma}-g^{\mu\gamma} {\delta X^\nu}_{;\gamma} -\frac{1}{\mathbb{D}-1} \left( g^{\gamma\nu}\delta^\mu_{\gamma} T_\eta \delta X^{{\eta}} -g^{\gamma\nu}\delta^\mu_{\eta} T_\gamma \delta X^{{\eta}} \right) -\frac{1}{\mathbb{D}-1} \left( g^{\mu\gamma}\delta^\nu_{\gamma} T_\eta \delta X^{{\eta}} -g^{\mu\gamma}\delta^\nu_{\eta} T_\gamma \delta X^{{\eta}} \right) +\frac{2}{\mathbb{D}-1}g^{\mu\nu}T_\gamma \delta X^\gamma \\ &=-g^{\gamma\nu} {\delta X^\mu}_{;\gamma}-g^{\mu\gamma} {\delta X^\nu}_{;\gamma} -\frac{1}{\mathbb{D}-1} \left( \cancel{g^{\mu\nu} T_\eta \delta X^{{\eta}}} -g^{\gamma\nu} T_\gamma \delta X^{{\mu}} \right) -\frac{1}{\mathbb{D}-1} \left( \cancel{g^{\mu\nu} T_\eta \delta X^{{\eta}}} -g^{\mu\gamma} T_\gamma \delta X^{{\nu}} \right) +\cancel{\frac{2}{\mathbb{D}-1}g^{\mu\nu}T_\gamma \delta X^\gamma} \\ &=-g^{\gamma\nu} {\delta X^\mu}_{;\gamma}-g^{\mu\gamma} {\delta X^\nu}_{;\gamma} +\frac{1}{\mathbb{D}-1} g^{\gamma\nu} T_\gamma \delta X^{{\mu}} +\frac{1}{\mathbb{D}-1} g^{\mu\gamma} T_\gamma \delta X^{{\nu}} \end{align*}\] \end{adjustwidth} Projective invariance\\ https://arxiv.org/pdf/1807.10168\\ https://arxiv.org/pdf/1812.03420 \[\begin{align*} \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}\delta g^{\mu\nu} &=\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}\left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu}\\ &=\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}\left( -g^{\gamma\nu} {\delta X^\mu}_{;\gamma}-g^{\mu\gamma} {\delta X^\nu}_{;\gamma} +\frac{1}{\mathbb{D}-1} g^{\gamma\nu} T_\gamma \delta X^{{\mu}} +\frac{1}{\mathbb{D}-1} g^{\mu\gamma} T_\gamma \delta X^{{\nu}} \right)\\ &= -\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{;\gamma} -\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\mu\gamma} {\delta X^\nu}_{;\gamma} +\frac{1}{\mathbb{D}-1}\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} g^{\gamma\nu} T_\gamma \delta X^{{\mu}} +\frac{1}{\mathbb{D}-1}\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} g^{\mu\gamma} T_\gamma \delta X^{{\nu}} \\ &= -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{;\gamma} +\frac{2}{\mathbb{D}-1}\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} g^{\gamma\nu} T_\gamma \delta X^{{\mu}} \end{align*}\] Using \[\begin{align*} \left(t_{GR}\right)^\gamma_{\varepsilon,\gamma} =\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} +\left(t_{GR}\right)^\eta_\nu \, \Gamma^\nu_{\eta\varepsilon} &\leftrightarrow \left(t_{GR}\right)^\gamma_{\varepsilon;\gamma} =\left\{ EoM_{\delta\Gamma^\alpha_{\nu\mu}}\right\} R^\alpha_{\nu\varepsilon\mu} \\ \left(t_{YM}\right)^\mu_{\gamma,\mu} =\left\{ \left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} \right]_{,\mu} -\frac{\partial \mathscr{L}_{YM}}{\partial( B^a_\nu)} \right\}G^a_{\gamma\nu} + \left( t_{YM}\right)^\eta_\mu \Gamma^\mu_{\eta\gamma} &\leftrightarrow \left(t_{YM}\right)^\mu_{\gamma;\mu} =\left\{ EoM_{\delta B^a_\nu} \right\}G^a_{\gamma\nu} \end{align*}\] We have \[\begin{align*} \frac{\partial\left({\mathscr{L}_{GR}}+{\mathscr{L}_{YM}}\right)}{\partial g_{\mu\nu}}\delta g_{\mu\nu} &=\left\{\left[{\left(t_{GR}\right)^\mu}_{\gamma}+{\left(t_{YM}\right)^\mu}_{\gamma}\right]{\delta X^\gamma}\right\}_{,\mu} \, \, -\left[{\left(t_{GR}\right)^\mu}_{\gamma}+{\left(t_{YM}\right)^\mu}_{\gamma}\right]_{;\mu}{\delta X^\gamma}\\ &=\left\{\left[{\left(t_{GR}\right)^\mu}_{\gamma}+{\left(t_{YM}\right)^\mu}_{\gamma}\right]{\delta X^\gamma}\right\}_{,\mu} -\left\{ EoM_{\delta\Gamma^\alpha_{\nu\mu}}\right\} R^\alpha_{\nu\varepsilon\mu}{\delta X^\gamma} -\left\{ EoM_{\delta B^a_\nu} \right\}G^a_{\gamma\nu}{\delta X^\gamma} \end{align*}\] Move to left \[\begin{align*} \frac{\partial\left({\mathscr{L}_{GR}}+{\mathscr{L}_{YM}}\right)}{\partial g_{\mu\nu}}\delta g_{\mu\nu} +\left\{ EoM_{\delta\Gamma^\alpha_{\nu\mu}}\right\} R^\alpha_{\nu\varepsilon\mu}{\delta X^\gamma} +\left\{ EoM_{\delta B^a_\nu} \right\}G^a_{\gamma\nu}{\delta X^\gamma} -\left\{\left[{\left(t_{GR}\right)^\mu}_{\gamma}+{\left(t_{YM}\right)^\mu}_{\gamma}\right]{\delta X^\gamma}\right\}_{,\mu} &=0 \end{align*}\] Compare with \[\begin{align*} \Delta S &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\delta g^{\mu\nu} d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} F_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{EM} \right) \delta x^\gamma \right]d^4x \\ &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu}d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( t_{EM} \right)^\mu_\gamma \delta X^\gamma \right]d^4x \\ &= \int -2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} {\delta X^\nu}_{;\gamma}d^4x - \int \partial_\mu \left[ \sqrt{-g}\, \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right]d^4x \\ &= \int -2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} {\delta X^\nu}_{;\gamma}d^4x - \int \sqrt{-g}\,\nabla_\mu \left[ \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right]d^4x \\ &= \int -2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} {\delta X^\nu}_{;\gamma}d^4x - \int \sqrt{-g}\, \left[\nabla_\mu \left( T_{EM} \right)^\mu_\nu \right] \delta X^\nu d^4x - \int \sqrt{-g}\,\left( T_{EM} \right)^\mu_\nu \left(\nabla_\mu {\delta X^\nu}\right) d^4x \\ &= \int \left[-2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma}\right] {\delta X^\nu}_{;\gamma}d^4x - \int \sqrt{-g}\, \left[\nabla_\mu \left( T_{EM} \right)^\mu_\nu \right] \delta X^\nu d^4x - \int \left[ \sqrt{-g}\, \left( T_{EM} \right)^\gamma_\nu \right] {\delta X^\nu}_{;\gamma} d^4x \\ &= \int \left[-2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\nu \right] {\delta X^\nu}_{;\gamma}d^4x - \int \sqrt{-g}\, \left[\nabla_\mu \left( T_{EM} \right)^\mu_\nu \right] \delta X^\nu d^4x \end{align*}\] \[\begin{align*} -2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} &=\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\nu\\ \to \frac{1}{\sqrt{-g}}\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}&=-\frac{1}{2} g_{\mu\gamma} \left( T_{EM} \right)^\gamma_\nu\\ \end{align*}\] \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align*} \Delta S &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\delta g^{\mu\nu} d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} F_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{EM} \right) \delta x^\gamma \right]d^4x \\ &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu}d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( t_{EM} \right)^\mu_\gamma \delta X^\gamma \right]d^4x \\ &= \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} \right]d^4x - \int \partial_\mu \left[ \sqrt{-g}\, \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right]d^4x \\ &= \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} \right]d^4x - \int \left[ \sqrt{-g}_{\, ,\mu} \left( T_{EM} \right)^\mu_\nu \delta X^\nu +\sqrt{-g}\, \left( T_{EM} \right)^\mu_{\nu,\mu} \delta X^\nu +\sqrt{-g}\, \left( T_{EM} \right)^\mu_\nu {\delta X^\nu}_{,\mu} \right]d^4x \\ &= \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\gamma}}{g^{\mu\gamma}}_{,\nu} \delta X^\nu -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} -\sqrt{-g}_{\, ,\mu} \left( T_{EM} \right)^\mu_\nu \delta X^\nu -\sqrt{-g}\, \left( T_{EM} \right)^\mu_{\nu,\mu} \delta X^\nu -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\mu {\delta X^\mu}_{,\gamma} \right]d^4x \\ &= \int \left[ -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\mu \right]{\delta X^\mu}_{,\gamma}d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\gamma}}{g^{\mu\gamma}}_{,\nu} -\sqrt{-g}_{\, ,\mu} \left( T_{EM} \right)^\mu_\nu \right] \delta X^\nu d^4x - \int \sqrt{-g}\, \left( T_{EM} \right)^\mu_{\nu,\mu} \delta X^\nu d^4x \\ &= \int \left[ -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\mu \right]{\delta X^\mu}_{,\gamma}d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\gamma}}{g^{\mu\gamma}}_{,\nu} +\frac{1}{2}\sqrt{-g}\, g_{\psi\phi} {g^{\psi\phi}}_{ ,\mu} \left( T_{EM} \right)^\mu_\nu \right] \delta X^\nu d^4x - \int \sqrt{-g}\, \left( T_{EM} \right)^\mu_{\nu,\mu} \delta X^\nu d^4x \\ &= \int \left[ -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\mu \right]{\delta X^\mu}_{,\gamma}d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\psi\phi}}{g^{\psi\phi}}_{,\mu} \delta X^\mu +\frac{1}{2}\sqrt{-g}\, g_{\psi\phi} {g^{\psi\phi}}_{ ,\mu} \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right] d^4x - \int \sqrt{-g}\, \left( T_{EM} \right)^\mu_{\nu,\mu} \delta X^\nu d^4x \\ &= \int \left[ -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\mu \right]{\delta X^\mu}_{,\gamma}d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\psi\phi}} \delta X^\mu +\frac{1}{2}\sqrt{-g}\, g_{\psi\phi} \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right] {g^{\psi\phi}}_{ ,\mu} d^4x - \int \sqrt{-g}\, \left( T_{EM} \right)^\mu_{\nu,\mu} \delta X^\nu d^4x \\ &= \int \left[ -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\mu \right]{\delta X^\mu}_{,\gamma}d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\psi\phi}} \delta^\mu_\nu +\frac{1}{2}\sqrt{-g}\, g_{\psi\phi} \left( T_{EM} \right)^\mu_\nu \right] \delta X^\nu {g^{\psi\phi}}_{ ,\mu} d^4x - \int \sqrt{-g}\, \left( T_{EM} \right)^\mu_{\nu,\mu} \delta X^\nu d^4x \\ \end{align*}\] \end{adjustwidth} \[\begin{align*} \Delta S &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\delta g^{\mu\nu} d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} F_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{EM} \right) \delta x^\gamma \right]d^4x \\ &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu}d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( t_{EM} \right)^\mu_\gamma \delta X^\gamma \right]d^4x \\ &= \int -2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} {\delta X^\nu}_{;\gamma}d^4x - \int \partial_\mu \left[ \sqrt{-g}\, \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right]d^4x \\ &= \int -2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} {\delta X^\nu}_{;\gamma}d^4x - \int \sqrt{-g}\,\nabla_\mu \left[ \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right]d^4x \\ &= \int -2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} {\delta X^\nu}_{;\gamma}d^4x - \int \sqrt{-g}\, \left[\nabla_\mu \left( T_{EM} \right)^\mu_\nu \right] \delta X^\nu d^4x - \int \sqrt{-g}\,\left( T_{EM} \right)^\mu_\nu \left(\nabla_\mu {\delta X^\nu}\right) d^4x \\ &= \int \left[-2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma}\right] {\delta X^\nu}_{;\gamma}d^4x - \int \sqrt{-g}\, \left[\nabla_\mu \left( T_{EM} \right)^\mu_\nu \right] \delta X^\nu d^4x - \int \left[ \sqrt{-g}\, \left( T_{EM} \right)^\gamma_\nu \right] {\delta X^\nu}_{;\gamma} d^4x \\ &= \int \left[-2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\nu \right] {\delta X^\nu}_{;\gamma}d^4x - \int \sqrt{-g}\, \left[\nabla_\mu \left( T_{EM} \right)^\mu_\nu \right] \delta X^\nu d^4x \end{align*}\] \[\begin{align*} -2\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}g^{\mu\gamma} &=\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\nu\\ \to \frac{1}{\sqrt{-g}}\frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}&=-\frac{1}{2} g_{\mu\gamma} \left( T_{EM} \right)^\gamma_\nu\\ \end{align*}\] \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align*} \Delta S &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\delta g^{\mu\nu} d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} F_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{EM} \right) \delta x^\gamma \right]d^4x \\ &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu}d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( t_{EM} \right)^\mu_\gamma \delta X^\gamma \right]d^4x \\ &= \int \left[ \frac{\partial \mathscr{L}_{EM}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma -2\frac{\partial \mathscr{L}_{EM}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} \right]d^4x - \int \left( t_{EM} \right)^\mu_{\gamma,\mu} \delta X^\gamma d^4x - \int \left( t_{EM} \right)^\mu_\gamma {\delta X^\gamma}_{,\mu} d^4x \\ &= \int \left[ \cancel{ \frac{\partial \mathscr{L}_{EM}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma } -2\frac{\partial \mathscr{L}_{EM}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} \right]d^4x - \int \cancel{\frac{\partial \mathscr{L}_{EM}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma } d^4x - \int \left( t_{EM} \right)^\gamma_\mu {\delta X^\mu}_{,\gamma} d^4x \\ &= \int \left[ -2\frac{\partial \mathscr{L}_{EM}}{\partial g^{\mu\nu}}g^{\gamma\nu} + \left( t_{EM} \right)^\gamma_\mu \right]{\delta X^\mu}_{,\gamma} d^4x \end{align*}\] \end{adjustwidth} \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align*} \Delta S &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\delta g^{\mu\nu} d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} F_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{EM} \right) \delta x^\gamma \right]d^4x \\ &= \int \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\nu}}\left(\hat{\mathcal{L}}_{\delta X}\, g\right)^{\mu\nu}d^4x + \int \left[\text{EoM of }\delta A_\nu \right]\delta A_\nu d^4x - \int \partial_\mu \left[ \left( t_{EM} \right)^\mu_\gamma \delta X^\gamma \right]d^4x \\ &= \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} \right]d^4x - \int \partial_\mu \left[ \sqrt{-g}\, \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right]d^4x \\ &= \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} \right]d^4x - \int \left\{ \left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu +\left[ \sqrt{-g}\, \left( T_{EM} \right)^\mu_\nu {\delta X^\nu}_{,\mu} \right] \right\}d^4x \\ &= \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} -\left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu -\sqrt{-g}\, \left( T_{EM} \right)^\mu_\nu {\delta X^\nu}_{,\mu} \right]d^4x \\ &= \int \left[ -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} {\delta X^\mu}_{,\gamma} -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\mu {\delta X^\mu}_{,\gamma} +\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma -\left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu \right]d^4x \\ &= \int \left[ -2\frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}g^{\gamma\nu} -\sqrt{-g}\, \left( T_{EM} \right)^\gamma_\mu \right] {\delta X^\mu}_{,\gamma} d^4x + \int \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma d^4x - \int \left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu d^4x \\ &= \int -2\left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]g^{\gamma\nu} {\delta X^\mu}_{,\gamma} d^4x + \int \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}}{g^{\mu\nu}}_{,\gamma} \delta X^\gamma d^4x - \int \left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu d^4x \\ &= \int -2\left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]g^{\gamma\nu} {\delta X^\mu}_{,\gamma} d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} \textcolor{red}{+\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} -\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu}} \right]{g^{\mu\nu}}_{,\gamma} \delta X^\gamma d^4x\\ &\qquad\qquad - \int \left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu d^4x \\ &= \int -2\left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]g^{\gamma\nu} {\delta X^\mu}_{,\gamma} d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]{g^{\mu\nu}}_{,\gamma} \delta X^\gamma d^4x\\ &\qquad\qquad - \int \frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} {g^{\mu\nu}}_{,\gamma} \delta X^\gamma d^4x - \int \left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu d^4x \\ &= \int -2\left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]g^{\gamma\nu} {\delta X^\mu}_{,\gamma} d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]{g^{\mu\nu}}_{,\gamma} \delta X^\gamma d^4x\\ &\qquad\qquad - \int \frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \left( -g^{\xi\nu}\Gamma^\mu_{\xi\gamma} -g^{\mu\xi}\Gamma^\nu_{\xi\gamma} +\frac{2}{\mathbb{D}-1}g^{\mu\nu}T_\gamma \right) \delta X^\gamma d^4x - \int \left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu d^4x \\ &= \int -2\left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]g^{\gamma\nu} {\delta X^\mu}_{,\gamma} d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]{g^{\mu\nu}}_{,\gamma} \delta X^\gamma d^4x\\ &\qquad\qquad - \int \frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu \left( -g_{\sigma\nu}g^{\xi\nu}\Gamma^\mu_{\xi\gamma} -g_{\sigma\nu}g^{\mu\xi}\Gamma^\nu_{\xi\gamma} +\frac{2}{\mathbb{D}-1}g_{\sigma\nu}g^{\mu\nu}T_\gamma \right) \delta X^\gamma d^4x - \int \left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu d^4x \\ &= \int -2\left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]g^{\gamma\nu} {\delta X^\mu}_{,\gamma} d^4x + \int \left[ \frac{\partial\mathscr{L}}{\partial g^{\mu\nu}} +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu g_{\sigma\nu} \right]{g^{\mu\nu}}_{,\gamma} \delta X^\gamma d^4x\\ &\qquad\qquad - \int \frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\sigma_\mu \left( -\Gamma^\mu_{\sigma\gamma} -g_{\sigma\nu}g^{\mu\xi}\Gamma^\nu_{\xi\gamma} +\frac{2}{\mathbb{D}-1}\delta_{\sigma}^{\mu}T_\gamma \right) \delta X^\gamma d^4x - \int \left[ \sqrt{-g} \left( T_{EM} \right)^\mu_\nu \right]_{,\mu} \delta X^\nu d^4x \\ \end{align*}\] \end{adjustwidth} \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align*} &\left[ \frac{\partial\mathscr{L}}{\partial g^{\psi\phi}} \delta X^\mu +\frac{1}{2}\sqrt{-g}\, g_{\psi\phi} \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right] {g^{\psi\phi}}_{ ,\mu} \\ &=\left[ \frac{\partial\mathscr{L}}{\partial g^{\psi\phi}} \delta X^\mu +\frac{1}{2}\sqrt{-g}\, g_{\psi\phi} \left( T_{EM} \right)^\mu_\nu \delta X^\nu \right] \left( -g^{\sigma\phi}\Gamma^\psi_{\sigma\mu} -g^{\psi\sigma}\Gamma^\phi_{\sigma\mu} +\frac{2}{\mathbb{D}-1}g^{\psi\phi}T_\mu \right)\\ &= \frac{\partial\mathscr{L}}{\partial g^{\psi\phi}} \delta X^\mu\left( -g^{\sigma\phi}\Gamma^\psi_{\sigma\mu} -g^{\psi\sigma}\Gamma^\phi_{\sigma\mu} +\frac{2}{\mathbb{D}-1}g^{\psi\phi}T_\mu \right) +\frac{1}{2}\sqrt{-g}\, g_{\psi\phi} \left( T_{EM} \right)^\mu_\nu \delta X^\nu\left( -g^{\sigma\phi}\Gamma^\psi_{\sigma\mu} -g^{\psi\sigma}\Gamma^\phi_{\sigma\mu} +\frac{2}{\mathbb{D}-1}g^{\psi\phi}T_\mu \right) \\ &= \delta X^\mu\left( -2\frac{\partial\mathscr{L}}{\partial g^{\psi\phi}}g^{\sigma\phi}\Gamma^\psi_{\sigma\mu} +\frac{2}{\mathbb{D}-1}\frac{\partial\mathscr{L}}{\partial g^{\psi\phi}}g^{\psi\phi}T_\mu \right) +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\mu_\nu \delta X^\nu\left( - g_{\psi\phi}g^{\sigma\phi}\Gamma^\psi_{\sigma\mu} - g_{\psi\phi}g^{\psi\sigma}\Gamma^\phi_{\sigma\mu} +\frac{2}{\mathbb{D}-1}g_{\psi\phi}g^{\sigma\phi}T_\mu \right) \\ &= \delta X^\nu\left( -2\frac{\partial\mathscr{L}}{\partial g^{\psi\phi}}g^{\sigma\phi}\Gamma^\psi_{\sigma\nu} +\frac{2}{\mathbb{D}-1}\frac{\partial\mathscr{L}}{\partial g^{\psi\phi}}g^{\psi\phi}T_\nu \right) +\frac{1}{2}\sqrt{-g}\, \left( T_{EM} \right)^\mu_\nu \delta X^\nu\left( -2 g_{\psi\phi}g^{\sigma\phi}\Gamma^\psi_{\sigma\mu} +\frac{2}{\mathbb{D}-1} g_{\psi\phi}g^{\psi\phi} T_\mu \right) \\ \end{align*}\] \end{adjustwidth} \[\begin{align*} &\delta \mathscr{L}=\frac{\delta \mathscr{L}}{\delta g_{\mu\nu}}\hat{\mathcal{L}}_X g_{\mu\nu}+\cancelto{0}{\left[\text{EoM of }\delta A_\nu \right] }{\mathcal{L}}_X A_\nu -\partial_\mu\left( t^\mu_\gamma \delta X^\gamma \right)=0\\ &\to \delta \mathscr{L}=\frac{\delta \mathscr{L}}{\delta g_{\mu\nu}}\left(g_{\gamma\nu} {\delta X^\gamma}_{;\mu}+g_{\mu\gamma} {\delta X^\gamma}_{;\nu}\right) -\partial_\mu\left( t^\mu_\gamma \delta X^\gamma \right)=0 \\ &\to \delta \mathscr{L}=2\frac{\delta \mathscr{L}}{\delta g_{\mu\nu}}g_{\gamma\nu} {\delta X^\gamma}_{;\mu} -\partial_\mu\left( t^\mu_\gamma \delta X^\gamma \right)=0 \\ &\to \delta \mathscr{L}=\partial_\mu\left( 2\frac{\delta \mathscr{L}}{\delta g_{\mu\nu}}g_{\gamma\nu} {\delta X^\gamma}\right)_{,\mu} -\partial_\mu\left( t^\mu_\gamma \delta X^\gamma \right)=0 \\ &\to 2\frac{\delta \mathscr{L}}{\delta g_{\mu\nu}}g_{\gamma\nu}=t^\mu_\gamma \end{align*}\] \section{Relation between EoM and Conservation law} \subsection{EM conservation law and EoM} \[\begin{align*} \left(t_{EM}\right)^\mu_{\gamma,\mu} &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} F_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{EM} \right]_{,\mu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} F_{\gamma\nu,\mu} -\left( \mathscr{L}_{EM} \right)_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} -\frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta}F_{\gamma\nu,\mu} -\left( -\frac{1}{16\pi c} g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}F_{\mu\nu}\sqrt{-g} \right)_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} -\frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta}F_{\gamma\nu,\mu} + \frac{1}{8\pi c} g^{\mu\alpha}g^{\nu\beta}\sqrt{-g}F_{\alpha\beta}F_{\mu\nu,\gamma} + \frac{1}{16\pi c}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}F_{\alpha\beta}F_{\mu\nu} \end{align*}\] Using the Bianchi identity of \(F_{\mu\nu}\), \[\begin{align*} &F_{\mu\nu,\gamma}+F_{\nu\gamma,\mu}+F_{\gamma\mu,\nu}=0\\ \to &F_{\mu\nu,\gamma}=-F_{\nu\gamma,\mu}-F_{\gamma\mu,\nu} \end{align*}\] The term \[\begin{align*} g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu,\gamma} &=-g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\nu\gamma,\mu}-g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\gamma\mu,\nu}\\ &=-g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\nu\gamma,\mu}-g^{\textcolor{blue}{\nu}\textcolor{red}{\beta}}g^{\textcolor{blue}{\mu}\textcolor{red}{\alpha}}\sqrt{-g} F_{\textcolor{red}{\beta\alpha}}F_{\gamma\textcolor{blue}{\nu,\mu}}\\ &=\textcolor{blue}{+}g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\textcolor{blue}{\gamma\nu},\mu}\textcolor{red}{+}g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} F_{\textcolor{red}{\alpha\beta}}F_{\gamma\nu,\mu}\\ &=2g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\gamma\nu,\mu} \end{align*}\] \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align*} &\left(t_{EM}\right)^\mu_{\gamma,\mu} \\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} -\frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta}F_{\gamma\nu,\mu} + \frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}\sqrt{-g}F_{\alpha\beta}F_{\gamma\nu,\mu} + \frac{1}{16\pi c}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2{g^{\mu\alpha}}_{,\gamma}g^{\nu\beta}\sqrt{-g}+ g^{\mu\alpha}g^{\nu\beta}{\sqrt{-g}}_{,\gamma} \right)F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2{g^{\mu\alpha}}_{,\gamma}g^{\nu\beta}\sqrt{-g}-\frac{1}{2}g^{\mu\alpha}g^{\nu\beta}{\sqrt{-g}}g_{\phi\psi}{g^{\phi\psi}}_{,\gamma} \right)F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2{g^{\mu\alpha}}_{,\gamma}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu}-\frac{1}{2}g^{\mu\alpha}g^{\nu\beta}{\sqrt{-g}}g_{\phi\psi}{g^{\phi\psi}}_{,\gamma}F_{\alpha\beta}F_{\mu\nu} \right)\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2{g^{\mu\alpha}}_{,\gamma}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu}-\frac{1}{2}g^{\phi\psi}g^{\nu\beta}{\sqrt{-g}}g_{\mu\alpha}{g^{\mu\alpha}}_{,\gamma}F_{\psi\beta}F_{\phi\nu} \right)\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu}-\frac{1}{2}g^{\phi\psi}g^{\nu\beta}{\sqrt{-g}}g_{\mu\alpha}F_{\psi\beta}F_{\phi\nu} \right){g^{\mu\alpha}}_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu}-\frac{1}{2}g^{\phi\psi}g^{\nu\beta}{\sqrt{-g}}g_{\mu\alpha}F_{\psi\beta}F_{\phi\nu} \right)\delta^\mu_\sigma{g^{\sigma\alpha}}_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu}-\frac{1}{2}g^{\phi\psi}g^{\nu\beta}{\sqrt{-g}}g_{\mu\alpha}F_{\psi\beta}F_{\phi\nu} \right)g^{\mu\xi}g_{\xi\sigma}{g^{\sigma\alpha}}_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2g^{\mu\xi}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu}-\frac{1}{2}\delta^\xi_\alpha g^{\phi\psi}g^{\nu\beta}{\sqrt{-g}}F_{\psi\beta}F_{\phi\nu} \right)g_{\xi\sigma}{g^{\sigma\alpha}}_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} - \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha g_{\xi\sigma}{g^{\sigma\alpha}}_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} - \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha g_{\xi\sigma} \left( -g^{\mu\alpha}\Gamma^\sigma_{\mu\gamma} -g^{\sigma\mu}\Gamma^\alpha_{\mu\gamma} +\frac{2}{\mathbb{D}-1}g^{\sigma\alpha}T_{\gamma}\right)\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} - \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( -g_{\xi\sigma}g^{\mu\alpha}\Gamma^\sigma_{\mu\gamma} -g_{\xi\sigma}g^{\sigma\mu}\Gamma^\alpha_{\mu\gamma} +\frac{2}{\mathbb{D}-1}g_{\xi\sigma}g^{\sigma\alpha}T_{\gamma}\right)\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} - \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( -g_{\xi\sigma}g^{\mu\alpha}\Gamma^\sigma_{\mu\gamma} -\delta_\xi^\mu \Gamma^\alpha_{\mu\gamma} +\frac{2}{\mathbb{D}-1}\delta_\xi^\alpha T_{\gamma}\right)\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)\Gamma^\sigma_{\mu\gamma} -\frac{1}{\mathbb{D}-1} \left({t_{EM}}\right)^\alpha _\alpha T_{\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)\Gamma^\sigma_{\gamma\mu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)T^\sigma_{\mu\gamma} -\frac{1}{\mathbb{D}-1} \left({t_{EM}}\right)^\alpha _\alpha T_{\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)\Gamma^\sigma_{\gamma\mu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right) \frac{1}{\mathbb{D}-1} \left( \delta^\sigma_{\mu} T_\gamma -\delta^\sigma_{\gamma} T_\mu \right) -\frac{1}{\mathbb{D}-1} \left({t_{EM}}\right)^\alpha _\alpha T_{\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)\Gamma^\sigma_{\gamma\mu} + \frac{1}{2(\mathbb{D}-1)} \left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha}\delta^\sigma_{\mu} T_\gamma -g_{\xi\sigma}g^{\mu\alpha}\delta^\sigma_{\gamma} T_\mu +\delta_\xi^\mu \delta_\sigma^\alpha \delta^\sigma_{\mu} T_\gamma -\delta_\xi^\mu \delta_\sigma^\alpha \delta^\sigma_{\gamma} T_\mu \right)\\&\qquad\qquad\qquad-\frac{1}{\mathbb{D}-1} \left({t_{EM}}\right)^\alpha_\alpha T_{\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)\Gamma^\sigma_{\gamma\mu} + \frac{1}{2(\mathbb{D}-1)} \left({t_{EM}}\right)^\xi_\alpha \left( \delta^\alpha_\xi T_\gamma -g_{\xi\gamma}g^{\mu\alpha} T_\mu + \delta^\alpha_\xi T_\gamma - \delta^\alpha_{\gamma} T_\xi \right) -\frac{1}{\mathbb{D}-1} \left({t_{EM}}\right)^\alpha_\alpha T_{\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)\Gamma^\sigma_{\gamma\mu} + \frac{1}{2(\mathbb{D}-1)} \left( \cancel{\left({t_{EM}}\right)^\alpha_\alpha T_\gamma} -\left({t_{EM}}\right)^\xi_\alpha g_{\xi\gamma}g^{\mu\alpha} T_\mu + \cancel{\left({t_{EM}}\right)^\alpha_\alpha T_\gamma }- \left({t_{EM}}\right)^\xi_\alpha\delta^\alpha_{\gamma} T_\xi \right)\\&\qquad\qquad\qquad -\cancel{\frac{1}{\mathbb{D}-1} \left({t_{EM}}\right)^\alpha_\alpha T_{\gamma}} \end{align*}\] \end{adjustwidth} \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align*} &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)\Gamma^\sigma_{\gamma\mu} + \frac{1}{2(\mathbb{D}-1)} \left( -\left({t_{EM}}\right)^\xi_\alpha g_{\xi\gamma}g^{\mu\alpha} T_\mu - \left({t_{EM}}\right)^\xi_\alpha \delta^\alpha_{\gamma} \delta^\mu_\xi T_\mu \right)\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)\Gamma^\sigma_{\gamma\mu} - \frac{1}{2(\mathbb{D}-1)} \left({t_{EM}}\right)^\xi_\alpha T_\mu \left( g_{\xi\gamma}g^{\mu\alpha} + \delta^\alpha_{\gamma} \delta^\mu_\xi \right) \\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right)\Gamma^\sigma_{\gamma\mu} - \frac{1}{2(\mathbb{D}-1)} \left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} + \delta^\alpha_{\sigma} \delta^\mu_\xi \right)\delta^\sigma_{\gamma} T_\mu\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{2}\left({t_{EM}}\right)^\xi_\alpha \left( g_{\xi\sigma}g^{\mu\alpha} +\delta_\xi^\mu \delta_\sigma^\alpha \right) \left( \Gamma^\sigma_{\gamma\mu} - \frac{1}{\mathbb{D}-1}\delta^\sigma_{\gamma} T_\mu \right) \end{align*}\] \end{adjustwidth} \begin{adjustwidth}{-2.5cm}{-1cm} \[\begin{align*} &\left(t_{EM}\right)^\mu_{\gamma,\mu} \\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} -\frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta}F_{\gamma\nu,\mu} + \frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}\sqrt{-g}F_{\alpha\beta}F_{\gamma\nu,\mu} + \frac{1}{16\pi c}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2{g^{\mu\alpha}}_{,\gamma}g^{\nu\beta}\sqrt{-g}+ g^{\mu\alpha}g^{\nu\beta}{\sqrt{-g}}_{,\gamma} \right)F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2{g^{\mu\alpha}}_{,\gamma}g^{\nu\beta}\sqrt{-g}-\frac{1}{2}g^{\mu\alpha}g^{\nu\beta}{\sqrt{-g}}g_{\phi\psi}{g^{\phi\psi}}_{,\gamma} \right)F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2{g^{\mu\alpha}}_{,\gamma}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu}-\frac{1}{2}g^{\mu\alpha}g^{\nu\beta}{\sqrt{-g}}g_{\phi\psi}{g^{\phi\psi}}_{,\gamma}F_{\alpha\beta}F_{\mu\nu} \right)\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2{g^{\mu\alpha}}_{,\gamma}g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu}-\frac{1}{2}g^{\phi\psi}g^{\nu\beta}{\sqrt{-g}}g_{\mu\alpha}{g^{\mu\alpha}}_{,\gamma}F_{\psi\beta}F_{\phi\nu} \right)\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( 2g^{\nu\beta}\sqrt{-g} F_{\alpha\beta}F_{\mu\nu}-\frac{1}{2}g^{\phi\psi}g^{\nu\beta}{\sqrt{-g}}g_{\mu\alpha}F_{\psi\beta}F_{\phi\nu} \right){g^{\mu\alpha}}_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{\delta \mathscr{L}_{EM}}{\delta g^{\mu\alpha}}{g^{\mu\alpha}}_{,\gamma}\\ \end{align*}\] \end{adjustwidth} \[\begin{align*} \to \left(t_{EM}\right)^\mu_{\gamma;\mu}&= \left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\frac{1}{\mathbb{D}-1}\left({t_{EM}}\right)^\xi_\xi T_{\gamma} \end{align*}\] \[\begin{align*} &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( -2 g^{\eta\alpha}\Gamma^\mu_{\eta\gamma}g^{\nu\beta}\sqrt{-g} -2 g^{\mu\eta}\Gamma^\alpha_{\eta\gamma}g^{\nu\beta}\sqrt{-g}+ g^{\mu\alpha}g^{\nu\beta}\sqrt{-g}\Gamma_{\eta\gamma}^{\eta} \right)F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} + \frac{1}{16\pi c}\left( -2 g^{\eta\alpha}g^{\nu\beta}\Gamma^\mu_{\eta\gamma} -2 g^{\mu\eta}g^{\nu\beta}\Gamma^\alpha_{\eta\gamma}+ g^{\mu\alpha}g^{\nu\beta}\Gamma_{\eta\gamma}^{\eta} \right)\sqrt{-g}F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\frac{1}{16\pi c}\left( -2 g^{\eta\alpha}g^{\nu\beta}\Gamma^\mu_{\eta\gamma} -2 g^{\mu\eta}g^{\nu\beta}\Gamma^\alpha_{\eta\gamma}+ g^{\mu\alpha}g^{\nu\beta}\Gamma_{\eta\gamma}^{\eta} \right)\sqrt{-g}F_{\alpha\beta}F_{\mu\nu}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\frac{1}{16\pi c}\left( -2 g^{\eta\alpha}g^{\nu\beta}\Gamma^\mu_{\eta\gamma}F_{\alpha\beta}F_{\mu\nu} -2 g^{\mu\eta}g^{\nu\beta}\Gamma^\alpha_{\eta\gamma}F_{\alpha\beta}F_{\mu\nu}+ g^{\mu\alpha}g^{\nu\beta}\Gamma_{\eta\gamma}^{\eta} F_{\alpha\beta}F_{\mu\nu} \right)\sqrt{-g}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\frac{1}{16\pi c}\left( -2 g^{\eta\alpha}g^{\nu\beta}\Gamma^\mu_{\eta\gamma}F_{\alpha\beta}F_{\mu\nu} -2 g^{\textcolor{red}{\alpha}\eta}g^{\textcolor{blue}{\beta\nu}}\Gamma^{\textcolor{red}{\mu}}_{\eta\gamma}F_{\textcolor{red}{\mu}\textcolor{blue}{\nu}}F_{\textcolor{red}{\alpha}\textcolor{blue}{\beta}}+ g^{\mu\alpha}g^{\nu\beta}\Gamma_{\eta\gamma}^{\eta} F_{\alpha\beta}F_{\mu\nu} \right)\sqrt{-g}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\frac{1}{16\pi c}\left( -4 g^{\eta\alpha}g^{\nu\beta}\Gamma^\mu_{\eta\gamma}F_{\alpha\beta}F_{\mu\nu} + g^{\mu\alpha}g^{\nu\beta}\Gamma_{\eta\gamma}^{\eta} F_{\alpha\beta}F_{\mu\nu} \right)\sqrt{-g}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\frac{1}{16\pi c}\left( -4 g^{\eta\alpha}g^{\nu\beta}F_{\alpha\beta}F_{\mu\nu}\Gamma^\mu_{\eta\gamma} +\delta^\eta_{\textcolor{red}{\mu}} g^{\mu\alpha}g^{\nu\beta} F_{\alpha\beta}F_{\mu\nu} \Gamma_{\eta\gamma}^{\textcolor{red}{\mu}} \right)\sqrt{-g}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\frac{1}{16\pi c}\left( -4 g^{\eta\alpha}g^{\nu\beta}F_{\alpha\beta}F_{\mu\nu} +\delta^\eta_\mu g^{\mu\alpha}g^{\nu\beta} F_{\alpha\beta}F_{\mu\nu} \right)\Gamma^\mu_{\eta\gamma}\sqrt{-g}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\left[ -\frac{1}{4\pi c} g^{\eta\alpha}g^{\nu\beta}F_{\alpha\beta}\sqrt{-g}F_{\mu\nu} -\delta^\eta_\mu \left(-\frac{1}{16\pi c} g^{\mu\alpha}g^{\nu\beta} F_{\alpha\beta}F_{\mu\nu}\sqrt{-g} \right) \right]\Gamma^\mu_{\eta\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\eta A_\nu)}F_{\mu\nu} -\delta^\eta_\mu \mathscr{L}_{EM} \right]\Gamma^\mu_{\eta\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \right]_{,\mu}F_{\gamma\nu} +\left( t_{EM}\right)^\eta_\mu \Gamma^\mu_{\eta\gamma} \end{align*}\] Fact \(\text{Torsion-free Metric compatible connection:} \sqrt{-g}\ _{;\gamma} =0\) \[\begin{align*} &T^\mu_{\gamma;\nu}=T^\mu_{\gamma,\nu}+T^\eta_\gamma \Gamma^\mu_{\eta\nu}-T^\mu_\eta \Gamma^\eta_{\gamma\nu}\\ &T^\mu_{\gamma;\mu}=T^\mu_{\gamma,\mu}+T^\eta_\gamma \Gamma^\mu_{\eta\mu}-T^\mu_\eta \Gamma^\eta_{\gamma\mu}\\ &T^\mu_{\gamma;\mu} \sqrt{-g}=T^\mu_{\gamma,\mu} \sqrt{-g}+T^\eta_\gamma \sqrt{-g}\Gamma^\mu_{\eta\mu} -T^\mu_\eta \sqrt{-g} \ \Gamma^\eta_{\gamma\mu}\\ &T^\mu_{\gamma;\mu} \sqrt{-g}=T^\mu_{\gamma,\mu} \sqrt{-g}+T^\eta_\gamma \sqrt{-g}\Gamma^\mu_{\mu\eta} +T^\eta_\gamma \sqrt{-g}T^\mu_{\eta\mu}-T^\mu_\eta \sqrt{-g} \ \Gamma^\eta_{\gamma\mu}\\ &\left(T^\mu_{\gamma} \sqrt{-g}\right)_{;\mu}=T^\mu_{\gamma,\mu} \sqrt{-g}+T^\mu_\gamma \sqrt{-g}\ _{,\mu} -T^\eta_\gamma \sqrt{-g}T^\mu_{\mu\eta}-T^\mu_\eta \sqrt{-g} \ \Gamma^\eta_{\gamma\mu}\\ &\left(T^\mu_{\gamma} \sqrt{-g}\right)_{;\mu}=\left(T^\mu_\gamma \sqrt{-g}\right)_{,\mu} -\left(T^\eta_\gamma \sqrt{-g}\right)T_{\eta}- \left(T^\mu_\eta \sqrt{-g}\right)\Gamma^\eta_{\gamma\mu}\\ &t^\mu_{\gamma;\mu}=t^\mu_{\gamma,\mu} - t^\mu_\eta \Gamma^\eta_{\gamma\mu}- t^\mu_\gamma T_{\mu} \end{align*}\] Fact: for any antisymmetric 2-tensor \(F^{\mu\gamma}=-F^{\gamma\mu}\): \[\begin{align*} {F^{\mu\gamma}}_{;\nu}&={F^{\mu\gamma}}_{,\nu}+F^{\eta\gamma} \Gamma^\mu_{\eta\nu}+F^{\mu\eta} \Gamma^\gamma_{\eta\nu}\\ {F^{\mu\gamma}}_{;\mu}&={F^{\mu\gamma}}_{,\mu}+\underbrace{F^{\eta\gamma} \Gamma^\mu_{\eta\mu}}_{=F^{\eta\gamma}\Gamma^\mu_{\mu\eta}+F^{\eta\gamma}T^\mu_{\eta\mu}}+\underbrace{F^{\mu\eta} \Gamma^\gamma_{\eta\mu}}_{=\frac{1}{2}F^{\mu\eta} T^\gamma_{\eta\mu}}\\ &={F^{\mu\gamma}}_{,\mu}+F^{\eta\gamma}\Gamma^\mu_{\mu\eta}+F^{\eta\gamma}T^\mu_{\eta\mu}+\frac{1}{2\left(\mathbb{D}-1\right)}F^{\mu\eta} \left(\delta^\gamma_\eta T_\mu-\delta^\gamma_\mu T_\eta \right)\\ &={F^{\mu\gamma}}_{,\mu}+F^{\eta\gamma} \Gamma^\mu_{\mu\eta} -F^{\eta\gamma}T^\mu_{\mu\eta}-\frac{1}{\left(\mathbb{D}-1\right)}F^{\mu\eta} \delta^\gamma_\eta T_\mu\\ &={F^{\mu\gamma}}_{,\mu}+F^{\eta\gamma} \Gamma^\mu_{\mu\eta}-F^{\eta\gamma}T_{\eta}-\frac{1}{\left(\mathbb{D}-1\right)}F^{\mu\gamma} T_\mu\\ &={F^{\mu\gamma}}_{,\mu}+F^{\eta\gamma} \Gamma^\mu_{\mu\eta}-\frac{\mathbb{D}}{\left(\mathbb{D}-1\right)}F^{\mu\gamma} T_\mu\\ {F^{\mu\gamma}}_{;\mu}\sqrt{-g}&={F^{\mu\gamma}}_{,\mu}\sqrt{-g}+F^{\eta\gamma} \sqrt{-g}\Gamma^\mu_{\eta\mu}\\ &={F^{\mu\gamma}}_{,\mu}\sqrt{-g}+F^{\eta\gamma} \sqrt{-g} \ _{,\mu}\\ {\left(F^{\mu\gamma}\sqrt{-g} \right)}_{;\mu}&={\left(F^{\mu\gamma}\sqrt{-g} \right)}_{,\mu}\\ \end{align*}\] \subsection{YM conservation law and EoM} \begin{adjustwidth}{-1.5cm}{-1cm} \[\begin{align*} \left(t_{YM}\right)^\mu_{\gamma,\mu} &=\left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} G^a_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{YM} \right]_{,\mu}\\ &=\left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} \right]_{,\mu}G^a_{\gamma\nu} + \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} G^a_{\gamma\nu,\mu} -\left( \mathscr{L}_{YM} \right)_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} \right]_{,\mu}G^a_{\gamma\nu} -\frac{1}{4\pi c}K_{ad} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} G^d_{\alpha\beta}G^a_{\gamma\nu,\mu} -\left( -\frac{1}{16\pi c} K_{ad}g^{\mu\alpha}g^{\nu\beta}G^a_{\mu\nu}G^d_{\alpha\beta}\sqrt{-g} \right)_{,\gamma}\\ &=\left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} \right]_{,\mu}G^a_{\gamma\nu} -\frac{1}{4\pi c}K_{ad} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} G^d_{\alpha\beta}G^a_{\gamma\nu,\mu} + \frac{1}{8\pi c}K_{ad} g^{\mu\alpha}g^{\nu\beta}\sqrt{-g}G^d_{\alpha\beta}G^a_{\mu\nu,\gamma} + \frac{1}{16\pi c}K_{ad}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}G^a_{\mu\nu}G^d_{\alpha\beta}\\ &=\left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} \right]_{,\mu}G^a_{\gamma\nu} -\frac{1}{4\pi c}K_{da}g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} G^d_{{\alpha\beta}} \lambda f^a_{cb}G^c_{{\gamma\nu}}B^b_{\mu} + \frac{1}{16\pi c}K_{ad}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}G^a_{\mu\nu}G^d_{\alpha\beta}\\ &=\left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} \right]_{,\mu}G^a_{\gamma\nu} -\left(\textcolor{orange}{-}\frac{1}{4\pi c}K_{d\textcolor{red}{c}}g^{\nu\beta}g^{\mu\alpha}\sqrt{-g} G^d_{{\textcolor{orange}{\beta\alpha}}} \lambda f^{\textcolor{red}{c}}_{\textcolor{red}{a}b}B^b_{\mu} \right) G^{\textcolor{red}{a}}_{{\gamma\nu}} + \frac{1}{16\pi c}K_{ad}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}G^a_{\mu\nu}G^d_{\alpha\beta}\\ &=\left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} \right]_{,\mu}G^a_{\gamma\nu} -\frac{\partial \mathscr{L}_{YM}}{\partial( B^a_\nu)} G^{\textcolor{red}{a}}_{{\gamma\nu}} + \frac{1}{16\pi c}K_{ad}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}G^a_{\mu\nu}G^d_{\alpha\beta}\\ &=\left\{ \left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} \right]_{,\mu} -\frac{\partial \mathscr{L}_{YM}}{\partial( B^a_\nu)} \right\}G^a_{\gamma\nu} + \frac{1}{16\pi c}K_{ad}\left( g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} \right)_{,\gamma}G^a_{\mu\nu}G^d_{\alpha\beta}\\ &=\left\{ \left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} \right]_{,\mu} -\frac{\partial \mathscr{L}_{YM}}{\partial( B^a_\nu)} \right\}G^a_{\gamma\nu} + \left( t_{YM}\right)^\eta_\mu \Gamma^\mu_{\eta\gamma} \end{align*}\] \end{adjustwidth} \subsection{Gauge Bianchi Identity} \[\begin{align*} &D_{[\gamma} G^a_{\mu\nu]} =0\\ & D_{\gamma} G^a_{\mu\nu} +D_{\mu} G^a_{\nu\gamma} +D_{\nu} G^a_{\gamma\mu} =0\\ & \left(\partial_{\gamma}G^a_{\mu\nu}+\lambda f^a_{bc}B^b_\gamma G^c_{\mu\nu} \right) +\left(\partial_{\mu} G^a_{\nu\gamma} +\lambda f^a_{bc}B^b_{\mu} G^c_{\nu\gamma}\right) +\left(\partial_{\nu} G^a_{\gamma\mu}+\lambda f^a_{bc}B^b_{\nu} G^c_{\gamma\mu}\right) =0\\ & \left(G^a_{\mu\nu,\gamma}-\lambda f^a_{cb}G^c_{\mu\nu} B^b_\gamma \right) +\left( G^a_{\nu\gamma,\mu}-\lambda f^a_{cb}G^c_{\nu\gamma}B^b_{\mu} \right) +\left( G^a_{\gamma\mu,\nu}-\lambda f^a_{cb} G^c_{\gamma\mu}B^b_{\nu}\right) =0\\ & G^a_{\mu\nu,\gamma} =\lambda f^a_{cb}G^c_{\mu\nu} B^b_\gamma -\left( G^a_{\nu\gamma,\mu}-\lambda f^a_{cb}G^c_{\nu\gamma}B^b_{\mu} \right) -\left( G^a_{\gamma\mu,\nu}-\lambda f^a_{cb} G^c_{\gamma\mu}B^b_{\nu}\right) \end{align*}\] \begin{adjustwidth}{-1.5cm}{-1cm} \[\begin{align*} K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta} G^a_{\mu\nu,\gamma} &=K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta}\lambda f^a_{cb}G^c_{\mu\nu} B^b_\gamma -K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta} \left( G^a_{\nu\gamma,\mu}-\lambda f^a_{cb}G^c_{\nu\gamma}B^b_{\mu} \right) -K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta} \left( G^a_{\gamma\mu,\nu}-\lambda f^a_{cb} G^c_{\gamma\mu}B^b_{\nu}\right)\\ &=K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta}\lambda f^a_{cb}G^c_{\mu\nu} B^b_\gamma -K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta} \left( G^a_{\nu\gamma,\mu}-\lambda f^a_{cb}G^c_{\nu\gamma}B^b_{\mu} \right) -K_{da}g^{\textcolor{blue}{\nu}\textcolor{red}{\beta}}g^{\textcolor{blue}{\mu}\textcolor{red}{\alpha}} G^d_{\textcolor{red}{\beta\alpha}} \left( G^a_{\gamma\textcolor{blue}{\nu,\mu}}-\lambda f^a_{cb} G^c_{\gamma\textcolor{blue}{\nu}}B^b_{\textcolor{blue}{\mu}}\right)\\ &=K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta}\lambda f^a_{cb}G^c_{\mu\nu} B^b_\gamma \textcolor{red}{+}2K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{\textcolor{red}{\alpha\beta}} \left( G^a_{\textcolor{red}{\gamma\nu},\mu}-\lambda f^a_{cb}G^c_{\textcolor{red}{\gamma\nu}}B^b_{\mu} \right) \\ &=\textcolor{red}{0} +2K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{{\alpha\beta}} \left( G^a_{{\gamma\nu},\mu}-\lambda f^a_{cb}G^c_{{\gamma\nu}}B^b_{\mu} \right) \\ \end{align*}\] \end{adjustwidth} \[\begin{align*} \frac{1}{8\pi c} K_{da}g^{\mu\alpha}g^{\nu\beta} \sqrt{-g}G^d_{\alpha\beta} G^a_{\mu\nu,\gamma} &= \frac{1}{4\pi c}K_{da}g^{\mu\alpha}g^{\nu\beta}\sqrt{-g} G^d_{{\alpha\beta}} \left( G^a_{{\gamma\nu},\mu}-\lambda f^a_{cb}G^c_{{\gamma\nu}}B^b_{\mu} \right) \end{align*}\] \[\begin{align*} K_{da}f^a_{bc}&=f^f_{de} f^e_{af} f^a_{bc}\\ &=-f^f_{de} f^e_{fa} f^a_{bc}\\ &=f^f_{de}\left( f^e_{ba} f^a_{cf}+f^e_{ca} f^a_{fb}\right)\\ &=f^f_{de} f^e_{ba} f^a_{cf}\textcolor{red}{-} f^f_{de}f^e_{ca} f^a_{\textcolor{red}{bf}} \end{align*}\] \[\begin{align*} K_{da}g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta}\lambda f^a_{cb}G^c_{\mu\nu} B^b_\gamma &= \lambda K_{da}f^a_{cb}g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta}G^c_{\mu\nu} B^b_\gamma\\ &=\lambda \left(f^f_{de} f^e_{ba} f^a_{cf}- f^f_{de}f^e_{ca} f^a_{bf} \right) g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta}G^c_{\mu\nu} B^b_\gamma\\ &=\lambda f^f_{de} f^e_{ba} f^a_{cf} g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta}G^c_{\mu\nu} B^b_\gamma -\lambda f^f_{{\textcolor{red}{c}}e}f^e_{{\textcolor{red}{d}}a} f^a_{bf} g^{\textcolor{orange}{\alpha\mu}}g^{\textcolor{blue}{\beta\nu}} G^{\textcolor{red}{c}}_{\textcolor{orange}{\mu}\textcolor{blue}{\nu}}G^{\textcolor{red}{d}}_{\textcolor{orange}{\alpha}\textcolor{blue}{\beta}}B^b_\gamma\\ &=\lambda f^f_{de} f^e_{ba} f^a_{cf} g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta}G^c_{\mu\nu} B^b_\gamma -\lambda f^e_{{{d}}a} f^a_{bf} f^f_{ce} g^{\mu\alpha}g^{\nu\beta} G^{{d}}_{{\alpha}{\beta}}G^{{c}}_{{\mu}{\nu}}B^b_\gamma\\ &=\lambda f^f_{de} f^e_{ba} f^a_{cf} g^{\mu\alpha}g^{\nu\beta} G^d_{\alpha\beta}G^c_{\mu\nu} B^b_\gamma -\lambda f^{\textcolor{orange}{f}}_{{{d}}\textcolor{red}{e}} f^{\textcolor{red}{e}}_{b\textcolor{blue}{a}} f^{\textcolor{blue}{a}}_{{{c}}\textcolor{orange}{f}} g^{\mu\alpha}g^{\nu\beta} G^{{d}}_{{\alpha}{\beta}}G^{{c}}_{{\mu}{\nu}}B^b_\gamma\\ &=0 \end{align*}\] \subsection{GR conservation law and EoM} \[\begin{align*} \left(t_{GR}\right)^\gamma_\varepsilon &=\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \mathscr{R}^b_{c\varepsilon\mu} - \delta^\gamma_\varepsilon \mathscr{L}_{GR}\\ &=\frac{1}{2\kappa} e\,\eta^{ce}\left(e^\mu_e e^\gamma_b -e^\gamma_e e^\mu_b \right)\mathscr{R}^b_{c\varepsilon\mu}-\delta^\gamma_\varepsilon\frac{1}{2\kappa}e\, \eta^{ae}e^\sigma_e e^\omega_c \mathscr{R}^c_{a\omega\sigma}\\ &=e^c_\nu e^\alpha_b \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}} \mathscr{R}^b_{c\varepsilon\mu} - \delta^\gamma_\varepsilon \mathscr{L}_{GR}\\ &= \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}} R^\alpha_{\nu\varepsilon\mu} - \delta^\gamma_\varepsilon \mathscr{L}_{GR}\\ \end{align*}\] \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\nu\mu,\gamma}^\alpha} &= \frac{\partial}{\partial \Gamma_{\nu\mu,\gamma}^\alpha} \left(\frac{1}{2\kappa} \sqrt{-g} \, g^{\kappa\sigma} \delta_\varepsilon^\omega R_{\kappa\omega\sigma}^\varepsilon \right) \nonumber\\ &=\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu} \delta_\alpha^\gamma - g^{\nu\gamma} \delta_\alpha^\mu \right) \end{align*}\] \begin{adjustwidth}{-2.55cm}{-1cm} \[\begin{align*} \left(t_{GR}\right)^\gamma_{\varepsilon,\gamma} &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} +\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}} R^\alpha_{\nu\varepsilon\mu,\gamma} - \delta^\gamma_\varepsilon \left(\mathscr{L}_{GR}\right)_{,\gamma}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} +\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu} \delta_\alpha^\gamma - g^{\nu\gamma} \delta_\alpha^\mu \right) R^\alpha_{\nu\varepsilon\mu,\gamma} - \left(\frac{1}{2\kappa} \sqrt{-g} \, g^{\nu\mu} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \right)_{,\varepsilon}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} +\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu} \delta_\alpha^\gamma - g^{\nu\gamma} \delta_\alpha^\mu \right) R^\alpha_{\nu\varepsilon\mu,\gamma} - \frac{1}{2\kappa} \sqrt{-g} \, g^{\nu\mu} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu,\varepsilon} - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g} \, g^{\nu\mu}\right)_{,\varepsilon}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} +\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu} \delta_\alpha^\gamma R^\alpha_{\nu\varepsilon\mu,\gamma} - g^{\nu\textcolor{red}{\mu}} \delta_\alpha^{\textcolor{red}{\gamma}} R^\alpha_{\nu\varepsilon\textcolor{red}{\gamma},\textcolor{red}{\mu}} -g^{\nu\mu} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu,\varepsilon} \right) - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g} \, g^{\nu\mu}\right)_{,\varepsilon}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} +\frac{1}{2\kappa} \sqrt{-g} g^{\nu\mu} \delta_\alpha^\gamma \left( R^\alpha_{\nu\varepsilon\mu,\gamma} \textcolor{red}{+}R^\alpha_{\nu\textcolor{red}{\gamma\varepsilon},\mu} \textcolor{red}{+}R^\alpha_{\nu\textcolor{red}{\mu\gamma},\varepsilon} \right) - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g} \, g^{\nu\mu}\right)_{,\varepsilon}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} -\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g}\,_{,\varepsilon} g^{\nu\mu}+\sqrt{-g} \, {g^{\nu\mu}}_{,\varepsilon}\right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} R_{\nu\mu} \left( \sqrt{-g}\, \Gamma^\eta_{\eta\varepsilon} g^{\nu\mu} -\sqrt{-g} \, g^{\eta\mu}\Gamma^\nu_{\eta\varepsilon} -\sqrt{-g} \, g^{\nu\eta}\Gamma^\mu_{\eta\varepsilon} \right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \left( \sqrt{-g}\,g^{\nu\mu} R_{\nu\mu} \Gamma^\eta_{\eta\varepsilon} -\sqrt{-g} \, g^{\eta\mu} R_{\nu\mu} \Gamma^\nu_{\eta\varepsilon} -\sqrt{-g} \, g^{\nu\eta} R_{\nu\mu} \Gamma^\mu_{\eta\varepsilon} \right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \left( \sqrt{-g}\,R\, \Gamma^\eta_{\eta\varepsilon} -\sqrt{-g} \, g^{\eta\mu} R_{\nu\mu} \Gamma^\nu_{\eta\varepsilon} -\sqrt{-g} \, g^{\textcolor{red}{\mu}\eta} R_{\textcolor{red}{\mu\nu}} \Gamma^{\textcolor{red}{\nu}}_{\eta\varepsilon} \right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \left( \sqrt{-g}\, \textcolor{red}{\delta^\eta_\nu} R\, \Gamma^{\textcolor{red}{\nu}}_{\eta\varepsilon} -2\sqrt{-g} \, g^{\eta\mu} R_{\mu\nu} \Gamma^\nu_{\eta\varepsilon} \right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} + \frac{1}{\kappa} \sqrt{-g} \left( g^{\eta\mu} R_{\mu\nu} -\frac{1}{2} \, \delta^\eta_\nu R \right)\Gamma^\nu_{\eta\varepsilon}\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} + \frac{1}{\kappa} \sqrt{-g} \, \mathbb{G}^\eta_\nu \, \Gamma^\nu_{\eta\varepsilon} \end{align*}\] \end{adjustwidth} \[\begin{align*} R^\alpha_{\nu\varepsilon\mu;\gamma} +R^\alpha_{\nu\gamma\varepsilon;\mu} +R^\alpha_{\nu\mu\gamma;\varepsilon} =0 \end{align*}\] \begin{adjustwidth}{-2.55cm}{-1cm} \[\begin{align*} &\left(R^\alpha_{\nu\varepsilon\mu,\gamma} +R^\eta_{\nu\varepsilon\mu} \Gamma^\alpha_{\eta\gamma} -R^\alpha_{\eta\varepsilon\mu} \Gamma^\eta_{\nu\gamma} -R^\alpha_{\nu\eta\mu} \Gamma^\eta_{\varepsilon\gamma} -R^\alpha_{\nu\varepsilon\eta} \Gamma^\eta_{\mu\gamma} \right)\\ &\quad\quad+\left(R^\alpha_{\nu\gamma\varepsilon,\mu} +R^\eta_{\nu\gamma\varepsilon} \Gamma^\alpha_{\eta\mu} -R^\alpha_{\eta\gamma\varepsilon} \Gamma^\eta_{\nu\mu} -R^\alpha_{\nu\eta\varepsilon} \Gamma^\eta_{\gamma\mu} -R^\alpha_{\nu\gamma\eta} \Gamma^\eta_{\varepsilon\mu} \right)\\ &\quad\quad\quad\quad +\left(R^\alpha_{\nu\mu\gamma,\varepsilon} +R^\eta_{\nu\mu\gamma} \Gamma^\alpha_{\eta\varepsilon} -R^\alpha_{\eta\mu\gamma} \Gamma^\eta_{\nu\varepsilon} -R^\alpha_{\nu\eta\gamma} \Gamma^\eta_{\mu\varepsilon} -R^\alpha_{\nu\mu\eta} \Gamma^\eta_{\gamma\varepsilon} \right) =0\\ \end{align*}\] \end{adjustwidth} \[\begin{align*} R^\alpha_{\nu\varepsilon\mu,\gamma} +R^\alpha_{\nu\gamma\varepsilon,\mu} +R^\alpha_{\nu\mu\gamma,\varepsilon} =&-\left(R^\eta_{\nu\varepsilon\mu} \Gamma^\alpha_{\eta\gamma} -R^\alpha_{\eta\varepsilon\mu} \Gamma^\eta_{\nu\gamma} -R^\alpha_{\nu\eta\mu} \Gamma^\eta_{\varepsilon\gamma} -R^\alpha_{\nu\varepsilon\eta} \Gamma^\eta_{\mu\gamma} \right)\\ &\quad\quad -\left(R^\eta_{\nu\gamma\varepsilon} \Gamma^\alpha_{\eta\mu} -R^\alpha_{\eta\gamma\varepsilon} \Gamma^\eta_{\nu\mu} -R^\alpha_{\nu\eta\varepsilon} \Gamma^\eta_{\gamma\mu} -R^\alpha_{\nu\gamma\eta} \Gamma^\eta_{\varepsilon\mu} \right)\\ &\quad\quad\quad\quad -\left(R^\eta_{\nu\mu\gamma} \Gamma^\alpha_{\eta\varepsilon} -R^\alpha_{\eta\mu\gamma} \Gamma^\eta_{\nu\varepsilon} -R^\alpha_{\nu\eta\gamma} \Gamma^\eta_{\mu\varepsilon} -R^\alpha_{\nu\mu\eta} \Gamma^\eta_{\gamma\varepsilon} \right) \end{align*}\] \[\begin{align*} &-\frac{1}{2\kappa} \sqrt{-g} g^{\nu\mu} \delta_\alpha^\gamma \left(R^\eta_{\nu\varepsilon\mu} \Gamma^\alpha_{\eta\gamma} -R^\alpha_{\eta\varepsilon\mu} \Gamma^\eta_{\nu\gamma} -R^\alpha_{\nu\eta\mu} \Gamma^\eta_{\varepsilon\gamma} -R^\alpha_{\nu\varepsilon\eta} \Gamma^\eta_{\mu\gamma} \right)\\ &\quad\quad -\frac{1}{2\kappa} \sqrt{-g} g^{\nu\mu} \delta_\alpha^\gamma \left(R^\eta_{\nu\gamma\varepsilon} \Gamma^\alpha_{\eta\mu} -R^\alpha_{\eta\gamma\varepsilon} \Gamma^\eta_{\nu\mu} -R^\alpha_{\nu\eta\varepsilon} \Gamma^\eta_{\gamma\mu} -R^\alpha_{\nu\gamma\eta} \Gamma^\eta_{\varepsilon\mu} \right)\\ &\quad\quad\quad\quad -\frac{1}{2\kappa} \sqrt{-g} g^{\nu\mu} \delta_\alpha^\gamma \left(R^\eta_{\nu\mu\gamma} \Gamma^\alpha_{\eta\varepsilon} -R^\alpha_{\eta\mu\gamma} \Gamma^\eta_{\nu\varepsilon} -R^\alpha_{\nu\eta\gamma} \Gamma^\eta_{\mu\varepsilon} -R^\alpha_{\nu\mu\eta} \Gamma^\eta_{\gamma\varepsilon} \right)\\ =&-\frac{1}{2\kappa} \sqrt{-g} g^{\nu\mu} \left(R^\eta_{\nu\varepsilon\mu} \Gamma^\alpha_{\eta\alpha} -R^\alpha_{\eta\varepsilon\mu} \Gamma^\eta_{\nu\alpha} -R^\alpha_{\nu\eta\mu} \Gamma^\eta_{\varepsilon\alpha} -R^\alpha_{\nu\varepsilon\eta} \Gamma^\eta_{\mu\alpha} \right)\\ &\quad\quad -\frac{1}{2\kappa} \sqrt{-g} g^{\nu\mu} \left(R^\eta_{\nu\alpha\varepsilon} \Gamma^\alpha_{\eta\mu} -R^\alpha_{\eta\alpha\varepsilon} \Gamma^\eta_{\nu\mu} -R^\alpha_{\nu\eta\varepsilon} \Gamma^\eta_{\alpha\mu} -R^\alpha_{\nu\alpha\eta} \Gamma^\eta_{\varepsilon\mu} \right)\\ &\quad\quad\quad\quad -\frac{1}{2\kappa} \sqrt{-g} \left(R^\eta_{\nu\mu\alpha} \Gamma^\alpha_{\eta\varepsilon} -R^\alpha_{\eta\mu\alpha} \Gamma^\eta_{\nu\varepsilon} -R^\alpha_{\nu\eta\alpha} \Gamma^\eta_{\mu\varepsilon} -R^\alpha_{\nu\mu\eta} \Gamma^\eta_{\alpha\varepsilon} \right)\\ =&-\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu}R^\eta_{\nu\varepsilon\mu} \Gamma^\alpha_{\eta\alpha} -g^{\nu\mu}R^\alpha_{\eta\varepsilon\mu} \Gamma^\eta_{\nu\alpha} -g^{\nu\mu}R^\alpha_{\nu\eta\mu} \Gamma^\eta_{\varepsilon\alpha} -g^{\nu\mu}R^\alpha_{\nu\varepsilon\eta} \Gamma^\eta_{\mu\alpha} \right)\\ &\quad\quad -\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu}R^\eta_{\nu\alpha\varepsilon} \Gamma^\alpha_{\eta\mu} -g^{\nu\mu}R^\alpha_{\eta\alpha\varepsilon} \Gamma^\eta_{\nu\mu} -g^{\nu\mu}R^\alpha_{\nu\eta\varepsilon} \Gamma^\eta_{\alpha\mu} -g^{\nu\mu}R^\alpha_{\nu\alpha\eta} \Gamma^\eta_{\varepsilon\mu} \right)\\ &\quad\quad\quad\quad -\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu}R^\eta_{\nu\mu\alpha} \Gamma^\alpha_{\eta\varepsilon} -g^{\nu\mu}R^\alpha_{\eta\mu\alpha} \Gamma^\eta_{\nu\varepsilon} -g^{\nu\mu}R^\alpha_{\nu\eta\alpha} \Gamma^\eta_{\mu\varepsilon} -g^{\nu\mu}R^\alpha_{\nu\mu\eta} \Gamma^\eta_{\alpha\varepsilon} \right)\\ =&-\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu}R^{\textcolor{red}{\alpha}}_{\nu\varepsilon\mu} \Gamma^{\textcolor{red}{\eta}}_{\textcolor{red}{\alpha\eta}} -g^{\textcolor{red}{\eta}\mu}R^\alpha_{\textcolor{red}{\nu}\varepsilon\mu} \Gamma^{\textcolor{red}{\nu}}_{\textcolor{red}{\eta}\alpha} -g^{\nu\mu}R^\alpha_{\nu\eta\mu} \Gamma^\eta_{\varepsilon\alpha} -g^{\nu\textcolor{red}{\eta}}R^\alpha_{\nu\varepsilon\textcolor{red}{\mu}} \Gamma^{\textcolor{red}{\mu}}_{\textcolor{red}{\eta}\alpha} \right)\\ &\quad\quad -\frac{1}{2\kappa} \sqrt{-g} \left( \cancel{g^{\nu\textcolor{red}{\eta}}R^{\textcolor{red}{\alpha}}_{\nu\textcolor{red}{\mu}\varepsilon} \Gamma^{\textcolor{red}{\mu}}_{\textcolor{red}{\alpha\eta}}} -g^{\textcolor{red}{\eta\gamma}}\textcolor{red}{\delta^\mu_\alpha}R^\alpha_{\textcolor{red}{\nu\mu}\varepsilon} \Gamma^{\textcolor{red}{\nu}}_{\textcolor{red}{\eta\gamma}} -\cancel{g^{\nu\textcolor{red}{\eta}}R^\alpha_{\nu\textcolor{red}{\mu}\varepsilon} \Gamma^{\textcolor{red}{\mu}}_{\alpha\textcolor{red}{\eta}}} -\xcancel{g^{\nu\mu}R^\alpha_{\nu\alpha\eta} \Gamma^\eta_{\varepsilon\mu}} \right)\\ &\quad\quad\quad\quad -\frac{1}{2\kappa} \sqrt{-g} \left( \bcancel{g^{\nu\mu}R^{\textcolor{red}{\alpha}}_{\nu\mu\textcolor{red}{\eta}} \Gamma^{\textcolor{red}{\eta}}_{\textcolor{red}{\alpha}\varepsilon}} -g^{\nu\mu}R^\alpha_{\eta\mu\alpha} \Gamma^\eta_{\nu\varepsilon} -\xcancel{g^{\nu\mu}R^\alpha_{\nu\eta\alpha} \Gamma^\eta_{\mu\varepsilon}} -\bcancel{g^{\nu\mu}R^\alpha_{\nu\mu\eta} \Gamma^\eta_{\alpha\varepsilon}} \right)\\ =&-\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu}R^{{\alpha}}_{\nu\varepsilon\mu} \Gamma^{{\eta}}_{{\alpha\eta}} -g^{{\eta}\mu}R^\alpha_{{\nu}\varepsilon\mu} \Gamma^{{\nu}}_{{\eta}\alpha} -g^{\nu\mu}R^\alpha_{\nu\eta\mu} \Gamma^\eta_{\varepsilon\alpha} -g^{\nu{\eta}}R^\alpha_{\nu\varepsilon{\mu}} \Gamma^{{\mu}}_{{\eta}\alpha} \textcolor{red}{+}g^{{\eta\gamma}}{\delta^\mu_\alpha}R^\alpha_{\nu\textcolor{red} {\varepsilon\mu}}\Gamma^{{\nu}}_{{\eta\gamma}} -g^{\nu\mu}R^\alpha_{\eta\mu\alpha} \Gamma^\eta_{\nu\varepsilon} \right)\\ =&-\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu}R^{{\alpha}}_{\nu\varepsilon\mu} \Gamma^{{\eta}}_{{\alpha\eta}} -g^{{\eta}\mu}R^\alpha_{{\nu}\varepsilon\mu} \Gamma^{{\nu}}_{{\eta}\alpha} -g^{\nu{\eta}}R^\alpha_{\nu\varepsilon{\mu}} \Gamma^{{\mu}}_{{\eta}\alpha} +g^{{\eta\gamma}}{\delta^\mu_\alpha}R^\alpha_{\nu\varepsilon\mu}\Gamma^{{\nu}}_{{\eta\gamma}} \right) -\frac{1}{2\kappa} \sqrt{-g} \left( -g^{\nu\mu}R^\alpha_{\nu\eta\mu} \Gamma^\eta_{\varepsilon\alpha} -g^{\nu\mu}R^\alpha_{\eta\mu\alpha} \Gamma^\eta_{\nu\varepsilon} \right)\\ =&-\frac{1}{2\kappa} \sqrt{-g} \left( g^{\eta\gamma}{\delta^\mu_\alpha}\Gamma^{{\nu}}_{{\eta\gamma}}+g^{\nu\mu} \Gamma^{{\eta}}_{{\alpha\eta}} -g^{\eta\mu}\Gamma^{{\nu}}_{\eta\alpha} -g^{\nu\eta} \Gamma^{\mu}_{\eta\alpha} \right) R^{{\alpha}}_{\nu\varepsilon\mu} -\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu}g^{\alpha\gamma}R_{\gamma\nu\textcolor{red}{\mu\eta}} \Gamma^\eta_{\varepsilon\alpha} \textcolor{red}{+}g^{\nu\mu}R^\alpha_{\eta\textcolor{red}{\alpha\mu}} \Gamma^\eta_{\nu\varepsilon} \right)\\ =&-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} -\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu}g^{\alpha\gamma}R_{\mu\eta\gamma\nu} \Gamma^\eta_{\varepsilon\alpha} +g^{\nu\mu}R_{\eta\mu} \Gamma^\eta_{\nu\varepsilon} \right)\\ =&-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} -\frac{1}{2\kappa} \sqrt{-g} \left( \textcolor{red}{-}g^{\nu\mu}g^{\alpha\gamma}R_{\mu\eta\textcolor{red}{\nu\gamma}} \Gamma^\eta_{\varepsilon\alpha} +g^{\nu\mu}R_{\eta\mu} \Gamma^\eta_{\nu\varepsilon} \right)\\ =&-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} -\frac{1}{2\kappa} \sqrt{-g} \left( -g^{\alpha\gamma}R^\nu_{\eta{\nu\gamma}} \Gamma^\eta_{\varepsilon\alpha} +g^{\nu\mu}R_{\eta\mu} \Gamma^\eta_{\nu\varepsilon} \right)\\ =&-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} -\frac{1}{2\kappa} \sqrt{-g} \left( -g^{\alpha\gamma}R_{\eta\gamma} \Gamma^\eta_{\varepsilon\alpha} +g^{\nu\mu}R_{\eta\mu} \Gamma^\eta_{\nu\varepsilon} \right)\\ =&-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} -\frac{1}{2\kappa} \sqrt{-g} \left( -\cancel{g^{\textcolor{red}{\nu\mu}}R_{\eta\textcolor{red}{\mu}} \Gamma^\eta_{\varepsilon\textcolor{red}{\nu}}} +\cancel{g^{\nu\mu}R_{\eta\mu} \Gamma^\eta_{\nu\varepsilon}} \right)\\ =&-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} \end{align*}\] \begin{adjustwidth}{-2.55cm}{-1cm} \[\begin{align*} &\left(t_{GR}\right)^\gamma_{\varepsilon,\gamma}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} +\frac{1}{2\kappa} \sqrt{-g} g^{\nu\mu} \delta_\alpha^\gamma \left( R^\alpha_{\nu\varepsilon\mu,\gamma} \textcolor{red}{+}R^\alpha_{\nu\textcolor{red}{\gamma\varepsilon},\mu} \textcolor{red}{+}R^\alpha_{\nu\textcolor{red}{\mu\gamma},\varepsilon} \right) - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g} \, g^{\nu\mu}\right)_{,\varepsilon}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} -\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g}\,_{,\varepsilon} g^{\nu\mu}+\sqrt{-g} \, {g^{\nu\mu}}_{,\varepsilon}\right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left( -\frac{1}{2}\sqrt{-g}\, g_{\phi\psi} {g^{\phi\psi}}_{,\varepsilon} g^{\nu\mu}+\sqrt{-g} \, {g^{\nu\mu}}_{,\varepsilon}\right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g}\, \delta^\gamma_\alpha \left( -\frac{1}{2} R^\alpha_{\nu\gamma\mu} g_{\phi\psi} {g^{\phi\psi}}_{,\varepsilon} g^{\nu\mu} + R^\alpha_{\nu\gamma\mu} {g^{\nu\mu}}_{,\varepsilon} \right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g}\, \delta^\gamma_\alpha \left( -\frac{1}{2} R^\alpha_{\phi\gamma\psi} g_{\nu\mu} {g^{\nu\mu}}_{,\varepsilon} g^{\phi\psi} + R^\alpha_{\nu\gamma\mu} {g^{\nu\mu}}_{,\varepsilon} \right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g}\, \delta^\gamma_\alpha \left( R^\alpha_{\nu\gamma\mu} -\frac{1}{2} R^\alpha_{\phi\gamma\psi} g_{\nu\mu} g^{\phi\psi} \right){g^{\nu\mu}}_{,\varepsilon}\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g}\, \left( \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} -\frac{1}{2} R g_{\nu\mu} \right) \left( -g^{\sigma\mu}\Gamma^\nu_{\sigma\varepsilon} -g^{\nu\sigma}\Gamma^\mu_{\sigma\varepsilon} +\frac{2}{\mathbb{D}-1}g^{\nu\mu}T_{\varepsilon} \right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} \\&\qquad\qquad - \frac{1}{2\kappa} \sqrt{-g}\, \left( -\delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} g^{\sigma\mu}\Gamma^\nu_{\sigma\varepsilon} -\delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} g^{\nu\sigma}\Gamma^\mu_{\sigma\varepsilon} +\frac{2}{\mathbb{D}-1}\delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} g^{\nu\mu}T_{\varepsilon} +\frac{1}{2} R g_{\nu\mu}g^{\sigma\mu}\Gamma^\nu_{\sigma\varepsilon} +\frac{1}{2} R g_{\nu\mu}g^{\nu\sigma}\Gamma^\mu_{\sigma\varepsilon} -\frac{1}{2} R g_{\nu\mu}\frac{2}{\mathbb{D}-1}g^{\nu\mu}T_{\varepsilon} \right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} \\&\qquad\qquad - \frac{1}{2\kappa} \sqrt{-g}\, \left( -\delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} g^{\sigma\mu}\Gamma^\nu_{\sigma\varepsilon} -\delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} g^{\nu\sigma}\Gamma^\mu_{\sigma\varepsilon} +\frac{2}{\mathbb{D}-1}R T_{\varepsilon} +\frac{1}{2}R \Gamma^\sigma_{\sigma\varepsilon} +\frac{1}{2}R \Gamma^\sigma_{\sigma\varepsilon} -\frac{\mathbb{D}}{\mathbb{D}-1} R T_{\varepsilon} \right)\\ \end{align*}\] \end{adjustwidth} \begin{adjustwidth}{-2.55cm}{-1cm} \[\begin{align*} &\left(t_{GR}\right)^\gamma_{\varepsilon,\gamma}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} +\frac{1}{2\kappa} \sqrt{-g} g^{\nu\mu} \delta_\alpha^\gamma \left( R^\alpha_{\nu\varepsilon\mu,\gamma} \textcolor{red}{+}R^\alpha_{\nu\textcolor{red}{\gamma\varepsilon},\mu} \textcolor{red}{+}R^\alpha_{\nu\textcolor{red}{\mu\gamma},\varepsilon} \right) - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g} \, g^{\nu\mu}\right)_{,\varepsilon}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} -\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g} \, {g^{\nu\mu}}_{,\varepsilon} +\sqrt{-g}\,_{,\varepsilon} g^{\nu\mu}\right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left( \sqrt{-g} \, {g^{\nu\mu}}_{,\varepsilon}-\frac{1}{2}\sqrt{-g}\, g_{\phi\psi} {g^{\phi\psi}}_{,\varepsilon} g^{\nu\mu} \right) \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} {g^{\nu\mu}}_{,\varepsilon} -\frac{1}{2} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} g_{\phi\psi} {g^{\phi\psi}}_{,\varepsilon} g^{\nu\mu} \right) \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( R_{\nu\mu}{g^{\nu\mu}}_{,\varepsilon} -\frac{1}{2} R\, g_{\phi\psi}{g^{\phi\psi}}_{,\varepsilon} \right) \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( R_{\nu\mu} -\frac{1}{2} R\, g_{\nu\mu} \right) {g^{\nu\mu}}_{,\varepsilon} \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( \delta^\sigma_\nu R_{\sigma\mu} -\frac{1}{2} R\, g_{\nu\mu} \right) {g^{\nu\mu}}_{,\varepsilon} \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( g^{\sigma\gamma} g_{\gamma\nu} R_{\sigma\mu} -\frac{1}{2} R\,\delta^\gamma_\mu g_{\gamma\nu} \right) {g^{\nu\mu}}_{,\varepsilon} \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( g^{\sigma\gamma} R_{\sigma\mu} -\frac{1}{2} \delta^\gamma_\mu R \right)g_{\gamma\nu}{g^{\nu\mu}}_{,\varepsilon} \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( \textcolor{red}{\frac{1}{2} g^{\sigma\beta} R^\gamma_{\sigma\mu\beta}} \textcolor{red}{-\frac{1}{2} g^{\sigma\beta} R^\gamma_{\sigma\mu\beta}} + g^{\sigma\gamma} R_{\sigma\mu} -\frac{1}{2} \delta^\gamma_\mu R \right)g_{\gamma\nu}{g^{\nu\mu}}_{,\varepsilon} \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} -\frac{1}{2} \frac{1}{2\kappa} \sqrt{-g} \, \left( g^{\sigma\beta} R^\gamma_{\sigma\mu\beta} + g^{\sigma\gamma} R_{\sigma\mu} - \delta^\gamma_\mu R \right)g_{\gamma\nu}{g^{\nu\mu}}_{,\varepsilon} -\frac{1}{2} \frac{1}{2\kappa} \sqrt{-g} \, \left( - g^{\sigma\beta} R^\gamma_{\sigma\mu\beta} + g^{\sigma\gamma} R_{\sigma\mu} \right)g_{\gamma\nu}{g^{\nu\mu}}_{,\varepsilon} \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} -\frac{1}{2} t^\gamma_\mu g_{\gamma\nu}{g^{\nu\mu}}_{,\varepsilon} -\frac{1}{2} \frac{1}{2\kappa} \sqrt{-g} \, \left( - g^{\sigma\beta} R^\gamma_{\sigma\mu\beta} + g^{\sigma\gamma} R_{\sigma\mu} \right)g_{\gamma\nu}{g^{\nu\mu}}_{,\varepsilon} \\ \\ \end{align*}\] \end{adjustwidth} \begin{adjustwidth}{-2.55cm}{-1cm} \[\begin{align*} &\left(t_{GR}\right)^\gamma_{\varepsilon,\gamma}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} +\frac{1}{2\kappa} \sqrt{-g} g^{\nu\mu} \delta_\alpha^\gamma \left( R^\alpha_{\nu\varepsilon\mu,\gamma} \textcolor{red}{+}R^\alpha_{\nu\textcolor{red}{\gamma\varepsilon},\mu} \textcolor{red}{+}R^\alpha_{\nu\textcolor{red}{\mu\gamma},\varepsilon} \right) - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g} \, g^{\nu\mu}\right)_{,\varepsilon}\\ &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma} R^\alpha_{\nu\varepsilon\mu} -\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} R^{{\alpha}}_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left(\sqrt{-g} \, {g^{\nu\mu}}_{,\varepsilon} +\sqrt{-g}\,_{,\varepsilon} g^{\nu\mu}\right)\\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} \left( \sqrt{-g} \, {g^{\nu\mu}}_{,\varepsilon}-\frac{1}{2}\sqrt{-g}\, g_{\phi\psi} {g^{\phi\psi}}_{,\varepsilon} g^{\nu\mu} \right) \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} {g^{\nu\mu}}_{,\varepsilon} -\frac{1}{2} \delta^\gamma_\alpha R^\alpha_{\nu\gamma\mu} g_{\phi\psi} {g^{\phi\psi}}_{,\varepsilon} g^{\nu\mu} \right) \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( R_{\nu\mu}{g^{\nu\mu}}_{,\varepsilon} -\frac{1}{2} R\, g_{\phi\psi}{g^{\phi\psi}}_{,\varepsilon} \right) \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( R_{\nu\mu} -\frac{1}{2} R\, g_{\nu\mu} \right) {g^{\nu\mu}}_{,\varepsilon} \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( R_{\nu\mu} -\frac{1}{2} R\, g_{\nu\mu} \right) \left( -g^{\sigma\mu}\Gamma^\nu_{\sigma\varepsilon} -g^{\nu\sigma}\Gamma^\mu_{\sigma\varepsilon} +\frac{2}{\mathbb{D}-1} {g^{\nu\mu}} T_{\varepsilon} \right) \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} - \frac{1}{2\kappa} \sqrt{-g} \, \left( R_{\nu\mu} -\frac{1}{2} R\, g_{\nu\mu} \right) \left( -g^{\sigma\mu}\Gamma^\nu_{\sigma\varepsilon} -g^{\nu\sigma}\Gamma^\mu_{\sigma\varepsilon} \right) - \frac{1}{\kappa} \sqrt{-g} \, \left( R_{\nu\mu} -\frac{1}{2} R\, g_{\nu\mu} \right) \frac{1}{\mathbb{D}-1} {g^{\nu\mu}} T_{\varepsilon} \\ &=\left\{ \left[\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu,\gamma}}\right]_{,\gamma}-\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma^\alpha_{\nu\mu}} \right\} R^\alpha_{\nu\varepsilon\mu} + \frac{1}{2\kappa} \sqrt{-g} \, \left( R_{\nu\mu} +R_{\mu\nu} - R\, g_{\nu\mu} \right) g^{\sigma\mu}\Gamma^\nu_{\sigma\varepsilon} - \frac{1}{\kappa} \sqrt{-g} \, \left( R -\frac{1}{2} \mathbb{D} R\, \right) \frac{1}{\mathbb{D}-1} T_{\varepsilon} \end{align*}\] \end{adjustwidth} \section{Direct Variation} \[\begin{align*} F^d_{\varepsilon\gamma}=A^d_{\gamma,\varepsilon}-A^d_{\varepsilon,\gamma}+f^d_{bc}A^b_\varepsilon A^c_\gamma \end{align*}\] \[\begin{align*} \frac{\delta}{\delta A^a_\nu} &=\frac{\delta F^d_{\varepsilon\gamma}}{\delta A^a_\nu}\frac{\delta}{\delta F^d_{\varepsilon\gamma}} =\frac{\delta }{\delta A^a_\nu} \left( A^d_{\gamma,\varepsilon}-A^d_{\varepsilon,\gamma}+f^d_{bc}A^b_\varepsilon A^c_\gamma \right) \frac{\delta}{\delta F^d_{\varepsilon\gamma}}\\ &= \left( f^d_{bc}\delta^\nu_\varepsilon\delta^b_a A^c_\gamma +f^d_{bc}A^b_\varepsilon\delta^\nu_\gamma\delta^c_a \right) \frac{\delta}{\delta F^d_{\varepsilon\gamma}}\\ &= f^d_{ac} A^c_\gamma \frac{\delta}{\delta F^d_{\nu\gamma}} +f^d_{ba}A^b_\varepsilon \frac{\delta}{\delta F^d_{\varepsilon\nu}}\\ &= f^d_{ac} A^c_\gamma \frac{\delta}{\delta F^d_{\nu\gamma}} +f^d_{ab}A^b_\varepsilon \frac{\delta}{\delta F^d_{\nu\varepsilon}} \\ &= f^d_{ac} A^c_\mu*2 \frac{\delta}{\delta F^d_{\nu\mu}} \end{align*}\] \[\begin{align*} \frac{\delta}{\delta A^a_{\nu,\mu}} &=\frac{\delta F^d_{\varepsilon\gamma}}{\delta A^a_{\nu,\mu}}\frac{\delta}{\delta F^d_{\varepsilon\gamma}} =\frac{\delta }{\delta A^a_{\nu,\mu}} \left( A^d_{\gamma,\varepsilon}-A^d_{\varepsilon,\gamma}+f^d_{bc}A^b_\varepsilon A^c_\gamma \right) \frac{\delta}{\delta F^d_{\varepsilon\gamma}}\\ &= \left( \delta^\nu_\gamma\delta^\mu_\varepsilon\delta^d_a-\delta^\nu_\varepsilon\delta^\mu_\gamma\delta^d_a \right) \frac{\delta}{\delta F^d_{\varepsilon\gamma}}\\ &= \frac{\delta}{\delta F^a_{\mu\nu}} -\frac{\delta}{\delta F^d_{\nu\mu}}\\ &= -2\frac{\delta}{\delta F^a_{\nu\mu}} =-\frac{\delta}{\delta A^a_{\mu,\nu}} \end{align*}\] \[\begin{align*} \frac{\delta}{\delta A^a_{\nu}}= f^d_{ac} A^c_\mu*2 \frac{\delta}{\delta F^d_{\nu\mu}} =-f^d_{ac} A^c_\mu\frac{\delta}{\delta A^d_{\nu,\mu}} \end{align*}\] \[\begin{align*} A^a_\eta \frac{\delta}{\delta A^a_{\nu}} &=-f^d_{ac} A^a_\eta A^c_\mu\frac{\delta}{\delta A^d_{\nu,\mu}}\\ &=f^a_{dc} A^d_\eta A^c_\mu\frac{\delta}{\delta A^a_{\mu,\nu}} \end{align*}\] \[\begin{align*} \partial_\mu\left(\frac{\delta}{\delta A^a_{\nu,\mu}}\right)-\frac{\delta}{\delta A^a_{\nu}} &=\partial_\mu\left(2\frac{\delta}{\delta F^a_{\mu\nu}}\right)+f^d_{ac} A^c_\mu*2 \frac{\delta}{\delta F^d_{\mu\nu}}\\ &=2\left(\delta^d_a\partial_\mu+f^d_{ac} A^c_\mu\right)\frac{\delta}{\delta F^d_{\mu\nu}}\\ &=2\left(\delta^d_a\partial_\mu-f^d_{ca} A^c_\mu\right)\frac{\delta}{\delta F^d_{\mu\nu}} \end{align*}\] \[\begin{align*} \left(t_{YM}\right)^\mu_{\gamma,\mu} &=\left[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_{\nu,\mu}} G^a_{\gamma\nu} - \delta^\mu_\gamma \mathscr{L}_{YM} \right]_{,\mu}\\ &=\left\{ \left[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_{\nu,\mu}} \right]_{,\mu} -\frac{\partial \mathscr{L}_{YM}}{\partial( B^a_\nu)} \right\}G^a_{\gamma\nu} + \left( t_{YM}\right)^\eta_\mu \Gamma^\mu_{\eta\gamma}\\\\ \to \nabla_{\mu} \left[ \frac{\partial \mathscr{L}_{YM}}{\partial G^a_{\mu\nu}} G^a_{\gamma\nu} -\frac{1}{2} \delta^\mu_\gamma \mathscr{L}_{YM} \right] &= \left[ \left(\delta^d_a\partial_\mu-f^d_{ca} A^c_\mu\right)\frac{\partial \mathscr{L}_{YM}}{\partial G^d_{\mu\nu}} \right] G^a_{\gamma\nu} \end{align*}\] \section{Self-dual} \[\begin{align*} F_{\alpha\beta}&=g_{\alpha\gamma}g_{\beta\sigma}\, \frac{\varepsilon^{\gamma\sigma\mu\nu}}{\sqrt{-g}}F_{\mu\nu}\\ \sqrt{-g}\,g^{\alpha\gamma}g^{\beta\sigma}F_{\alpha\beta}&=\varepsilon^{\gamma\sigma\mu\nu}F_{\mu\nu}\\ \left(\sqrt{-g}\,g^{\alpha\gamma}g^{\beta\sigma}F_{\alpha\beta}\right)_{;\sigma}&=\left(\varepsilon^{\gamma\sigma\mu\nu}F_{\mu\nu}\right)_{;\sigma}\\ \left(\sqrt{-g}\,F^{\gamma\sigma}\right)_{;\sigma}&= \left(\sqrt{-g}\,F^{\gamma\sigma}\right)_{,\sigma}= \varepsilon^{\gamma\sigma\mu\nu}F_{\mu\nu;\sigma} \end{align*}\] \[\begin{align*} \left(t_{GR}\right)^\gamma_{\varepsilon;\gamma} &=\left[\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \mathscr{R}^b_{c\varepsilon\mu} - \delta^\gamma_\varepsilon \mathscr{L}_{GR}\right]_{;\gamma} \\ &=\left[\frac{1}{2\kappa} \left(g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\sqrt{-g}\right]_{;\gamma}\\ &=\frac{1}{2\kappa} \left( g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu;\gamma} +g^{\beta\gamma}R_{\beta\varepsilon;\gamma} -\delta^\gamma_\varepsilon R_{;\gamma} \right)\sqrt{-g} \end{align*}\] \[\begin{align*} R^\phi_{\psi\mu\nu;\sigma} +R^\phi_{\psi\nu\sigma;\mu} +R^\phi_{\psi\sigma\mu;\nu} &=0\\ \delta_\phi^\sigma R^\phi_{\psi\mu\nu;\sigma} +\delta_\phi^\sigma R^\phi_{\psi\nu\sigma;\mu} +\delta_\phi^\sigma R^\phi_{\psi\sigma\mu;\nu} &=0\\ R^\sigma_{\psi\mu\nu;\sigma} -R_{\psi\nu;\mu} +R_{\psi\mu;\nu} &=0\\ g^{\psi\nu} R^\sigma_{\psi\mu\nu;\sigma} -g^{\psi\nu}R_{\psi\nu;\mu} +g^{\psi\nu}R_{\psi\mu;\nu} &=0\\ g^{\psi\nu} R^\sigma_{\psi\mu\nu;\sigma} -\delta^\sigma_\mu R_{;\sigma} +g^{\psi\sigma}R_{\psi\mu;\sigma} &=0 \end{align*}\] \[\begin{align*} R^\phi_{\psi\mu\nu;\sigma} +R^\phi_{\psi\nu\sigma;\mu} +R^\phi_{\psi\sigma\mu;\nu} &=0\\ g^{\psi\nu}\delta_\phi^\sigma \left(R^\phi_{\psi\mu\nu;\sigma} +R^\phi_{\psi\nu\sigma;\mu} +R^\phi_{\psi\sigma\mu;\nu}\right) &=0\\ g^{\psi\nu}g^{\sigma\xi}g_{\xi\phi} \left(R^\phi_{\psi\mu\nu;\sigma} +R^\phi_{\psi\nu\sigma;\mu} +R^\phi_{\psi\sigma\mu;\nu}\right) &=0\\ g^{\psi\nu}g_{\xi\phi} \left(R^\phi_{\psi\mu\nu;\sigma} +R^\phi_{\psi\nu\sigma;\mu} +R^\phi_{\psi\sigma\mu;\nu}\right) &=0\\ \end{align*}\] \[\begin{align*} \varepsilon^{\gamma\sigma\mu\nu}R^\phi_{\psi\mu\nu}&\\ g^{\psi\nu}\delta_\phi^\sigma \varepsilon^{\gamma\sigma\mu\nu}R^\phi_{\psi\mu\nu}=0&=g^{\psi\nu}\delta_\phi^\sigma\left(\sqrt{-g}\,g^{\alpha\gamma}g^{\beta\sigma}R^\phi_{\psi\alpha\beta}\right)_{;\sigma}\\ &=g^{\psi\nu}\delta_\phi^\sigma \sqrt{-g}\,g^{\alpha\gamma}g^{\beta\sigma}R^\phi_{\psi\alpha\beta;\sigma}\\ \end{align*}\] \section{Summary} The natural symmetrical, gauge-invariant canonical energy-momentum tensor for the abelian gauge field is derived. This derivation does not depend on flat spacetime geometry, hence is background independent. This method has potential to cover the non-abelian field theory and general relativity. \printbibliography \newpage \section{Supplementary Derivation} \subsection{Abelian Case (Electromagnetic Field)} \textbf{Eq.\eqref{eq:ab_lag}} \[\begin{align*} \Delta \mathscr{L}_{EM} &= \mathscr{L}_{EM}[\tilde{A}_\nu(\tilde{x}^\gamma), \tilde{A}_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] - \mathscr{L}_{EM}[A_\nu(x^\gamma), A_{\nu,\mu}(x^\gamma), x^\gamma] \\ &= \mathscr{L}_{EM}[\tilde{A}_\nu(\tilde{x}^\gamma), \tilde{A}_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] - \mathscr{L}_{EM}[A_\nu(\tilde{x}^\gamma), A_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] + \mathscr{L}_{EM}[A_\nu(\tilde{x}^\gamma), A_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] - \mathscr{L}_{EM}[A_\nu(x^\gamma), A_{\nu,\mu}(x^\gamma), x^\gamma]\\ &= \left[ \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu} \delta A_\nu + \frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)} \delta (\partial_\mu A_\nu) \right] (\tilde{x}^\gamma) + \left[ \mathscr{L}_{EM}_{, \gamma} \delta x^\gamma \right] (x^\gamma) \\ & = \left[ \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu} \delta A_\nu + \frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)} \delta (\partial_\mu A_\nu) \right] (x^\gamma) + O(\delta^2) + \left[ \mathscr{L}_{EM}_{, \gamma} \delta x^\gamma \right] (x^\gamma) \end{align*}\] \\ \textbf{Eq.\eqref{eq:Delta S_ab}} \[\begin{align*} \Delta S &= \int d^4x \cdot \underbracket[0.4pt][0pt]{\mathscr{L}_{EM} \delta x^\gamma_{,\gamma}}_{(A)} + \int d^4x \left[ \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu}\delta A_\nu + \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}\delta(\partial_\mu A_\nu) + \underbracket[0.4pt][0pt]{\mathscr{L}_{EM}_{,\gamma} \delta x^\gamma}_{(B)} \right] \\ &= \int d^4 x \cdot \left[ \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu}\delta A_\nu + \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}\partial_\mu(\delta A_\nu) + \underbracket[0.4pt][0pt]{(\mathscr{L}_{EM} \delta x^\gamma)_{,\gamma}}_{(A)+(B)} \right]\\ &= \int \left[ \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu} - \partial_\mu\left(\frac{\partial\mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)}\right) \right] \delta A_\nu d^4 x + \int \left[ \partial_\mu\left(\frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)}\delta A_\nu\right) + (\mathscr{L}_{EM} \delta x^\gamma)_{,\gamma}\right] d^4 x \end{align*}\] \textbf{Eq.\eqref{eq:ab_partial}} \[\begin{align*} \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} &= \frac{\partial}{\partial(\partial_\mu A_\nu)} \left( -\frac{1}{16\pi c} g^{\alpha\rho}g^{\beta\sigma}F_{\alpha\beta}F_{\rho\sigma}\sqrt{-g} \right) \\ &= -\frac{1}{16\pi c} g^{\alpha\rho}g^{\beta\sigma}\sqrt{-g} \frac{\partial}{\partial(\partial_\mu A_\nu)} (F_{\alpha\beta}F_{\rho\sigma}) \\ &= -\frac{1}{16\pi c} g^{\alpha\rho}g^{\beta\sigma}\sqrt{-g} \left[ \frac{\partial F_{\alpha\beta}}{\partial(\partial_\mu A_\nu)}F_{\rho\sigma} + F_{\alpha\beta}\frac{\partial F_{\rho\sigma}}{\partial(\partial_\mu A_\nu)} \right] \\ &= -\frac{1}{16\pi c} g^{\alpha\rho}g^{\beta\sigma}\sqrt{-g} \left[ (\delta^\mu_\alpha \delta^\nu_\beta - \delta^\nu_\alpha \delta^\mu_\beta)F_{\rho\sigma} + F_{\alpha\beta}(\delta^\mu_\rho \delta^\nu_\sigma - \delta^\nu_\rho \delta^\mu_\sigma) \right] \\ &= -\frac{1}{16\pi c} \sqrt{-g} \left[ (g^{\mu\alpha}g^{\nu\beta} - g^{\nu\alpha}g^{\mu\beta})F_{\alpha\beta} + F_{\alpha\beta}(g^{\mu\alpha}g^{\nu\beta} - g^{\nu\alpha}g^{\mu\beta}) \right] \\ &= -\frac{1}{8\pi c} \sqrt{-g} \left[ (g^{\mu\alpha}g^{\nu\beta} - g^{\nu\alpha}g^{\mu\beta})F_{\alpha\beta} \right]\\ &= -\frac{1}{8\pi c} \sqrt{-g} \left[ g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta} - g^{\nu\alpha}g^{\mu\beta}F_{\alpha\beta} \right]\\ &= -\frac{1}{8\pi c} \sqrt{-g} \left[ g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta} - g^{\nu\textcolor{red}{\beta}}g^{\mu\textcolor{red}{\alpha}}F_{\textcolor{red}{\beta\alpha}} \right] \end{align*}\] Using the antisymmetric property: \(F_{\beta\alpha} = -F_{\alpha\beta}\), we get \[\begin{align*} \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} &= -\frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta} \end{align*}\] \textbf{Eq.\eqref{eq:anti_ab}} \[\begin{align*} \left[ \frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)} A_\gamma \delta x^\gamma\right]_{,\nu \mu} &= \left[ -\frac{1}{4\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta} A_\gamma \delta x^\gamma \right]_{,\nu \mu} \\ &= \left[ -\frac{1}{8\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta} A_\gamma \delta x^\gamma \right]_{,\nu \mu} + \left[ -\frac{1}{8\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta} A_\gamma \delta x^\gamma \right]_{,\nu \mu} \\ &= \left[ -\frac{1}{8\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta} A_\gamma \delta x^\gamma \right]_{,\nu \mu} + \left[ -\frac{1}{8\pi c} g^{\textcolor{red}{\nu}\textcolor{blue}{\beta}}g^{\textcolor{red}{\mu}\textcolor{blue}{\alpha}} \sqrt{-g} F_{\textcolor{blue}{\beta\alpha}} A_\gamma \delta x^\gamma \right]_{,\textcolor{red}{ \mu\nu}} \\ &= \left[ -\frac{1}{8\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\alpha\beta} A_\gamma \delta x^\gamma \right]_{,\nu \mu} \textcolor{red}{-} \left[ -\frac{1}{8\pi c} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F_{\textcolor{red}{\alpha\beta}} A_\gamma \delta x^\gamma \right]_{,\nu \mu} \\ &=0 \end{align*}\] \textbf{Eq.\eqref{eq:*2_ab}}\\ The EoM is : \[\begin{align*} \left( \frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)} \right)_{,\mu} =\frac{\partial \mathscr{L}_{EM}}{\partial A_\nu} \end{align*}\] However, in free Abelian field case, LHS is \[\begin{align*} \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu}=0 \end{align*}\] Hence \[\begin{align} \label{eq:ab=0} \left( \frac{\partial \mathscr{L}_{EM}}{\partial (\partial_\mu A_\nu)} \right)_{,\mu} =0 \left(= \frac{\partial \mathscr{L}_{EM}}{\partial A_\nu} \right) \end{align}\]. On the other hand, by Eq.\eqref{eq:ab_partial}, we will derive \[\begin{align} \label{ab_SWAP} \frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\nu A_\mu)} &=-\frac{1}{4\pi c} g^{\nu{\alpha}} g^{\mu{\beta}}\sqrt{-g} F_{{\alpha\beta}}\nonumber=-\left[-\frac{1}{4\pi c}g^{\mu{\beta}} g^{\nu{\alpha}} \sqrt{-g} F_{{\beta\alpha}}\right]\nonumber\\ &=-\left[-\frac{1}{4\pi c}g^{\mu\textcolor{red}{\alpha}} g^{\nu\textcolor{red}{\beta}} \sqrt{-g} F_{{\textcolor{red}{\alpha\beta}}}\right] \underbrace{=}_{Eq.\eqref{eq:ab_partial}}-\frac{\partial \mathscr{L}_{EM}}{\partial(\partial_\mu A_\nu)} \end{align}\] Thus \((*2)\) is \[\begin{align*} \underbracket[0.4pt][0pt]{ \left( \frac{\partial \mathscr{L}_{EM}}{ \partial (\partial_\mu A_\nu)}\right)_{,\nu} A_\gamma \delta x^\gamma}_{(*2)} \,\, \underbrace{=}_{Eq.\eqref{ab_SWAP}}\,\, \underbrace{ \left(\textcolor{red}{-} \frac{\partial \mathscr{L}_{EM}}{ \partial (\partial_{\textcolor{red}{\nu}} A_{\textcolor{red}{\mu}})}\right)_{,\nu} }_{Eq.\eqref{eq:ab=0}=0} A_\gamma \delta x^\gamma =0 \end{align*}\] \newpage \subsection{Non-Abelian Case (Yang-Mills Theory)} \textbf{Eq.\eqref{eq:Guv}} \\ The derivation of Eq.\eqref{eq:Guv} \[\begin{align*} \mathbf{F} \rightarrow \mathbf{F}_{\mu\nu} &= \hat{T}_a \partial_\mu B_\nu^a - \hat{T}_a \partial_\nu B_\mu^a + \lambda [B_\mu^a \hat{T}_a, B_\nu^b \hat{T}_b] \\ &= \hat{T}_a \partial_\mu B_\nu^a - \hat{T}_a \partial_\nu B_\mu^a + \lambda B_\mu^a B_\nu^b [\hat{T}_a, \hat{T}_b] \\ &= \hat{T}_a \left( \partial_\mu B_\nu^a - \partial_\nu B_\mu^a + \lambda f_{bc}^a B_\mu^b B_\nu^c\right) \\ &= \hat{T}_a \left( F_{\mu\nu}^a + \lambda f_{bc}^a B_\mu^b B_\nu^c \right) \end{align*}\] \textbf{Eq.\eqref{eq:Lagrangian}} \\ The explicit expression of Lagrangian Eq.\eqref{eq:Lagrangian} is \[ K_{ab}g^{\mu\alpha}g^{\nu\beta}F^{a}_{\mu\nu}F^{b}_{\alpha\beta} = K_{ab}g^{\mu\alpha}g^{\nu\beta}\left(\partial_\mu B^a_{\nu} - \partial_{\nu}B^a_{\mu} + \lambda f^{a}_{cd}B^{c}_{\mu}B^{d}_{\nu}\right) \left(\partial_{\alpha} B^{b}_{\beta}-\partial_{\beta}B^b_{\alpha}+\lambda f^b_{ef}B^e_{\alpha}B^f_{\beta}\right) \] \textbf{Eq.\eqref{eq:Delta_L}} \\ The derivation of \(\Delta \mathscr{L}_{YM}\) is \[\begin{align*} \Delta \mathscr{L}_{YM} &= \mathscr{L}_{YM}[\tilde{B}^a_\nu(\tilde{x}^\gamma), \tilde{B}^a_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] - \mathscr{L}_{YM}[B^a_\nu(x^\gamma), B^a_{\nu,\mu}(x^\gamma), x^\gamma] \\ &= \mathscr{L}_{YM}[\tilde{B}^a_\nu(\tilde{x}^\gamma), \tilde{B}^a_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] - \mathscr{L}_{YM}[B^a_\nu(\tilde{x}^\gamma), B^a_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] + \mathscr{L}_{YM}[B^a_\nu(\tilde{x}^\gamma), B^a_{\nu,\mu}(\tilde{x}^\gamma), \tilde{x}^\gamma] - \mathscr{L}_{YM}[B^a_\nu(x^\gamma), B^a_{\nu,\mu}(x^\gamma), x^\gamma]\\ &= \left[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_\nu} \delta B^a_\nu + \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)} \delta (\partial_\mu B^a_\nu) \right] (\tilde{x}^\gamma) + \left[ \mathscr{L}_{YM}_{, \gamma} \delta x^\gamma \right] (x^\gamma) \\ & = \left[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_\nu} \delta B^a_\nu + \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)} \delta (\partial_\mu B^a_\nu) \right] (x^\gamma) + O(\delta^2) + \left[ \mathscr{L}_{YM}_{, \gamma} \delta x^\gamma \right] (x^\gamma) \end{align*}\] \textbf{Eq.\eqref{eq:Delta_S}} \\ The derivation of \(\Delta S\) is \[\begin{align*} \Delta S &= \int d^4x \cdot \mathscr{L}_{YM} \delta x^\gamma_{,\gamma} + \int d^4x \left[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_\nu}\delta B^a_\nu + \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}\delta(\partial_\mu B^a_\nu) + \mathscr{L}_{YM}_{,\gamma} \delta x^\gamma \right] \\ &= \int d^4 x \cdot \left[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_\nu}\delta B^a_\nu + \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}\partial_\mu(\delta B^a_\nu) + (\mathscr{L}_{YM} \delta x^\gamma)_{,\gamma} \right]\\ &= \int \left[ \frac{\partial \mathscr{L}_{YM}}{\partial B^a_\nu} - \partial_\mu\left(\frac{\partial\mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}\right) \right] \delta B^a_\nu d^4 x + \int \left[ \partial_\mu\left(\frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)}\delta B^a_\nu\right) + (\mathscr{L}_{YM} \delta x^\gamma)_{,\gamma}\right] d^4 x \end{align*}\] \textbf{Eq.\eqref{eq:Non-ab_partial}} \[\begin{align*} \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B_\nu^a)} = -\frac{1}{4\pi c} K_{ab} g^{\alpha\mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} \end{align*}\] \[\begin{align*} \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} &= \frac{\partial}{\partial(\partial_\mu B^a_\nu)} \left( -\frac{1}{16\pi c} K_{cb} g^{\alpha\rho}g^{\beta\sigma}F^c_{\alpha\beta}F^b_{\rho\sigma}\sqrt{-g} \right) \\ &= -\frac{1}{16\pi c} K_{cb} g^{\alpha\rho}g^{\beta\sigma}\sqrt{-g} \frac{\partial}{\partial(\partial_\mu B^a_\nu)} (F^c_{\alpha\beta}F^b_{\rho\sigma}) \\ &= -\frac{1}{16\pi c} K_{cb} g^{\alpha\rho}g^{\beta\sigma}\sqrt{-g} \left[ \frac{\partial F^c_{\alpha\beta}}{\partial(\partial_\mu B^a_\nu)}F^b_{\rho\sigma} + F^c_{\alpha\beta}\frac{\partial F^b_{\rho\sigma}}{\partial(\partial_\mu B^a_\nu)} \right] \\ &= -\frac{1}{16\pi c} K_{cb} g^{\alpha\rho}g^{\beta\sigma}\sqrt{-g} \left[ (\delta^\mu_\alpha \delta^\nu_\beta - \delta^\nu_\alpha \delta^\mu_\beta) \delta^c_a F^b_{\rho\sigma} + F^c_{\alpha\beta}(\delta^\mu_\rho \delta^\nu_\sigma - \delta^\nu_\rho \delta^\mu_\sigma)\delta^b_a \right] \\ &= -\frac{1}{16\pi c} \sqrt{-g} \left[K_{ab} (g^{\mu\alpha}g^{\nu\beta} - g^{\nu\alpha}g^{\mu\beta})F^b_{\alpha\beta} + K_{ab} F^b_{\alpha\beta}(g^{\mu\alpha}g^{\nu\beta} - g^{\nu\alpha}g^{\mu\beta}) \right] \\ &= -\frac{1}{8\pi c} K_{ab} \sqrt{-g} \left[ (g^{\mu\alpha}g^{\nu\beta} - g^{\nu\alpha}g^{\mu\beta})F^b_{\alpha\beta} \right]\\ &= -\frac{1}{8\pi c} K_{ab} \sqrt{-g} \left[ g^{\mu\alpha}g^{\nu\beta}F^b_{\alpha\beta} - g^{\nu\alpha}g^{\mu\beta}F^c_{\alpha\beta} \right]\\ &= -\frac{1}{8\pi c} K_{ab} \sqrt{-g} \left[ g^{\mu\alpha}g^{\nu\beta}F^b_{\alpha\beta} - g^{\nu\textcolor{red}{\beta}}g^{\mu\textcolor{red}{\alpha}}F^b_{\textcolor{red}{\beta\alpha}} \right] \end{align*}\] Using the antisymmetric property: \(F^b_{\beta\alpha} = -F^b_{\alpha\beta}\), we get \[\begin{align*} \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} &= -\frac{1}{4\pi c} K_{ab} g^{\mu\alpha}g^{\nu\beta} \sqrt{-g} F^b_{\alpha\beta} \end{align*}\] \textbf{Eq.\eqref{eq:(*)}} \[\begin{align*} \underbracket[0.4pt][0pt]{\partial_\mu \left[ \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)} B^a_{\gamma} \delta x^\gamma_{,\nu} \right]}_{(*)} &= \partial_\mu \left[ \left( \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)} B^a_\gamma \delta x^\gamma\right)_{,\nu} - \left( \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)} \right)_{,\nu} B^a_\gamma \delta x^\gamma - \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)} B^a_{\gamma,\nu} \delta x^\gamma \right] \\ &= \underbracket[0.4pt][0pt]{\left[ \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B^a_\nu)} B^a_\gamma \delta x^\gamma \right]_{,\nu \mu}}_{(1)} - \underbracket[0.4pt][0pt]{\partial_\mu \left[ \left( \frac{\partial \mathscr{L}_{YM}}{ \partial (\partial_\mu B^a_\nu)}\right)_{,\nu} B^a_\gamma \delta x^\gamma\right]}_{(2)} - \underbracket[0.4pt][0pt]{\partial_\mu \left[\frac{\partial \mathscr{L}_{YM}}{\partial(\partial_\mu B^a_\nu)}B^a_{\gamma,\nu} \delta x^\gamma \right]}_{(3)} \end{align*}\] \textbf{Eq.\eqref{eq:(*1)}} \[\begin{align*} \left( \frac{\partial \mathscr{L}_{YM}}{\partial (\partial_\mu B_\nu^a)} B_\gamma^a \delta x^\nu \right)_{,\nu\mu} &=\left( -\frac{1}{4\pi c} K_{ab} g^{\alpha\mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} B_\gamma^a \delta x^\nu \right)_{,\nu\mu} \\ &= \left( -\frac{1}{8\pi c} K_{ab} g^{\alpha\mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} B_\gamma^a \delta x^\nu \right)_{,\nu\mu} + \left( -\frac{1}{8\pi c} K_{ab} g^{\alpha\mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} B_\gamma^a \delta x^\nu \right)_{,\nu\mu}\\ &= \left( -\frac{1}{8\pi c} K_{ab} g^{\alpha \mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} B_\gamma^a \delta x^\nu \right)_{,\nu\mu} + \left( -\frac{1}{8\pi c} K_{ab} g^{\textcolor{blue}{\beta}\textcolor{red}{\nu}} g^{\textcolor{blue}{\alpha}\textcolor{red}{\mu}} F_{\textcolor{blue}{\beta\alpha}}^b \sqrt{-g} B_\gamma^a \delta x^\nu \right)_{\textcolor{red}{,\mu\nu}} \\ &= \left( -\frac{1}{8\pi c} K_{ab} g^{\alpha\mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} B_\gamma^a \delta x^\nu \right)_{,\nu\mu} - \left( -\frac{1}{8\pi c} K_{ab} g^{\alpha\mu} g^{\beta\nu} F_{\alpha\beta}^b \sqrt{-g} B_\gamma^a \delta x^\nu \right)_{,\nu\mu} = 0 \end{align*}\] \\ \textbf{Eq.\eqref{eq:(*2)}} \\ The \((*2)\) term rely on the EoM: \[ \left( \frac{\partial \mathscr{L}_{YM}}{\partial(\partial_{\mu}B^{a}_{\nu})} \right)_{,\mu}= \frac{\partial \mathscr{L}_{YM}}{\partial B_{\nu}^{a}} \] \begin{table}[h] \centering \begin{tabular}{|>{\centering\arraybackslash}m{0.8\textwidth}|} % Use 'm' column type for vertical centering and width control \hline \[\begin{align*} \left(\frac{\partial \mathcal{L}}{\partial (\partial_\phi B_\varepsilon^a)} \right)_{,\phi} = \left( -\frac{1}{4\pi c} K_{ab} g^{\alpha\phi} g^{\beta\varepsilon} F_{\alpha\beta}^b \sqrt{-g} \right)_{,\phi} = \frac{\partial \mathcal{L}}{\partial B_\varepsilon^a} \text{ is EoM} \end{align*}\]\\ \hline \[\begin{align*} \left(\frac{\partial \mathcal{L}}{\partial(\partial_{\varepsilon}B^{a}_{\phi})}\right)_{,\phi} = \left( -\frac{1}{4\pi c} K_{ab} g^{\alpha\varepsilon} g^{\beta\phi} F_{\alpha\beta}^b \sqrt{-g} \right)_{,\phi}= \left( -\frac{1}{4\pi c} K_{ab} g^{\textcolor{red}{\beta}\varepsilon} g^{\textcolor{red}{\alpha}\phi} F_{\textcolor{red}{\beta\alpha}}^b \sqrt{-g} \right)_{,\phi} \\ = -\left( -\frac{1}{4\pi c} K_{ab} g^{\alpha\varepsilon} g^{\beta\phi} F_{\textcolor{red}{\alpha\beta}}^b \sqrt{-g} \right)_{,\phi}= -\frac{\partial \mathcal{L}}{\partial B^a_\varepsilon} \end{align*}\] \\ \hline \end{tabular} %\caption{A simple 2 row, 1 column table} % Add caption if you need it \label{tab:two_row_one_column} \end{table} The explicit form of \(\frac{\partial \mathscr{L}_{YM}}{\partial B^a_\mu}\) is: \[\begin{align*} \frac{\partial \mathscr{L}_{YM}}{\partial B^k_\varepsilon} &= -\frac{1}{8\pi c} K_{ab} g^{\mu\alpha} g^{\nu\beta} F_{\alpha\beta}^a \sqrt{-g}\frac{\partial }{\partial B^k_\varepsilon} \left(F_{\mu\nu}^b + \lambda f_{m n}^b B^m_\mu B^n_\nu \right) \\ &=\left(- \frac{1}{8\pi c} K_{ab} g^{\mu\alpha} g^{\nu\beta} F_{\alpha\beta}^a \sqrt{-g}\right) \left( \lambda f_{m n}^b \delta^\varepsilon_\mu \delta^m_k B^n_\nu + \lambda f_{m n}^b B^m_\mu \delta^\varepsilon_\nu \delta^n_k \right)\\ &=\left(- \frac{1}{8\pi c} K_{ab} g^{\mu\alpha} g^{\nu\beta} F_{\alpha\beta}^a \sqrt{-g}\right) \left( \lambda f_{k n}^b \delta^\varepsilon_\mu B^n_\nu + \lambda f_{m k}^b B^m_\mu \delta^\varepsilon_\nu\right)\\ &=\left(- \frac{1}{8\pi c} K_{ab} g^{\mu\alpha} g^{\nu\beta} F_{\alpha\beta}^a \sqrt{-g}\right) \left( \lambda f_{k n}^b \delta^\varepsilon_\mu B^n_\nu + \lambda f_{{\textcolor{red}{n}}k}^b B^{\textcolor{red}{n}}_\mu \delta^\varepsilon_\nu\right)\\ &=\left(- \frac{1}{8\pi c} K_{ab} g^{\mu\alpha} g^{\nu\beta} F_{\alpha\beta}^a \sqrt{-g}\right) \left( \lambda f_{k n}^b \delta^\varepsilon_\mu B^n_\nu \textcolor{red}{-} \lambda f_{{\textcolor{red}{kn}}}^b B^{{n}}_\mu \delta^\varepsilon_\nu\right)\\ &=\left(- \frac{1}{8\pi c} K_{ab} \lambda f_{kn}^b \sqrt{-g} \right) \left( g^{\mu\alpha} g^{\nu\beta} F_{\alpha\beta}^a \delta^\varepsilon_\mu B^n_\nu - g^{\mu\alpha} g^{\nu\beta} F_{\alpha\beta}^a B^{{n}}_\mu \delta^\varepsilon_\nu \right)\\ &=\left(- \frac{1}{8\pi c} K_{ab} \lambda f_{kn}^b \sqrt{-g} \right) \left( g^{\varepsilon\alpha} g^{\nu\beta} F_{\alpha\beta}^a B^n_\nu - g^{\mu\alpha} g^{\varepsilon\beta} F_{\alpha\beta}^a B^{{n}}_\mu \right)\\ &=\left(- \frac{1}{8\pi c} K_{ab} \lambda f_{kn}^b \sqrt{-g} \right) \left( g^{\varepsilon\alpha} g^{\textcolor{red}{\mu}\beta} F_{\alpha\beta}^a B^n_{\textcolor{red}{\mu}} - g^{\mu\textcolor{red}{\beta}} g^{\varepsilon\textcolor{red}{\alpha}} F_{\textcolor{red}{\beta\alpha}}^a B^{{n}}_\mu \right)\\ &=\left(- \frac{1}{8\pi c} K_{ab} \lambda f_{kn}^b \sqrt{-g} \right) \left( g^{\varepsilon\alpha} g^{{\mu}\beta} F_{\alpha\beta}^a \textcolor{red}{+} g^{\varepsilon{\alpha}}g^{\mu{\beta}} F_{\textcolor{red}{\alpha\beta}}^a \right)B^{{n}}_\mu\\ &=\left(- \frac{1}{4\pi c} K_{ab} \lambda f_{kn}^b \sqrt{-g} \right) \left( g^{\varepsilon\alpha} g^{{\mu}\beta} F_{\alpha\beta}^a \right)B^{{n}}_\mu\\ &=\left(- \frac{1}{4\pi c} K_{ab} g^{\varepsilon\alpha} g^{{\mu}\beta} F_{\alpha\beta}^a \sqrt{-g} \right) \lambda f_{kn}^b B^{{n}}_\mu\\ &=\frac{\partial\mathscr{L}_{YM}}{\partial(\partial_{\varepsilon}B_{\mu}^{b})} \lambda f_{kn}^b B^{{n}}_\mu \end{align*}\] The \((*2)\) term is: \[\begin{align*} \underbracket[0.4pt][0pt]{ \left( \frac{\partial \mathscr{L}_{YM}}{ \partial (\partial_\mu B^a_\nu)}\right)_{,\nu} B^a_\gamma \delta x^\gamma}_{(*2)} &=-\frac{\partial \mathcal{L}}{\partial B^a_\mu } B^a_\gamma \delta x^\gamma\\ &=-\left[ \frac{\partial\mathscr{L}_{YM}}{\partial(\partial_{\mu}B_{\nu}^{b})}(\lambda f_{an}^{b}B_{\nu}^{n}) \right] B^a_\gamma \delta x^\gamma\\ &= -\frac{\partial\mathscr{L}_{YM}}{\partial(\partial_{\mu}B_{\nu}^{\textcolor{red}{a}})}\left( \lambda f_{\textcolor{red}{b}n}^{\textcolor{red}{a}}B_{\nu}^{n} B^{\textcolor{red}{b}}_\gamma \delta x^\gamma \right) \end{align*}\] \[\begin{align*} \left(\frac{\partial \mathcal{L}}{\partial (\partial_\phi B_\varepsilon^a)} \right)_{,\phi} = \left( -\frac{1}{4\pi c} K_{ab} g^{\alpha\phi} g^{\beta\varepsilon} F_{\alpha\beta}^b \sqrt{-g} \right)_{,\phi} &= \frac{\partial \mathcal{L}}{\partial B_\varepsilon^a} =\left(- \frac{1}{4\pi c} K_{kb} g^{\varepsilon\alpha} g^{{ \phi}\beta} F_{\alpha\beta}^k \sqrt{-g} \right) \lambda f_{an}^b B^{{n}}_ \phi\\ \left( K_{ab} g^{\alpha\phi} g^{\beta\varepsilon} F_{\alpha\beta}^b \sqrt{-g} \right)_{,\phi} &=\left( K_{\textcolor{red}{kb}} g^{\varepsilon\alpha} g^{{ \phi}\beta} F_{\textcolor{blue}{\beta\alpha}}^{\textcolor{red}{b}} \sqrt{-g} \right) \lambda f_{\textcolor{blue}{na}}^{\textcolor{red}{k}} B^{{n}}_ \phi\\ \left( K_{ab} g^{\varepsilon\beta}g^{\phi\alpha} F_{\alpha\beta}^b \sqrt{-g} \right)_{,\phi} &=\left( K_{{kb}} g^{\varepsilon\textcolor{blue}{\beta}} g^{{ \phi}\textcolor{blue}{\alpha}} F_{\textcolor{blue}{\alpha\beta}}^{{b}} \sqrt{-g} \right) \lambda f_{{na}}^{{k}} B^{{n}}_ \phi\\ \end{align*}\] \newpage \subsection{GR-Palatini Action} \textbf{Eq.\eqref{eq:Einstein}} \\ \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial g^{\mu\nu}} &= \frac{1}{2\kappa}\frac{\partial }{\partial g^{\mu\nu}}\left(\sqrt{-g}g^{\kappa\sigma}\delta^\omega_\varepsilon R^\varepsilon_{\kappa\omega\sigma}\right) = \frac{1}{2\kappa} \sqrt{-g} \left( \delta^\omega_\varepsilon \delta^\kappa_\mu \delta^\sigma_\nu R^\varepsilon_{\kappa\omega\sigma} - \frac{1}{2} g_{\mu\nu} g^{\kappa\sigma} \delta^\omega_\varepsilon R^\varepsilon_{\kappa\omega\sigma} \right) \nonumber \\ &= \frac{1}{2\kappa} \sqrt{-g}\left( \delta^\omega_\varepsilon \delta^\kappa_\mu \delta^\sigma_\nu R^\varepsilon_{\kappa\omega\sigma} - \frac{1}{2} g_{\mu\nu} g^{\kappa\sigma} R_{\kappa\sigma} \right) \nonumber \\ &= \frac{1}{2\kappa} \sqrt{-g}\left( R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \right) \nonumber\\ &= \frac{1}{2\kappa} \sqrt{-g}\left( R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \right) \end{align*}\] \textbf{Eq.\eqref{eq:Fact_1}} \\ \[\begin{align*} \left( \sqrt{-g} \right)_{,\gamma} &= -\frac{1}{2} \sqrt{-g} \, g_{\phi\psi} {g^{\phi\psi}}_{, \gamma} \\ \left( \sqrt{-g} \right)_{;\gamma} &= -\frac{1}{2} \sqrt{-g} \, g_{\phi\psi} {g^{\phi\psi}}_{; \gamma} \\ \end{align*}\] \[\begin{align*} {g^{\mu\nu}}_{;\gamma} &= {g^{\mu\nu}}_{,\gamma} +g^{\eta\nu}\Gamma_{\eta\gamma}^{\mu} +g^{\mu\eta}\Gamma_{\eta\gamma}^{\nu}\\ \to g_{\mu\nu}{g^{\mu\nu}}_{;\gamma} &= g_{\mu\nu}{g^{\mu\nu}}_{,\gamma} +g_{\mu\nu}g^{\eta\nu}\Gamma_{\eta\gamma}^{\mu} +g_{\mu\nu}g^{\mu\eta}\Gamma_{\eta\gamma}^{\nu}\\ &= g_{\mu\nu}{g^{\mu\nu}}_{,\gamma} +\delta^\eta_\mu \Gamma_{\eta\gamma}^{\mu} +\delta^\eta_\nu \Gamma_{\eta\gamma}^{\nu}\\ &= g_{\mu\nu}{g^{\mu\nu}}_{,\gamma} +\Gamma_{\eta\gamma}^{\eta} +\Gamma_{\eta\gamma}^{\eta}\\ &= g_{\mu\nu}{g^{\mu\nu}}_{,\gamma} +2 \Gamma_{\eta\gamma}^{\eta}\\ \to \left( \sqrt{-g} \right)_{;\gamma}= -\frac{1}{2} \sqrt{-g} \, g_{\mu\nu} {g^{\mu\nu}}_{; \gamma} &= -\frac{1}{2} \sqrt{-g} \,\left( g_{\mu\nu}{g^{\mu\nu}}_{,\gamma} +2 \Gamma_{\eta\gamma}^{\eta} \right)\\ &= -\frac{1}{2} \sqrt{-g} g_{\mu\nu}{g^{\mu\nu}}_{,\gamma} - \sqrt{-g} \Gamma_{\eta\gamma}^{\eta}\\ &= \left( \sqrt{-g} \right)_{,\gamma} - \sqrt{-g} \Gamma_{\eta\gamma}^{\eta}\\ \to \frac{1}{\sqrt{-g}}\left( \sqrt{-g} \right)_{;\gamma}&= \frac{1}{\sqrt{-g}}\left( \sqrt{-g} \right)_{,\gamma}-\Gamma_{\eta\gamma}^{\eta} \end{align*}\] \[\begin{align*} \frac{1}{\sqrt{-g}}\left( \sqrt{-g} \, g^{\mu\nu}\right)_{;\gamma} &=\frac{1}{\sqrt{-g}}\left( \sqrt{-g} \, \right)_{;\gamma}g^{\mu\nu} +\frac{1}{\sqrt{-g}} \sqrt{-g} \, {g^{\mu\nu}}_{;\gamma}\\ &=\left( \frac{1}{\sqrt{-g}}\left( \sqrt{-g} \right)_{,\gamma}-\Gamma_{\eta\gamma}^{\eta} \right)g^{\mu\nu} +\frac{1}{\sqrt{-g}} \sqrt{-g} \, \left({g^{\mu\nu}}_{,\gamma} +g^{\eta\nu}\Gamma_{\eta\gamma}^{\mu} +g^{\mu\eta}\Gamma_{\eta\gamma}^{\nu} \right)\\ &=\frac{1}{\sqrt{-g}}\left( \sqrt{-g} \right)_{,\gamma}g^{\mu\nu}-\Gamma_{\eta\gamma}^{\eta}g^{\mu\nu} +\frac{1}{\sqrt{-g}} \sqrt{-g} \, {g^{\mu\nu}}_{,\gamma} +g^{\eta\nu}\Gamma_{\eta\gamma}^{\mu} +g^{\mu\eta}\Gamma_{\eta\gamma}^{\nu}\\ &=\frac{1}{\sqrt{-g}}\left( \sqrt{-g} \, g^{\mu\nu} \right)_{,\gamma}-\Gamma_{\eta\gamma}^{\eta}g^{\mu\nu} +g^{\eta\nu}\Gamma_{\eta\gamma}^{\mu} +g^{\mu\eta}\Gamma_{\eta\gamma}^{\nu}\\ \end{align*}\] \textbf{Eq.\eqref{eq:EoM2_R}} \\ Evaluate \(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu}^\alpha}\): \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu}^\alpha} &= \frac{\partial}{\partial \Gamma_{\mu\nu}^\alpha} \left(\frac{1}{2\kappa} \sqrt{-g} \, g^{\kappa\sigma} \delta_\varepsilon^\omega R_{\kappa\omega\sigma}^\varepsilon \right) \nonumber \\ &=\frac{1}{2\kappa} \sqrt{-g} \, g^{\kappa\sigma} \delta_\varepsilon^\omega \frac{\partial}{\partial \Gamma_{\mu\nu}^\alpha} \left( \Gamma_{\kappa\sigma,\omega}^\varepsilon - \Gamma_{\kappa\omega,\sigma}^\varepsilon + \Gamma_{\eta\omega}^\varepsilon \Gamma_{\kappa\sigma}^\eta - \Gamma_{\eta\sigma}^\varepsilon \Gamma_{\kappa\omega}^\eta \right) \nonumber \\ &= \frac{1}{2\kappa} \sqrt{-g} \, g^{\kappa\sigma} \delta_\varepsilon^\omega \left( \delta_\alpha^\varepsilon \delta_\eta^\mu \delta_\omega^\nu \Gamma_{\kappa\sigma}^\eta + \Gamma_{\eta\omega}^\varepsilon \delta_\alpha^\eta \delta_\kappa^\mu \delta_\sigma^\nu - \delta_\alpha^\varepsilon \delta_\eta^\mu \delta_\sigma^\nu \Gamma_{\kappa\omega}^\eta - \Gamma_{\eta\sigma}^\varepsilon \delta_\alpha^\eta \delta_\kappa^\mu \delta_\omega^\nu \right) \nonumber \\ &=\frac{1}{2\kappa}\left( \sqrt{-g} \, g^{\kappa\sigma} \delta_\alpha^\nu \Gamma_{\kappa\sigma}^\mu +\sqrt{-g}\, g^{\mu\nu} \Gamma_{\alpha\omega}^\omega -\sqrt{-g}\, g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu -\sqrt{-g}\, g^{\mu\sigma} \Gamma_{\alpha\sigma}^\nu \right) \end{align*}\] \textbf{Eq.\eqref{eq:EoM2_D}} \\ Evaluate \(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha}\): \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha} &= \frac{\partial}{\partial \Gamma_{\mu\nu,\gamma}^\alpha} \left(\frac{1}{2\kappa} \sqrt{-g} \, g^{\kappa\sigma} \delta_\varepsilon^\omega R_{\kappa\omega\sigma}^\varepsilon \right) \nonumber\\ &= \frac{1}{2\kappa} \sqrt{-g} \, g^{\kappa\sigma} \delta_\varepsilon^\omega \frac{\partial}{\partial \Gamma_{\mu\nu,\gamma}^\alpha} \left( \Gamma_{\kappa\sigma,\omega}^\varepsilon - \Gamma_{\kappa\omega,\sigma}^\varepsilon + \Gamma_{\eta\omega}^\varepsilon \Gamma_{\kappa\sigma}^\eta - \Gamma_{\eta\sigma}^\varepsilon \Gamma_{\kappa\omega}^\eta \right) \nonumber\\ &=\frac{1}{2\kappa} \sqrt{-g} \, g^{\kappa\sigma} \delta_\varepsilon^\omega \left( \delta_\alpha^\varepsilon \delta_\kappa^\mu \delta_\sigma^\nu \delta_\omega^\gamma - \delta_\alpha^\varepsilon \delta_\kappa^\mu \delta_\omega^\nu \delta_\sigma^\gamma \right) \nonumber\\ &=\frac{1}{2\kappa} \sqrt{-g} \left( g^{\mu\nu} \delta_\alpha^\gamma - g^{\mu\gamma} \delta_\alpha^\nu \right) \end{align*}\] \textbf{Eq.\eqref{eq:EoM2_DD}} \\ Evaluate \( \left( \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha} \right)_{,\gamma} \): \[\begin{align*} \left(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha}\right)_{,\gamma} &=\frac{1}{2\kappa} \left[\sqrt{-g} \left( g^{\mu\nu} \delta_\alpha^\gamma - g^{\mu\gamma} \delta_\alpha^\nu \right)\right]_{,\gamma} \nonumber \\ &=\frac{1}{2\kappa} (\sqrt{-g} \, g^{\mu\nu})_{,\gamma} \delta_\alpha^\gamma- (\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} \delta_\alpha^\nu \nonumber\\ &=\frac{1}{2\kappa} (\sqrt{-g} \, g^{\mu\nu})_{,\alpha} - \frac{1}{2\kappa} (\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} \delta_\alpha^\nu \end{align*}\] \textbf{Eq.\eqref{eq:EoM2_cal}} \\ \begin{adjustwidth}{-2cm}{-1.5cm} \[\begin{align} \label{eq:EoM2_cal_2} \left( \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha} \right)_{,\gamma} &= \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu}^\alpha} \nonumber \\ \underbracket[0.4pt][0pt]{(\sqrt{-g} \, g^{\mu\nu})_{,\alpha}}_{(A)} - \underbracket[0.4pt][0pt]{(\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} \delta_\alpha^\nu}_{(B)} &= \underbracket[0.4pt][0pt]{\sqrt{-g} \, g^{\kappa\sigma} \delta_\alpha^\nu \Gamma_{\kappa\sigma}^\mu}_{(C)} + \underbracket[0.4pt][0pt]{\sqrt{-g}\, g^{\mu\nu} \Gamma_{\alpha\omega}^\omega}_{(D)} -\underbracket[0.4pt][0pt]{\sqrt{-g}\, g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu}_{(E)} -\underbracket[0.4pt][0pt]{\sqrt{-g}\, g^{\mu\sigma} \Gamma_{\alpha\sigma}^\nu}_{(F)} \nonumber \\ \underbracket[0.4pt][0pt]{(\sqrt{-g} \, g^{\mu\nu})_{,\alpha}}_{(A)} +\underbracket[0.4pt][0pt]{\sqrt{-g}\, g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu}_{(E)} &=\underbracket[0.4pt][0pt]{(\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} \delta_\alpha^\nu}_{(B)}+ \underbracket[0.4pt][0pt]{\sqrt{-g} \, g^{\kappa\sigma} \delta_\alpha^\nu \Gamma_{\kappa\sigma}^\mu}_{(C)} + \underbracket[0.4pt][0pt]{\sqrt{-g}\, g^{\mu\nu} \Gamma_{\alpha\omega}^\omega}_{(D)} -\underbracket[0.4pt][0pt]{\sqrt{-g}\, g^{\mu\sigma} \Gamma_{\alpha\sigma}^\nu}_{(F)} \nonumber \\ \frac{1}{\sqrt{-g}}(\sqrt{-g} \, g^{\mu\nu})_{,\alpha} + g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu &= \underbracket[0.4pt][0pt]{ \left[\frac{1}{\sqrt{-g}} (\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} + g^{\kappa\sigma} \Gamma_{\kappa\sigma}^\mu\right]}_{(B)+(C)} \delta_\alpha^\nu+g^{\mu\nu} \Gamma_{\alpha\textcolor{red}{\gamma}}^{\textcolor{red}{\gamma}} - g^{\mu\textcolor{red}{\kappa}} \Gamma_{\alpha\textcolor{red}{\kappa}}^\nu \nonumber\\ \underbrace{\frac{1}{\sqrt{-g}}(\sqrt{-g} \, g^{\mu\nu})_{,\alpha} \textcolor{blue}{-g^{\mu\nu}\Gamma^\gamma_{\gamma\alpha}} + g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu + \textcolor{red}{ g^{\mu\kappa}\Gamma^\nu_{\kappa\alpha}}}_{\eqref{eq:Fact_1}={\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha}} &=\underbrace{\left[ \frac{1}{\sqrt{-g}} (\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} + g^{\textcolor{red}{\eta\gamma}} \Gamma_{\textcolor{red}{\eta\gamma}}^\mu \right] }_{(\#)} \delta_\alpha^\nu + \underbrace{g^{\mu\nu} \Gamma_{\alpha\gamma}^{\gamma} \textcolor{blue}{-g^{\mu\nu}\Gamma^\gamma_{\gamma\alpha}}}_{g^{\mu\nu} T_{\alpha\gamma}^{\gamma} } + \underbrace{ \textcolor{red}{ g^{\mu\kappa}\Gamma^\nu_{\kappa\alpha}} - g^{\mu\kappa} \Gamma_{\alpha\kappa}^\nu }_{g^{\mu\kappa}T^\nu_{\kappa\alpha}} \end{align}\] \end{adjustwidth} %\hspace*{-2.5cm} Calculate \((\#)\) term: \[\begin{align*} \underbrace{\left[ \frac{1}{\sqrt{-g}} (\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} + g^{\eta\gamma} \Gamma_{\eta\gamma}^\mu \right]}_{(\#)} &= \underbrace{\frac{1}{\sqrt{-g}} (\sqrt{-g} \, g^{\mu\gamma})_{,\gamma} \textcolor{blue}{-g^{\mu\gamma}\Gamma^\eta_{\eta\gamma}} + g^{\eta\gamma} \Gamma_{\eta\gamma}^\mu \textcolor{red}{+ g^{\mu\eta}\Gamma^\gamma_{\eta\gamma}} }_{\eqref{eq:Fact_1}={\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\gamma}\right)}_{;\gamma}} \textcolor{blue}{+}\underbrace{\textcolor{blue}{g^{\mu\gamma}\Gamma^\eta_{\eta\gamma}} \textcolor{red}{- g^{\mu\eta}\Gamma^\gamma_{\eta\gamma}}}_{g^{\mu\gamma}\, T^\eta_{\eta\gamma}}\\ &= {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\gamma}\right)}_{;\gamma}+g^{\mu\gamma}\, T^\eta_{\eta\gamma} \end{align*}\] Eq.\eqref{eq:EoM2_cal_2} become \[\begin{align} {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha} &=\underbrace{\left[ {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\gamma}\right)}_{;\gamma}+g^{\mu\gamma}\, T^\eta_{\eta\gamma}\right] }_{(\#)} \delta_\alpha^\nu + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\kappa}T^\nu_{\kappa\alpha} \end{align}\] \textbf{Eq.\eqref{eq:EoM2_result}}\\ Starting from Eq.\eqref{eq:EoM2_cal} \[\begin{align*} {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha} &=\left[ {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\gamma}\right)}_{;\gamma}+g^{\mu\gamma}\, T^\eta_{\eta\gamma}\right] \delta_\alpha^\nu + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\kappa}T^\nu_{\kappa\alpha}\\ {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha} -\delta_\alpha^\nu {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\gamma}\right)}_{;\gamma} &= \delta_\alpha^\nu g^{\mu\gamma}\, T^\eta_{\eta\gamma} + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\kappa}T^\nu_{\kappa\alpha}\\ \delta^\alpha_\nu {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha} -\delta^\alpha_\nu \delta_\alpha^\nu {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\gamma}\right)}_{;\gamma} &= \delta^\alpha_\nu\delta_\alpha^\nu g^{\mu\gamma}\, T^\eta_{\eta\gamma} + \delta^\alpha_\nu g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + \delta^\alpha_\nu g^{\mu\kappa}T^\nu_{\kappa\alpha}\\ {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\nu} -\mathbb{D} {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\gamma}\right)}_{;\gamma} &= \mathbb{D}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} + g^{\mu\nu} T_{\nu\gamma}^{\gamma} + g^{\mu\kappa}T^\nu_{\kappa\nu}\\ \left(1-\mathbb{D}\right){\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\nu} &= \left(\mathbb{D}-2\right)\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \\ {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\nu} &=-\frac{\left(\mathbb{D}-2\right)}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \end{align*}\] Substituting back into Eq.\eqref{eq:EoM2_cal} \[\begin{align*} {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha} &=\left( -\frac{\left(\mathbb{D}-2\right)}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} +g^{\mu\gamma}\, T^\eta_{\eta\gamma} \right) \delta_\alpha^\nu + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\kappa}T^\nu_{\kappa\alpha}\\ &=\frac{1}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\gamma}T^\nu_{\gamma\alpha} \end{align*}\] Furthermore, this equation must be symmetric on \(\mu\leftrightarrow\nu\): \begin{adjustwidth}{-2cm}{-1.5cm} \[\begin{align*} {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha} -{\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\nu\mu}\right)}_{;\alpha} =0 &=\left( \frac{1}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu + \cancel{g^{\mu\nu} T_{\alpha\gamma}^{\gamma}} + g^{\mu\gamma}T^\nu_{\gamma\alpha} \right) -\left( \frac{1}{\left(\mathbb{D}-1\right)}\, g^{\nu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\mu + \cancel{g^{\nu\mu} T_{\alpha\gamma}^{\gamma} }+ g^{\nu\gamma}T^\mu_{\gamma\alpha} \right)\\ &= \frac{1}{\left(\mathbb{D}-1\right)}\, \left(g^{\mu\gamma}\delta_\alpha^\nu -g^{\nu\gamma}\, \delta_\alpha^\mu \right)\, T^\eta_{\eta\gamma} \, +g^{\mu\gamma}T^\nu_{\gamma\alpha} -g^{\nu\gamma}T^\mu_{\gamma\alpha}\\ &= \frac{1}{\left(\mathbb{D}-1\right)}\, \left(g^{\mu\gamma}\delta_\alpha^\nu -g^{\nu\gamma}\, \delta_\alpha^\mu \right)\, T^\eta_{\eta\gamma} \, +\left( g^{\mu\gamma}\delta^\nu_\eta -g^{\nu\gamma}\delta^\mu_\eta \right)T^\eta_{\gamma\alpha}\\ &= \frac{1}{\left(\mathbb{D}-1\right)}\, \left(g^{\mu\gamma}\delta_\beta^\nu -g^{\nu\gamma}\, \delta_\beta^\mu \right)\delta^\beta_\alpha T^\eta_{\eta\gamma} \, -\left( g^{\mu\gamma}\delta^\nu_\beta -g^{\nu\gamma}\delta^\mu_\beta \right)T^\beta_{\alpha\gamma}\\ &= \left(g^{\mu\gamma}\delta_\beta^\nu -g^{\nu\gamma}\, \delta_\beta^\mu \right) \left( \frac{1}{\left(\mathbb{D}-1\right)} \delta^\beta_\alpha T^\eta_{\eta\gamma} \, -T^\beta_{\alpha\gamma} \right) \end{align*}\] \end{adjustwidth} \[\begin{align*} 0&= \left(g^{\mu\gamma}\delta_\beta^\nu -g^{\nu\gamma}\, \delta_\beta^\mu \right) \left( \frac{1}{\left(\mathbb{D}-1\right)} \delta^\beta_\alpha T^\eta_{\eta\gamma} \, -T^\beta_{\alpha\gamma} \right)\\ 0&= g_{\mu\phi}g_{\nu\psi}\left(g^{\mu\gamma}\delta_\beta^\nu -g^{\nu\gamma}\, \delta_\beta^\mu \right) \left( \frac{1}{\left(\mathbb{D}-1\right)} \delta^\beta_\alpha T^\eta_{\eta\gamma} \, -T^\beta_{\alpha\gamma} \right)\\ &=\left( g_{\nu\psi}g^{\mu\gamma} \delta^\gamma_\phi \delta_\beta^\nu -g_{\mu\phi}g_{\nu\psi} g^{\nu\gamma}\, \delta_\beta^\mu \right) \left( \frac{1}{\left(\mathbb{D}-1\right)} \delta^\beta_\alpha T^\eta_{\eta\gamma} \, -T^\beta_{\alpha\gamma} \right)\\ &=\left( g_{\beta\psi} \delta^\gamma_\phi -g_{\beta\phi}\delta^\gamma_\psi \right) \left( \frac{1}{\left(\mathbb{D}-1\right)} \delta^\beta_\alpha T^\eta_{\eta\gamma} \, -T^\beta_{\alpha\gamma} \right)\\ \left( g_{\beta\psi} \delta^\gamma_\phi -g_{\beta\phi}\delta^\gamma_\psi \right) T^\beta_{\alpha\gamma} &=\left( g_{\beta\psi} \delta^\gamma_\phi -g_{\beta\phi}\delta^\gamma_\psi \right) \frac{1}{\left(\mathbb{D}-1\right)} \delta^\beta_\alpha T_{\gamma} \\ g_{\beta\psi} \delta^\gamma_\phi T^\beta_{\alpha\gamma} -g_{\beta\phi}\delta^\gamma_\psi T^\beta_{\alpha\gamma} &= g_{\beta\psi} \delta^\gamma_\phi \frac{1}{\left(\mathbb{D}-1\right)} \delta^\beta_\alpha T_{\gamma} -g_{\beta\phi}\delta^\gamma_\psi \frac{1}{\left(\mathbb{D}-1\right)} \delta^\beta_\alpha T_{\gamma} \\ T_{\psi\alpha\phi} - T_{\phi\alpha\psi} &= \frac{1}{\left(\mathbb{D}-1\right)} \left(g_{\alpha\psi} T_{\phi} -g_{\alpha\phi} T_{\psi} \right) \end{align*}\] !!!!!!!!!!!!!!!!!!!!!!!!! \subsection*{第 1 步：證明您的方程式隱含 \(T_{[\psi\alpha\phi]} = 0\)} 我們的目標是證明這個方程式的輪換和（cyclic sum）為零，即： \[ (T_{\psi\alpha\phi} - T_{\phi\alpha\psi}) + (T_{\alpha\phi\psi} - T_{\psi\phi\alpha}) + (T_{\phi\psi\alpha} - T_{\alpha\psi\phi}) = 0 \] 原始方程式: \begin{equation} T_{\psi\alpha\phi} - T_{\phi\alpha\psi} = \frac{1}{\mathbb{D}-1} (g_{\alpha\psi} T_{\phi} - g_{\alpha\phi} T_{\psi}) \label{eq:original} \end{equation} 第一次輪換 (\(\psi \to \alpha \to \phi \to \psi\)): \begin{equation} T_{\alpha\phi\psi} - T_{\psi\phi\alpha} = \frac{1}{\mathbb{D}-1} (g_{\phi\alpha} T_{\psi} - g_{\phi\psi} T_{\alpha}) \label{eq:perm1} \end{equation} 第二次輪換 (\(\alpha \to \phi \to \psi \to \alpha\)): \begin{equation} T_{\phi\psi\alpha} - T_{\alpha\psi\phi} = \frac{1}{\mathbb{D}-1} (g_{\psi\phi} T_{\alpha} - g_{\psi\alpha} T_{\phi}) \label{eq:perm2} \end{equation} 現在，我們將這三個方程式 \eqref{eq:original}, \eqref{eq:perm1}, \eqref{eq:perm2} 全部相加。 \paragraph{觀察右側 (RHS) 的和：} \[ \begin{split} \text{RHS Sum} = \frac{1}{\mathbb{D}-1} [ &(g_{\alpha\psi} T_{\phi} - g_{\alpha\phi} T_{\psi}) \\ + &(g_{\phi\alpha} T_{\psi} - g_{\phi\psi} T_{\alpha}) \\ + &(g_{\psi\phi} T_{\alpha} - g_{\psi\alpha} T_{\phi}) ] \end{split} \] 由於度規張量是對稱的 (\(g_{\alpha\phi} = g_{\phi\alpha}\))，我們可以重新組合這些項： \[ \text{RHS Sum} = \frac{1}{\mathbb{D}-1} [ (g_{\alpha\psi} - g_{\psi\alpha}) T_{\phi} + (g_{\phi\alpha} - g_{\alpha\phi}) T_{\psi} + (g_{\psi\phi} - g_{\phi\psi}) T_{\alpha} ] = 0 \] 右側所有項完全抵消，其和為零。 \paragraph{觀察左側 (LHS) 的和：} 由於右側的和為零，左側的和也必須為零： \[ (T_{\psi\alpha\phi} - T_{\phi\alpha\psi}) + (T_{\alpha\phi\psi} - T_{\psi\phi\alpha}) + (T_{\phi\psi\alpha} - T_{\alpha\psi\phi}) = 0 \] 現在，我們利用撓率張量在其最後兩個指標上的基本反對稱性 (\(T_{abc} = -T_{acb}\)) 來改寫帶有負號的項： \[\begin{align*} -T_{\phi\alpha\psi} &= +T_{\phi\psi\alpha} \\ -T_{\psi\phi\alpha} &= +T_{\psi\alpha\phi} \\ -T_{\alpha\psi\phi} &= +T_{\alpha\phi\psi} \end{align*}\] 將這些代回 LHS 的和中： \[ (T_{\psi\alpha\phi} + T_{\phi\psi\alpha}) + (T_{\alpha\phi\psi} + T_{\psi\alpha\phi}) + (T_{\phi\psi\alpha} + T_{\alpha\phi\psi}) = 0 \] 整理後得到： \[ 2(T_{\psi\alpha\phi} + T_{\alpha\phi\psi} + T_{\phi\psi\alpha}) = 0 \] 這就直接證明了： \begin{equation} T_{\psi\alpha\phi} + T_{\alpha\phi\psi} + T_{\phi\psi\alpha} = 0 \label{eq:cyclic_sum_zero} \end{equation} 這意味著撓率張量的全反對稱部分 \(T_{[\psi\alpha\phi]}\) 必定為零。這不是一個假設，而是您所提供方程式的一個直接數學推論。 \subsection*{第 2 步：利用 \(T_{[\psi\alpha\phi]} = 0\) 解出半對稱形式} 現在我們有了兩個方程式： \begin{enumerate} \item 您的原始方程式（經過反對稱性改寫）： \begin{equation} T_{\psi\alpha\phi} + T_{\phi\psi\alpha} = \frac{1}{\mathbb{D}-1} (g_{\alpha\psi} T_{\phi} - g_{\alpha\phi} T_{\psi}) \label{eq:original_rewritten} \end{equation} \item 我們剛剛證明的輪換和為零： \begin{equation} T_{\psi\alpha\phi} + T_{\alpha\phi\psi} + T_{\phi\psi\alpha} = 0 \label{eq:cyclic_sum_zero_2} \end{equation} \end{enumerate} 從第二個方程式 \eqref{eq:cyclic_sum_zero_2}，我們可以得到： \[ T_{\phi\psi\alpha} = -T_{\psi\alpha\phi} - T_{\alpha\phi\psi} \] 將這個結果代入第一個方程式 \eqref{eq:original_rewritten} 的左側： \[ T_{\psi\alpha\phi} + (-T_{\psi\alpha\phi} - T_{\alpha\phi\psi}) = \frac{1}{\mathbb{D}-1} (g_{\alpha\psi} T_{\phi} - g_{\alpha\phi} T_{\psi}) \] \[ -T_{\alpha\phi\psi} = \frac{1}{\mathbb{D}-1} (g_{\alpha\psi} T_{\phi} - g_{\alpha\phi} T_{\psi}) \] 再次使用基本反對稱性改寫左側：\(-T_{\alpha\phi\psi} = +T_{\alpha\psi\phi}\)。 \[ T_{\alpha\psi\phi} = \frac{1}{\mathbb{D}-1} (g_{\alpha\psi} T_{\phi} - g_{\alpha\phi} T_{\psi}) \] 這個形式已經是最終答案了。為了使其看起來更符合標準寫法，我們將指標進行重新標記（\(\alpha \to \psi, \psi \to \alpha, \phi \to \phi\)）： \[ T_{\psi\alpha\phi} = \frac{1}{\mathbb{D}-1} (g_{\psi\alpha} T_{\phi} - g_{\psi\phi} T_{\alpha}) \] \[ T^\psi_{\alpha\phi} = \frac{1}{\mathbb{D}-1} (\delta^\psi_{\alpha} T_{\phi} - \delta^\psi_{\phi} T_{\alpha}) \] 是一個充要條件，它本身就足以將撓率張量的形式唯一地確定為半對稱撓率。 !!!!!!!!!!!!!!!!!!!!!!!!! \[\begin{align*} -\frac{1}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\gamma\eta} \, \delta_\alpha^\nu + g^{\mu\kappa}T^\nu_{\kappa\alpha} &=-\frac{1}{\left(\mathbb{D}-1\right)}\, g^{\nu\gamma}\, T^\eta_{\gamma\eta} \, \delta_\alpha^\mu + g^{\nu\kappa}T^\mu_{\kappa\alpha} \\ \end{align*}\] \[\begin{align*} {\frac{1}{\sqrt{-g}} g_{\mu\nu}\left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha} &=\frac{1}{\left(\mathbb{D}-1\right)}\, g_{\mu\nu} g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu + g_{\mu\nu} g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g_{\mu\nu}g^{\mu\kappa}T^\nu_{\kappa\alpha}\\ {\frac{1}{\sqrt{-g}} g_{\mu\nu} \left(\sqrt{-g}\,\right)}_{;\alpha} g^{\mu\nu} +\frac{1}{\sqrt{-g}} g_{\mu\nu} \sqrt{-g}\,{g^{\mu\nu}}_{;\alpha} &=\frac{1}{\left(\mathbb{D}-1\right)}\, \delta^\gamma_\nu \, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu + \mathbb{D} T_{\alpha\gamma}^{\gamma} + \delta^\kappa_\nu T^\nu_{\kappa\alpha}\\ {\frac{\mathbb{D}}{\sqrt{-g}} \left(\sqrt{-g}\,\right)}_{;\alpha} -\frac{2}{\sqrt{-g}} \left( -\frac{1}{2}\sqrt{-g}\, g_{\mu\nu} {g^{\mu\nu}}_{;\alpha} \right) &=\frac{1}{\left(\mathbb{D}-1\right)}T^\eta_{\eta\alpha} + \mathbb{D} T_{\alpha\gamma}^{\gamma} + T^\nu_{\nu\alpha}\\ {\frac{\mathbb{D}}{\sqrt{-g}} \left(\sqrt{-g}\,\right)}_{;\alpha} -\frac{2}{\sqrt{-g}} \left(\sqrt{-g}\,\right)_{;\alpha} &=-\frac{\mathbb{D}\left(\mathbb{D}-2\right)}{\mathbb{D}-1}T^\eta_{\eta\gamma} \\ {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,\right)}_{;\alpha} &=-\frac{\mathbb{D}}{\mathbb{D}-1}T^\eta_{\eta\gamma} \end{align*}\] \[\begin{align*} {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\,g^{\mu\nu}\right)}_{;\alpha} &=\frac{1}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\gamma}T^\nu_{\gamma\alpha} \\ {\frac{1}{\sqrt{-g}} \left(\sqrt{-g}\right)}_{;\alpha} g^{\mu\nu} +{g^{\mu\nu}}_{;\alpha} &=\frac{1}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\gamma}T^\nu_{\gamma\alpha} \\ -\frac{\mathbb{D}}{\mathbb{D}-1}T^\eta_{\eta\alpha} g^{\mu\nu} +{g^{\mu\nu}}_{;\alpha} &=\frac{1}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\gamma}T^\nu_{\gamma\alpha} \\ {g^{\mu\nu}}_{;\alpha} &=\frac{1}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu +\frac{\mathbb{D}}{\mathbb{D}-1}T^\eta_{\eta\alpha} g^{\mu\nu} + g^{\mu\nu} T_{\alpha\gamma}^{\gamma} + g^{\mu\gamma}T^\nu_{\gamma\alpha}\\ {g^{\mu\nu}}_{;\alpha} &=\frac{1}{\left(\mathbb{D}-1\right)}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu +\frac{1}{\left(\mathbb{D}-1\right)}T^\eta_{\eta\alpha} g^{\mu\nu}+ g^{\mu\gamma}T^\nu_{\gamma\alpha}\\ &= \frac{1}{\left(\mathbb{D}-1\right)}\, \cancel{g^{\mu\gamma}\, T_{\gamma} \, \delta_\alpha^\nu } +\frac{1}{\left(\mathbb{D}-1\right)}T_{\alpha} g^{\mu\nu}+ g^{\mu\gamma} \frac{1}{\mathbb{D}-1} \left( \delta^\nu_{\gamma} T_{\alpha} - \cancel{\delta^\nu_{\alpha} T_{\gamma}} \right)\\ &= \frac{1}{\left(\mathbb{D}-1\right)}T_{\alpha} g^{\mu\nu}+ \frac{1}{\mathbb{D}-1} g^{\mu\gamma} \delta^\nu_{\gamma} T_{\alpha} \\ &= \frac{2}{\left(\mathbb{D}-1\right)}T_{\alpha} g^{\mu\nu} \end{align*}\] \[ T^\psi_{\alpha\phi} = \frac{1}{\mathbb{D}-1} (\delta^\psi_{\alpha} T_{\phi} - \delta^\psi_{\phi} T_{\alpha}) \] \[ T^\nu_{\gamma\alpha} = \frac{1}{\mathbb{D}-1} (\delta^\nu_{\gamma} T_{\alpha} - \delta^\nu_{\alpha} T_{\gamma}) \] 驗證 \[\begin{align*} \sqrt{-g}_{\,;\gamma} &=-\frac{1}{2}\sqrt{-g}\,g_{\mu\nu} {g^{\mu\nu}}_{;\gamma}\\ &=-\frac{1}{2}\sqrt{-g}\,g_{\mu\nu} \frac{2}{\left(\mathbb{D}-1\right)}T_{\gamma} g^{\mu\nu}\\ &=-\frac{\mathbb{D}}{\left(\mathbb{D}-1\right)}\sqrt{-g}\, T_{\gamma} \\ \end{align*}\] Consist! If \[\begin{align*} \Gamma^\mu_{\alpha\nu}=\tilde{\Gamma}^\mu_{\alpha\nu}+C^\mu_{\alpha\nu} \end{align*}\] Define: \[\begin{align*} C^\mu_{\alpha\nu}&=\frac{1}{\mathbb{D}-1} \left( \delta^\mu_\alpha T_\nu +\delta^\mu_\nu T_\alpha -g_{\alpha\nu}g^{\mu\sigma} T_\sigma \right) \end{align*}\] \begin{adjustwidth}{-2cm}{-1.5cm} \[\begin{align*} &{g^{\mu\nu}}_{;\alpha} -\frac{2}{\left(\mathbb{D}-1\right)}T_{\alpha} g^{\mu\nu}=0 \\=& {g^{\mu\nu}}_{,\alpha} + g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu + g^{\mu\kappa}\Gamma^\nu_{\kappa\alpha} -\frac{2}{\left(\mathbb{D}-1\right)}T_{\alpha} g^{\mu\nu}\\ =&{g^{\mu\nu}}_{,\alpha} + g^{\kappa\nu} \tilde{\Gamma}_{\kappa\alpha}^\mu + g^{\kappa\nu} C_{\kappa\alpha}^\mu + g^{\mu\kappa}\tilde{\Gamma}^\nu_{\kappa\alpha} + g^{\mu\kappa}C^\nu_{\kappa\alpha} -\frac{2}{\left(\mathbb{D}-1\right)}T_{\alpha} g^{\mu\nu}\\ =&{g^{\mu\nu}}_{,\alpha} + g^{\kappa\nu} \tilde{\Gamma}_{\kappa\alpha}^\mu + g^{\kappa\nu} \frac{1}{\mathbb{D}-1} \left( \delta^\mu_\kappa T_\alpha +\delta^\mu_\alpha T_\kappa -g_{\kappa\alpha}g^{\mu\sigma} T_\sigma \right) + g^{\mu\kappa}\tilde{\Gamma}^\nu_{\kappa\alpha} + g^{\mu\kappa}\frac{1}{\mathbb{D}-1} \left( \delta^\nu_\kappa T_\alpha +\delta^\nu_\alpha T_\kappa -g_{\kappa\alpha}g^{\mu\sigma} T_\sigma \right) -\frac{2}{\left(\mathbb{D}-1\right)}T_{\alpha} g^{\mu\nu}\\ =&{g^{\mu\nu}}_{,\alpha} + g^{\kappa\nu} \tilde{\Gamma}_{\kappa\alpha}^\mu + \frac{1}{\mathbb{D}-1} \left( g^{\mu\nu} T_\alpha +\delta^\mu_\alpha T^\nu -g^{\kappa\nu}g_{\kappa\alpha} T^\mu \right) + g^{\mu\kappa}\tilde{\Gamma}^\nu_{\kappa\alpha} + \frac{1}{\mathbb{D}-1} \left( g^{\mu\nu} T_\alpha +\delta^\nu_\alpha T^\mu -g^{\mu\kappa}g_{\kappa\alpha} T^\mu \right) -\frac{2}{\left(\mathbb{D}-1\right)}T_{\alpha} g^{\mu\nu}\\ =&{g^{\mu\nu}}_{,\alpha} + g^{\kappa\nu} \tilde{\Gamma}_{\kappa\alpha}^\mu + \frac{1}{\mathbb{D}-1} \left( \xcancel{g^{\mu\nu} T_\alpha} +\cancel{\delta^\mu_\alpha T^\nu} -\bcancel{\delta^\nu_\alpha T^\mu} \right) + g^{\mu\kappa}\tilde{\Gamma}^\nu_{\kappa\alpha} + \frac{1}{\mathbb{D}-1} \left( \xcancel{g^{\mu\nu} T_\alpha} +\bcancel{\delta^\nu_\alpha T^\mu} -\cancel{\delta^\mu_\alpha T^\mu} \right) -\xcancel{\frac{2}{\left(\mathbb{D}-1\right)}T_{\alpha} g^{\mu\nu}}\\ =&{g^{\mu\nu}}_{,\alpha} + g^{\kappa\nu} \tilde{\Gamma}_{\kappa\alpha}^\mu + g^{\mu\kappa}\tilde{\Gamma}^\nu_{\kappa\alpha} \end{align*}\] \end{adjustwidth} If trosion-free, i.e., \(T^\alpha_{\beta\gamma}=0\), hence: \[\begin{align*} {g^{\mu\nu}}_{;\alpha} =0 \end{align*}\] is metric compatible. On the otherhand, if metric compatible, i.e., \({g^{\mu\nu}}_{;\alpha} =0\), hence: \[\begin{align} 0&=\frac{1}{3}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu + \frac{1}{3} g^{\mu\nu} T_{\gamma\alpha}^{\gamma} + g^{\mu\kappa}T^\nu_{\kappa\alpha} \label{eq:MT}\\ 0&=\delta^\alpha_\nu \frac{1}{3}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \, \delta_\alpha^\nu + \delta^\alpha_\nu \frac{1}{3} g^{\mu\nu} T_{\gamma\alpha}^{\gamma} + \delta^\alpha_\nu g^{\mu\kappa}T^\nu_{\kappa\alpha} \nonumber\\ 0&=\frac{4}{3}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} + \frac{1}{3} g^{\mu\nu} T_{\gamma\nu}^{\gamma} + g^{\mu\kappa}T^\nu_{\kappa\nu} \nonumber\\ 0&=- \frac{1}{3}\, g^{\mu\gamma}\, T^\eta_{\eta\gamma} \nonumber\\ 0&=T^\eta_{\eta\gamma} \nonumber \end{align}\] Put back into Eq.\eqref{eq:MT}: \[\begin{align*} 0&=0 + 0+g^{\mu\kappa}T^\nu_{\kappa\alpha} \\ 0&=T^\nu_{\kappa\alpha} \end{align*}\] imply torsion-free.\\ \subsection{Contorsion} \[\begin{align*} T^{\mu}_{\alpha\beta} &=\Gamma^{\mu}_{\alpha\beta} -\Gamma^{\mu}_{\beta\alpha} \\ \Gamma^{\mu}_{\alpha\beta} &=\left\{^{\mu}_{\alpha\beta}\right\}+K^{\mu}_{\alpha\beta} \\ K^{\mu}_{\alpha\beta} &= \frac{1}{2}g^{\mu\gamma} \left(T^{\xi}_{\alpha\beta}g_{\xi\gamma} + T^{\xi}_{\beta\gamma}g_{\xi\alpha} + T^{\xi}_{\alpha\gamma}g_{\xi\beta}\right) \\ K^{\mu}_{\beta\alpha} &= \frac{1}{2}g^{\mu\gamma} \left(T^{\xi}_{\beta\alpha}g_{\xi\gamma} + T^{\xi}_{\alpha\gamma}g_{\xi\beta} + T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right) \\ K^{\mu}_{\alpha\beta} -K^{\mu}_{\beta\alpha} &=\frac{1}{2}g^{\mu\gamma} \left(T^{\xi}_{\alpha\beta}g_{\xi\gamma} + \cancel{T^{\xi}_{\beta\gamma}g_{\xi\alpha} } + \bcancel{T^{\xi}_{\alpha\gamma}g_{\xi\beta}}\right) - \frac{1}{2}g^{\mu\gamma} \left(T^{\xi}_{\beta\alpha}g_{\xi\gamma} + \bcancel{T^{\xi}_{\alpha\gamma}g_{\xi\beta}} + \cancel{T^{\xi}_{\beta\gamma}g_{\xi\alpha} } \right)\\ &=g^{\mu\gamma}T^{\xi}_{\alpha\beta}g_{\xi\gamma} =T^{\mu}_{\alpha\beta} \\ \left\{^{\mu}_{\alpha\beta}\right\}-\left\{^{\mu}_{\beta\alpha}\right\} &= \Gamma^{\mu}_{\alpha\beta} -K^{\mu}_{\alpha\beta} -\Gamma^{\mu}_{\beta\alpha} +K^{\mu}_{\beta\alpha} =T^{\mu}_{\alpha\beta}-T^{\mu}_{\alpha\beta}=0 \end{align*}\] \[\begin{align*} {g^{\mu\nu}}_{;\alpha} &= {g^{\mu\nu}}_{,\alpha} + g^{\beta\nu}\Gamma^{\mu}_{\beta\alpha} + g^{\mu\beta}\Gamma^{\nu}_{\beta\alpha} \\ &= {g^{\mu\nu}}_{,\alpha} + g^{\beta\nu}\left(\left\{^{\mu}_{\beta\alpha} \right\} + K^{\mu}_{\beta\alpha}\right) + g^{\mu\beta}\left(\left\{^{\nu}_{\beta\alpha}\right\} + K^{\nu}_{\beta\alpha}\right) \\ &= {g^{\mu\nu}}_{,\alpha} + g^{\beta\nu}\left\{^{\mu}_{\beta\alpha}\right\} + g^{\mu\beta}\left\{^{\nu}_{\beta\alpha}\right\} + g^{\beta\nu}\left(\frac{1}{2}g^{\mu\gamma}\left( T^{\xi}_{\beta\alpha}g_{\xi\gamma} + T^{\xi}_{\alpha\gamma}g_{\xi\beta} + T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right) \right) \\ &\qquad + g^{\mu\beta}\left(\frac{1}{2}g^{\nu\gamma}\left( T^{\xi}_{\beta\alpha}g_{\xi\gamma} + T^{\xi}_{\alpha\gamma}g_{\xi\beta} + T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right)\right) \\ &= {g^{\mu\nu}}_{,\alpha} + g^{\beta\nu}\left\{^{\mu}_{\beta\alpha}\right\} + g^{\mu\beta}\left\{^{\nu}_{\beta\alpha}\right\} + \frac{1}{2}g^{\beta\nu}g^{\mu\gamma}\left( T^{\xi}_{\beta\alpha}g_{\xi\gamma} + T^{\xi}_{\alpha\gamma}g_{\xi\beta} + T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right) \\ &\qquad + \frac{1}{2}g^{\mu\beta}g^{\nu\gamma}\left( T^{\xi}_{\beta\alpha}g_{\xi\gamma} + T^{\xi}_{\alpha\gamma}g_{\xi\beta} + T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right)\\ &= {g^{\mu\nu}}_{,\alpha} + g^{\beta\nu}\left\{^{\mu}_{\beta\alpha}\right\} + g^{\mu\beta}\left\{^{\nu}_{\beta\alpha}\right\} + \frac{1}{2}\left( g^{\beta\nu}g^{\mu\gamma}T^{\xi}_{\beta\alpha}g_{\xi\gamma} + g^{\beta\nu}g^{\mu\gamma}T^{\xi}_{\alpha\gamma}g_{\xi\beta} + g^{\beta\nu}g^{\mu\gamma}T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right) \\ &\qquad + \frac{1}{2}\left( g^{\mu\beta}g^{\nu\gamma}T^{\xi}_{\beta\alpha}g_{\xi\gamma} + g^{\mu\beta}g^{\nu\gamma}T^{\xi}_{\alpha\gamma}g_{\xi\beta} + g^{\mu\beta}g^{\nu\gamma}T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right)\\ &= {g^{\mu\nu}}_{,\alpha} + g^{\beta\nu}\left\{^{\mu}_{\beta\alpha}\right\} + g^{\mu\beta}\left\{^{\nu}_{\beta\alpha}\right\} + \frac{1}{2}\left( g^{\beta\nu}\delta^\mu_\xi T^{\xi}_{\beta\alpha} + \delta^\nu_\xi g^{\mu\gamma}T^{\xi}_{\alpha\gamma} + g^{\beta\nu}g^{\mu\gamma}T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right) \\ &\qquad + \frac{1}{2}\left( g^{\mu\beta}\delta^\nu_\xi T^{\xi}_{\beta\alpha} + \delta^\mu_\xi g^{\nu\gamma}T^{\xi}_{\alpha\gamma} + g^{\mu\beta}g^{\nu\gamma}T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right)\\ &= {g^{\mu\nu}}_{,\alpha} + g^{\beta\nu}\left\{^{\mu}_{\beta\alpha}\right\} + g^{\mu\beta}\left\{^{\nu}_{\beta\alpha}\right\} + \frac{1}{2}\left( g^{\beta\nu} T^{\mu}_{\beta\alpha} + g^{\mu\gamma}T^{\nu}_{\alpha\gamma} + g^{\beta\nu}g^{\mu\gamma}T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right) \\ &\qquad + \frac{1}{2}\left( g^{\mu\beta} T^{\nu}_{\beta\alpha} + g^{\nu\gamma}T^{\mu}_{\alpha\gamma} + g^{\mu\beta}g^{\nu\gamma}T^{\xi}_{\beta\gamma}g_{\xi\alpha} \right)\\ &= {g^{\mu\nu}}_{,\alpha} + g^{\beta\nu}\left\{^{\mu}_{\beta\alpha}\right\} + g^{\mu\beta}\left\{^{\nu}_{\beta\alpha}\right\} + \frac{1}{2}\left( \cancel{g^{\beta\nu} T^{\mu}_{\beta\alpha} } +\bcancel{g^{\mu\textcolor{red}{\beta}}T^{\nu}_{\alpha\textcolor{red}{\beta}}} + \xcancel{g^{\beta\nu}g^{\mu\gamma}T^{\xi}_{\beta\gamma}g_{\xi\alpha} } \right) \\ &\qquad + \frac{1}{2}\left( \bcancel{g^{\mu\beta} T^{\nu}_{\beta\alpha} } + \cancel{g^{\nu\textcolor{red}{\beta}}T^{\mu}_{\alpha\textcolor{red}{\beta}}} + \xcancel{g^{\mu\textcolor{green}{\gamma}}g^{\nu\textcolor{red}{\beta}}T^{\xi}_{\textcolor{green}{\gamma}\textcolor{red}{\beta}}g_{\xi\alpha}} \right)\\ &= {g^{\mu\nu}}_{,\alpha} + g^{\beta\nu}\left\{^{\mu}_{\beta\alpha}\right\} + g^{\mu\beta}\left\{^{\nu}_{\beta\alpha}\right\} \end{align*}\] \begin{adjustwidth}{-2cm}{-1.5cm} \[\begin{align*} R^{\alpha}_{\beta\gamma\mu} &= \Gamma^{\alpha}_{\beta\mu,\gamma} - \Gamma^{\alpha}_{\beta\gamma,\mu} + \Gamma^{\alpha}_{\nu\gamma}\Gamma^{\nu}_{\beta\mu} - \Gamma^{\alpha}_{\nu\mu}\Gamma^{\nu}_{\beta\gamma} \\ &= (\{^{\alpha}_{\beta\mu}\} + K^{\alpha}_{\beta\mu})_{,\gamma} - (\{^{\alpha}_{\beta\gamma}\} + K^{\alpha}_{\beta\gamma})_{,\mu} + (\{^{\alpha}_{\nu\gamma}\} + K^{\alpha}_{\nu\gamma})(\{^{\nu}_{\beta\mu}\} + K^{\nu}_{\beta\mu}) - (\{^{\alpha}_{\nu\mu}\} + K^{\alpha}_{\nu\mu})(\{^{\nu}_{\beta\gamma}\} + K^{\nu}_{\beta\gamma}) \\ &= \{^{\alpha}_{\beta\mu}\}_{,\gamma} + K^{\alpha}_{\beta\mu,\gamma} - \{^{\alpha}_{\beta\gamma}\}_{,\mu} - K^{\alpha}_{\beta\gamma,\mu} + (\{^{\alpha}_{\nu\gamma}\}\{^{\nu}_{\beta\mu}\} + \{^{\alpha}_{\nu\gamma}\}K^{\nu}_{\beta\mu} + K^{\alpha}_{\nu\gamma}\{^{\nu}_{\beta\mu}\} + K^{\alpha}_{\nu\gamma}K^{\nu}_{\beta\mu}) \\ &\qquad - (\{^{\alpha}_{\nu\mu}\}\{^{\nu}_{\beta\gamma}\} + \{^{\alpha}_{\nu\mu}\}K^{\nu}_{\beta\gamma} + K^{\alpha}_{\nu\mu}\{^{\nu}_{\beta\gamma}\} + K^{\alpha}_{\nu\mu}K^{\nu}_{\beta\gamma}) \\ &= \{^{\alpha}_{\beta\mu}\}_{,\gamma} - \{^{\alpha}_{\beta\gamma}\}_{,\mu} + \{^{\alpha}_{\nu\gamma}\}\{^{\nu}_{\beta\mu}\} - \{^{\alpha}_{\nu\mu}\}\{^{\nu}_{\beta\gamma}\} + \textcolor{red}{K^{\alpha}_{\beta\mu,\gamma}} - K^{\alpha}_{\beta\gamma,\mu} + \left( \textcolor{red}{\{^{\alpha}_{\nu\gamma}\}K^{\nu}_{\beta\mu} }+ K^{\alpha}_{\nu\gamma}\{^{\nu}_{\beta\mu}\} + K^{\alpha}_{\nu\gamma}K^{\nu}_{\beta\mu} \right) \\ &\qquad + \left(\textcolor{red}{-K^{\alpha}_{\nu\mu}\{^{\nu}_{\beta\gamma}\} } - \{^{\alpha}_{\nu\mu}\}K^{\nu}_{\beta\gamma} - K^{\alpha}_{\nu\mu}K^{\nu}_{\beta\gamma}\right) \\ &= \{R\}^{\alpha}_{\beta\gamma\mu} + \textcolor{red}{\left( K^{\alpha}_{\beta\mu,\gamma} +\{^{\alpha}_{\nu\gamma}\}K^{\nu}_{\beta\mu} -\{^{\nu}_{\beta\gamma}\}K^{\alpha}_{\nu\mu} \right)} -\left( K^{\alpha}_{\beta\gamma,\mu} +\{^{\alpha}_{\nu\mu}\}K^{\nu}_{\beta\gamma} -\{^{\nu}_{\beta\mu}\}K^{\alpha}_{\nu\gamma} \right) +\left( K^{\alpha}_{\nu\gamma}K^{\nu}_{\beta\mu} -K^{\alpha}_{\nu\mu}K^{\nu}_{\beta\gamma} \right) \\ &= \{R\}^{\alpha}_{\beta\gamma\mu} + \textcolor{red}{\left( K^{\alpha}_{\beta\mu,\gamma} +\{^{\alpha}_{\nu\gamma}\}K^{\nu}_{\beta\mu} -\{^{\nu}_{\beta\gamma}\}K^{\alpha}_{\nu\mu} \textcolor{blue}{- \{^{\nu}_{\mu\gamma}\}K^{\alpha}_{\beta\nu}} \right)} -\left( K^{\alpha}_{\beta\gamma,\mu} +\{^{\alpha}_{\nu\mu}\}K^{\nu}_{\beta\gamma} -\{^{\nu}_{\beta\mu}\}K^{\alpha}_{\nu\gamma} \textcolor{blue}{- \{^{\nu}_{\gamma\mu}\}K^{\alpha}_{\beta\nu}} \right) +\left( K^{\alpha}_{\nu\gamma}K^{\nu}_{\beta\mu} -K^{\alpha}_{\nu\mu}K^{\nu}_{\beta\gamma} \right) \\ &= \{R\}^{\alpha}_{\beta\gamma\mu} + K^{\alpha}_{\beta\mu;\gamma} -K^{\alpha}_{\beta\gamma;\mu} +\left( K^{\alpha}_{\nu\gamma}K^{\nu}_{\beta\mu} -K^{\alpha}_{\nu\mu}K^{\nu}_{\beta\gamma} \right) \end{align*}\] \end{adjustwidth} \[\begin{align*} R&=g^{\beta\mu}\delta^\gamma_\alpha R^{\alpha}_{\beta\gamma\mu} =g^{\beta\mu}\delta^\gamma_\alpha \left(\{R\}^{\alpha}_{\beta\gamma\mu} + K^{\alpha}_{\beta\mu;\gamma} -K^{\alpha}_{\beta\gamma;\mu} +\left( K^{\alpha}_{\nu\gamma}K^{\nu}_{\beta\mu} -K^{\alpha}_{\nu\mu}K^{\nu}_{\beta\gamma} \right)\right)\\ &=g^{\beta\mu}\delta^\gamma_\alpha \{R\}^{\alpha}_{\beta\gamma\mu} + g^{\beta\mu}\delta^\gamma_\alpha K^{\alpha}_{\beta\mu;\gamma} -g^{\beta\mu}\delta^\gamma_\alpha K^{\alpha}_{\beta\gamma;\mu} +\left( g^{\beta\mu}\delta^\gamma_\alpha K^{\alpha}_{\nu\gamma}K^{\nu}_{\beta\mu} -g^{\beta\mu}\delta^\gamma_\alpha K^{\alpha}_{\nu\mu}K^{\nu}_{\beta\gamma} \right)\\ &=\{R\} + g^{\beta\mu} K^{\alpha}_{\beta\mu;\alpha} -g^{\beta\mu} K^{\alpha}_{\beta\alpha ;\mu} +\left( g^{\beta\mu} K^{\alpha}_{\nu\alpha}K^{\nu}_{\beta\mu} -g^{\beta\mu} K^{\alpha}_{\nu\mu}K^{\nu}_{\beta\alpha } \right)\\ \end{align*}\] \[\begin{align*} K^{\mu}_{\alpha\beta} &= \frac{1}{2}g^{\mu\gamma} \left(T^{\xi}_{\alpha\beta}g_{\xi\gamma} + T^{\xi}_{\beta\gamma}g_{\xi\alpha} + T^{\xi}_{\alpha\gamma}g_{\xi\beta}\right) \\ &=\frac{1}{2}\frac{1}{\mathbb{D}-1}g^{\mu\gamma} \left( \left(\delta^{\xi}_{\alpha}T_\beta-\delta^{\xi}_{\beta} T_\alpha\right)g_{\xi\gamma} + \left(\delta^{\xi}_{\beta}T_\gamma-\delta^{\xi}_{\gamma}T_\beta\right)g_{\xi\alpha} + \left(\delta^{\xi}_{\alpha}T_\gamma-\delta^{\xi}_{\gamma}T_\alpha\right)g_{\xi\beta} \right)\\ &=\frac{1}{2}\frac{1}{\mathbb{D}-1}g^{\mu\gamma} \left( \left(\cancel{g_{\alpha\gamma}}T_\beta-g_{\beta\gamma} T_\alpha\right) + \left(g_{\beta\alpha}T_\gamma-\cancel{g_{\gamma\alpha}T_\beta}\right) + \left(g_{\alpha\beta}T_\gamma-g_{\gamma\beta}T_\alpha\right) \right)\\ &=\frac{1}{\mathbb{D}-1}g^{\mu\gamma} \left(g_{\alpha\beta}T_\gamma-g_{\gamma\beta}T_\alpha\right)\\ &=\frac{1}{\mathbb{D}-1} \left(g^{\mu\gamma}g_{\alpha\beta}T_\gamma-\delta^\mu_\beta T_\alpha\right)\\ \end{align*}\] \[\begin{align*} K^{\alpha}_{\nu\gamma}K^{\nu}_{\beta\mu} -K^{\alpha}_{\nu\mu}K^{\nu}_{\beta\gamma} &=\frac{1}{\left(\mathbb{D}-1\right)^2} \left[ \left(g^{\alpha\xi}g_{\nu\gamma}T_\xi -\delta^\alpha_\nu T_\gamma\right) \left(g^{\nu\sigma}g_{\beta\mu}T_\sigma -\delta^\nu_\beta T_\mu\right) - \left(g^{\alpha\sigma}g_{\nu\mu}T_\sigma -\delta^\alpha_\nu T_\mu\right) \left(g^{\nu\xi}g_{\beta\gamma}T_\xi -\delta^\nu_\beta T_\gamma\right) \right]\\ &=\frac{1}{\left(\mathbb{D}-1\right)^2} \left[ \left( g^{\alpha\xi}g_{\nu\gamma}T_\xi -\delta^\alpha_\nu T_\gamma\right) \left(g^{\nu\sigma}g_{\beta\mu}T_\sigma -\delta^\nu_\beta T_\mu\right) - \left(g^{\alpha\sigma}g_{\nu\mu}T_\sigma -\delta^\alpha_\nu T_\mu\right) \left(g^{\nu\xi}g_{\beta\gamma}T_\xi -\delta^\nu_\beta T_\gamma\right) \right]\\ \end{align*}\] \[\begin{align*} &\left( g^{\alpha\xi}g_{\nu\gamma}T_\xi -\delta^\alpha_\nu T_\gamma\right) \left(g^{\nu\sigma}g_{\beta\mu}T_\sigma -\delta^\nu_\beta T_\mu\right) \\ &= \left(g^{\alpha\xi}g_{\nu\gamma}T_\xi\right) \left(g^{\nu\sigma}g_{\beta\mu}T_\sigma\right) - \left(g^{\alpha\xi}g_{\nu\gamma}T_\xi\right) \left(\delta^\nu_\beta T_\mu\right) - \left(\delta^\alpha_\nu T_\gamma\right) \left(g^{\nu\sigma}g_{\beta\mu}T_\sigma\right) + \left(\delta^\alpha_\nu T_\gamma\right) \left(\delta^\nu_\beta T_\mu\right) \\ &= g^{\alpha\xi} g_{\beta\mu} (g_{\nu\gamma} g^{\nu\sigma}) T_\xi T_\sigma - g^{\alpha\xi} (g_{\nu\gamma} \delta^\nu_\beta) T_\xi T_\mu - (\delta^\alpha_\nu g^{\nu\sigma}) g_{\beta\mu} T_\gamma T_\sigma + (\delta^\alpha_\nu \delta^\nu_\beta) T_\gamma T_\mu \\ &= g^{\alpha\xi} g_{\beta\mu} \delta^\sigma_\gamma T_\xi T_\sigma - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu - g^{\alpha\sigma} g_{\beta\mu} T_\gamma T_\sigma + \delta^\alpha_\beta T_\gamma T_\mu \\ &= g^{\alpha\xi} g_{\beta\mu} T_\xi T_\gamma - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu - g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma + \delta^\alpha_\beta T_\gamma T_\mu \\ &= g^{\alpha\xi} g_{\beta\mu} T_\xi T_\gamma - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu - g^{\alpha\xi} g_{\beta\mu} T_\xi T_\gamma + \delta^\alpha_\beta T_\gamma T_\mu \quad (\text{將啞指標 } \sigma \to \xi) \\ &= - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu + \delta^\alpha_\beta T_\gamma T_\mu \\ &= \delta^\alpha_\beta T_\gamma T_\mu - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu \end{align*}\] \[\begin{align*} &\left(g^{\alpha\sigma}g_{\nu\mu}T_\sigma -\delta^\alpha_\nu T_\mu\right) \left(g^{\nu\xi}g_{\beta\gamma}T_\xi -\delta^\nu_\beta T_\gamma\right) \\ &= \left(g^{\alpha\sigma}g_{\nu\mu}T_\sigma\right) \left(g^{\nu\xi}g_{\beta\gamma}T_\xi\right) - \left(g^{\alpha\sigma}g_{\nu\mu}T_\sigma\right) \left(\delta^\nu_\beta T_\gamma\right) - \left(\delta^\alpha_\nu T_\mu\right) \left(g^{\nu\xi}g_{\beta\gamma}T_\xi\right) + \left(\delta^\alpha_\nu T_\mu\right) \left(\delta^\nu_\beta T_\gamma\right) \\ &= g^{\alpha\sigma} g_{\beta\gamma} (g_{\nu\mu} g^{\nu\xi}) T_\sigma T_\xi - g^{\alpha\sigma} (g_{\nu\mu} \delta^\nu_\beta) T_\sigma T_\gamma - (\delta^\alpha_\nu g^{\nu\xi}) g_{\beta\gamma} T_\mu T_\xi + (\delta^\alpha_\nu \delta^\nu_\beta) T_\mu T_\gamma \\ &= g^{\alpha\sigma} g_{\beta\gamma} \delta^\xi_\mu T_\sigma T_\xi - g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma - g^{\alpha\xi} g_{\beta\gamma} T_\mu T_\xi + \delta^\alpha_\beta T_\mu T_\gamma \\ &= g^{\alpha\sigma} g_{\beta\gamma} T_\sigma T_\mu - g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma - g^{\alpha\xi} g_{\beta\gamma} T_\mu T_\xi + \delta^\alpha_\beta T_\mu T_\gamma \\ &= g^{\alpha\sigma} g_{\beta\gamma} T_\sigma T_\mu - g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma - g^{\alpha\sigma} g_{\beta\gamma} T_\mu T_\sigma + \delta^\alpha_\beta T_\mu T_\gamma \quad (\text{將啞指標 } \xi \to \sigma) \\ &= - g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma + \delta^\alpha_\beta T_\mu T_\gamma \\ &= \delta^\alpha_\beta T_\mu T_\gamma - g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma \end{align*}\] \[\begin{align*} &\left( g^{\alpha\xi}g_{\nu\gamma}T_\xi -\delta^\alpha_\nu T_\gamma\right) \left(g^{\nu\sigma}g_{\beta\mu}T_\sigma -\delta^\nu_\beta T_\mu\right) - \left(g^{\alpha\sigma}g_{\nu\mu}T_\sigma -\delta^\alpha_\nu T_\mu\right) \left(g^{\nu\xi}g_{\beta\gamma}T_\xi -\delta^\nu_\beta T_\gamma\right) \\ % &= \left( \delta^\alpha_\beta T_\gamma T_\mu - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu \right) - \left( \delta^\alpha_\beta T_\mu T_\gamma - g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma \right) \\ % &= \delta^\alpha_\beta T_\gamma T_\mu - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu - \delta^\alpha_\beta T_\mu T_\gamma + g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma \\ % &= \left( \delta^\alpha_\beta T_\gamma T_\mu - \delta^\alpha_\beta T_\mu T_\gamma \right) - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu + g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma \\ % &= 0 - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu + g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma \quad (\text{假設 } T \text{ 分量可交換}) \\ % &= g^{\alpha\sigma} g_{\beta\mu} T_\sigma T_\gamma - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu \\ % &= g^{\alpha\xi} g_{\beta\mu} T_\xi T_\gamma - g^{\alpha\xi} g_{\beta\gamma} T_\xi T_\mu \quad (\text{將啞指標 } \sigma \to \xi) \\ % &= g^{\alpha\xi} T_\xi (g_{\beta\mu} T_\gamma - g_{\beta\gamma} T_\mu) \end{align*}\] \[\begin{align*} \delta^\gamma_\alpha g^{\alpha\xi} T_\xi (g_{\beta\mu} T_\gamma - g_{\beta\gamma} T_\mu) &=g^{\gamma\xi} T_\xi (g_{\beta\mu} T_\gamma - g_{\beta\gamma} T_\mu)\\ &= g^{\gamma\xi} T_\xi g_{\beta\mu} T_\gamma - g^{\gamma\xi} T_\xi g_{\beta\gamma} T_\mu \end{align*}\] \[\begin{align*} g^{\beta\mu}\delta^\gamma_\alpha g^{\alpha\xi} T_\xi (g_{\beta\mu} T_\gamma - g_{\beta\gamma} T_\mu) &= g^{\beta\mu}\left(g^{\gamma\xi} T_\xi g_{\beta\mu} T_\gamma - g^{\gamma\xi} T_\xi g_{\beta\gamma} T_\mu\right)\\ &= g^{\beta\mu}g^{\gamma\xi} T_\xi g_{\beta\mu} T_\gamma -g^{\beta\mu} g^{\gamma\xi} T_\xi g_{\beta\gamma} T_\mu\\ &= \mathbb{D}g^{\gamma\xi} T_\xi T_\gamma - g^{\gamma\xi} T_\xi T_\gamma\\ &=\left( \mathbb{D}-1\right) g^{\gamma\xi} T_\xi T_\gamma \end{align*}\] \[\begin{align*} C^\mu_{\alpha\beta}&=\frac{1}{\mathbb{D}-1} \left( \delta^\mu_\alpha T_\beta +\delta^\mu_\beta T_\alpha -g_{\mu\alpha}g^{\mu\gamma} T_\gamma \right) \end{align*}\] \[\begin{align*} \Gamma^\mu_{\alpha\beta} &=\tilde{\Gamma}^\mu_{\alpha\beta}+C^\mu_{\alpha\beta} =\tilde{\left\{^\mu_{\alpha\beta}\right\}}+K^\mu_{\alpha\beta}+C^\mu_{\alpha\beta}\\ &=\tilde{\left\{^\mu_{\alpha\beta}\right\}}+\frac{1}{\mathbb{D}-1} \left( \cancel{g^{\mu\gamma}g_{\alpha\beta}T_\gamma}-\bcancel{\delta^\mu_\beta T_\alpha}\right) +\frac{1}{\mathbb{D}-1} \left( \delta^\mu_\alpha T_\beta +\bcancel{\delta^\mu_\beta T_\alpha} -\cancel{g_{\mu\alpha}g^{\mu\gamma} T_\gamma} \right)\\ &=\tilde{\left\{^\mu_{\alpha\beta}\right\}}+ \underbrace{\frac{1}{\mathbb{D}-1}\delta^\mu_\alpha T_\beta}_{D^\mu_{\alpha\beta}} \end{align*}\] \begin{adjustwidth}{-2cm}{-1.5cm} \[\begin{align*} R^{\alpha}_{\beta\gamma\mu} &=\{R\}^{\alpha}_{\beta\gamma\mu} + D^{\alpha}_{\beta\mu;\gamma} -D^{\alpha}_{\beta\gamma;\mu} +\left( D^{\alpha}_{\nu\gamma}D^{\nu}_{\beta\mu} -D^{\alpha}_{\nu\mu}D^{\nu}_{\beta\gamma} \right) \\ &= \{R\}^{\alpha}_{\beta\gamma\mu} + \frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\mu;\gamma} - \frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\gamma;\mu} +\frac{1}{\left(\mathbb{D}-1\right)^2}\left( \cancel{ \delta^\alpha_\nu T_\gamma \delta^\nu_\beta T_\mu} - \cancel{\delta^\alpha_\nu T_\mu \delta^\nu_\beta T_\gamma} \right) \end{align*}\] \end{adjustwidth} \[\begin{align*} R_{\beta\mu}&=\delta^\gamma_\alpha R^{\alpha}_{\beta\gamma\mu}\\ &=\delta^\gamma_\alpha \left(\{R\}^{\alpha}_{\beta\gamma\mu} + \frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\mu;\gamma} - \frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\gamma;\mu}\right)\\ &=\{R\}_{\beta\mu} + \delta^\gamma_\alpha\frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\mu;\gamma} - \delta^\gamma_\alpha\frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\gamma;\mu}\\ &=\{R\}_{\beta\mu} + \frac{1}{\mathbb{D}-1} T_{\mu;\beta} - \frac{1}{\mathbb{D}-1} T_{\beta;\mu} \end{align*}\] \[\begin{align*} R_{\beta\mu}-R_{\mu\beta} &=\left( \cancel{\{R\}_{\beta\mu}} + \frac{1}{\mathbb{D}-1} T_{\mu;\beta} - \frac{1}{\mathbb{D}-1} T_{\beta;\mu} \right) -\left( \cancel{\{R\}_{\mu\beta}} + \frac{1}{\mathbb{D}-1} T_{\beta;\mu} - \frac{1}{\mathbb{D}-1} T_{\mu;\beta} \right)\\ &=\frac{2}{\mathbb{D}-1} \left(T_{\mu;\beta} - T_{\beta;\mu} \right)\\ &=\frac{2}{\mathbb{D}-1} \left(T_{\mu,\beta} - T_{\beta,\mu} \right) \end{align*}\] \[\begin{align*} R&=g^{\beta\mu}\delta^\gamma_\alpha R^{\alpha}_{\beta\gamma\mu}\\ &=g^{\beta\mu}\delta^\gamma_\alpha \left(\{R\}^{\alpha}_{\beta\gamma\mu} + \frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\mu;\gamma} - \frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\gamma;\mu}\right)\\ &=\{R\} + g^{\beta\mu}\delta^\gamma_\alpha\frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\mu;\gamma} - g^{\beta\mu}\delta^\gamma_\alpha\frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\gamma;\mu}\\ &=\{R\} + \cancel{g^{\gamma\mu}\frac{1}{\mathbb{D}-1} T_{\mu;\gamma}} - \cancel{g^{\gamma\mu}\frac{1}{\mathbb{D}-1} T_{\gamma;\mu}}\\ &=\{R\} \end{align*}\] \[\begin{align*} \left(t_{GR}\right)^\gamma_\varepsilon &=\frac{1}{2\kappa} \left(g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu} +g^{\beta\gamma}\delta^\mu_\alpha R^\alpha_{\beta\mu\varepsilon}-\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left(g^{\beta\mu}\delta^\gamma_\alpha R^\alpha_{\beta\varepsilon\mu} -g^{\beta\gamma}\delta^\mu_\alpha R^\alpha_{\beta\varepsilon\mu}-\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left( \left( g^{\beta\mu}\delta^\gamma_\alpha -g^{\beta\gamma}\delta^\mu_\alpha \right)R^\alpha_{\beta\varepsilon\mu}-\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left( \left( g^{\beta\mu}\delta^\gamma_\alpha -g^{\beta\gamma}\delta^\mu_\alpha \right) \left( \{R\}^\alpha_{\beta\varepsilon\mu} +\frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\mu;\varepsilon} - \frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\varepsilon;\mu} \right) -\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left( \left( \left( g^{\beta\mu}\delta^\gamma_\alpha -g^{\beta\gamma}\delta^\mu_\alpha \right) \{R\}^\alpha_{\beta\varepsilon\mu} +\left( g^{\beta\mu}\delta^\gamma_\alpha -g^{\beta\gamma}\delta^\mu_\alpha \right) \frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\mu;\varepsilon} - \left( g^{\beta\mu}\delta^\gamma_\alpha -g^{\beta\gamma}\delta^\mu_\alpha \right) \frac{1}{\mathbb{D}-1}\delta^\alpha_\beta T_{\varepsilon;\mu} \right) -\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left( \left( 2g^{\beta\mu}\delta^\gamma_\alpha \{R\}^\alpha_{\beta\varepsilon\mu} +\left( g^{\beta\mu}\delta^\gamma_\beta -g^{\beta\gamma}\delta^\mu_\beta \right) \frac{1}{\mathbb{D}-1}T_{\mu;\varepsilon} - \left( g^{\beta\mu}\delta^\gamma_\beta -g^{\beta\gamma}\delta^\mu_\beta \right) \frac{1}{\mathbb{D}-1} T_{\varepsilon;\mu} \right) -\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left( \left( 2g^{\beta\mu}\delta^\gamma_\alpha \{R\}^\alpha_{\beta\varepsilon\mu} +\left( g^{\gamma\mu} -g^{\mu\gamma} \right) \frac{1}{\mathbb{D}-1}T_{\mu;\varepsilon} - \left( g^{\gamma\mu} -g^{\mu\gamma} \right) \frac{1}{\mathbb{D}-1} T_{\varepsilon;\mu} \right) -\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left( 2g^{\beta\mu}\delta^\gamma_\alpha \{R\}^\alpha_{\beta\varepsilon\mu} -\delta^\gamma_\varepsilon R \right)\sqrt{-g} \\ &=\frac{1}{2\kappa} \left( 2g^{\beta\mu} \{R\}^\gamma_{\beta\varepsilon\mu} -\delta^\gamma_\varepsilon R \right)\sqrt{-g} \end{align*}\] \subsection{GR-Vielbeins} The vielbein formalism chooses a new basis frame \(\{\hat{e}_a\}\) rather than a natural basis \(\{\partial_\alpha\}\), \[\begin{align} \label{eq:viel} \partial_\mu &=e_\mu^a \hat{e}_a \end{align}\] , where \(e_\mu^a\) is transformation called frame field (or vierbein field). Clearly, each point of frame field is an element in \(GL(n,\mathbb{R})\) group. We choose the frame field statisfies: \[\begin{align}\label{eq:g_viel} g_{\mu\nu}&= e_\mu^a e_\nu^b\eta_{ab} \end{align}\] The inverse transformation \(e^\mu_a\): \[\begin{align*} e^\mu_a e^a_\nu=\delta^\mu_\nu\,\,\, &and \,\,\,e^\mu_a e^b_\mu=\delta^a_b \end{align*}\] Define \(e=det(e_\mu^a)\), we have: \[\begin{align*} g=det(g_{\mu\nu})\underbrace{=}_{\eqref{eq:g_viel}}det(e_\mu^a e_\nu^b\eta_{ab})&=det(e_\mu^a )det(e_\nu^b)det(\eta_{ab})= e\, e\, (-1) \\ &\to e=\sqrt{-g} \end{align*}\] Recall the definition of tangent connection\(\Gamma_{\alpha \mu}^\beta\): \[\begin{align*} \nabla_\mu \partial_\alpha=\Gamma_{\alpha \mu}^\beta \partial_\beta \end{align*}\] We define the spin connection \(\omega_{a \mu}^b\) in similar way: \[\begin{align*} \nabla_\mu \hat{e}_a=\omega_{a \mu}^b \hat{e}_b \end{align*}\] \[\begin{align*} \mathfrak{g}&=\mathfrak{h}\oplus \mathfrak{p}\\ \omega&=W+\theta\\ \omega&:TM\to \mathfrak{g}, \text{eg, }\mathbb{R}^4\ltimes SO(1,3) \\ W&:TM\to \mathfrak{h}, \text{eg, }SO(1,3)\\ \theta&:TM\to \mathfrak{p}, \text{eg, }\mathbb{R}^4\\ \\\\ \nabla_\mu \hat{e}_a&=\omega_{a \mu}^b \hat{e}_b\\ &=\left(\omega_{\mu}\right)^b_a \hat{e}_b\\ &=\left(\omega_{\mu}^\mathfrak{i} \hat{T}_{\mathfrak{i}} \right)^b_a \hat{e}_b\\ &=\left(W_{\mu}^\mathfrak{ab} \hat{T}_{\mathfrak{ab}}+\theta_\mu^{\mathfrak{a}}\hat{T}_{\mathfrak{a}} \right)^b_a \hat{e}_b\\ &=W_{\mu}^\mathfrak{ab}\left( \hat{T}_{\mathfrak{ab}} \right)^b_a \hat{e}_b +\theta_\mu^{\mathfrak{a}}\left(\hat{T}_{\mathfrak{a}} \right)^b_a \hat{e}_b\\ \end{align*}\] \[\begin{align*} T^a&=d\theta^a +W^a_b\wedge e^b\\ T^a_{\mu\nu}&=e^a_{\nu,\mu}- +\omega^a_b\wedge e^b\\ \end{align*}\] The relation between \(\Gamma_{\alpha \mu}^\beta\) and \(\omega_{a \mu}^b\): \[\begin{align*} \nabla_\mu \partial_\alpha &\underbrace{=}_{\eqref{eq:viel}} \nabla_\mu \left( e^a_\alpha \hat{e}_a \right)=e^a_{\alpha,\mu} \hat{e}_a +e^a_\alpha\nabla_\mu\hat{e}_a=e^a_{\alpha,\mu} \hat{e}_a +e^a_\alpha\omega_{a \mu}^b \hat{e}_b \nonumber\\ &=\Gamma_{\alpha \mu}^\beta \partial_\beta=\Gamma_{\alpha \mu}^\beta e^b_\beta \hat{e}_b \nonumber \\ &\to \left(e^b_{\alpha,\mu} +e^a_\alpha\omega_{a \mu}^b - \Gamma_{\alpha \mu}^\beta e^b_\beta \right)\hat{e}_b=0 \nonumber \\ &\to e^b_{\alpha,\mu} +e^a_\alpha\omega_{a \mu}^b - \Gamma_{\alpha \mu}^\beta e^b_\beta=0 \end{align*}\] We can derive 3 useful formula: \[\begin{align} e^b_{\alpha,\mu}&=e^b_\beta \Gamma_{\alpha \mu}^\beta -e^a_\alpha\omega_{a \mu}^b \label{eq:frame,}\\ \Gamma_{\alpha \mu}^\beta&=e^\beta_b e^a_\alpha\omega_{a \mu}^b +e^\beta_b e^b_{\alpha,\mu} \label{eq:Gamma_omega}\\ \omega_{a \mu}^b&=e_a^\alpha e^b_\beta \Gamma_{\alpha \mu}^\beta -e_a^\alpha e^b_{\alpha,\mu} \label{eq:omega_Gamma} \end{align}\] Since \(e_a^\alpha e^b_{\alpha}=\delta^b_a\), then \(-e_a^\alpha e^b_{\alpha,\mu}=e_{a,\mu}^\alpha e^b_{\alpha}\). The Eq.\eqref{eq:omega_Gamma} can be rewritten: \[\begin{align} \omega_{a \mu}^b&=e_a^\alpha e^b_\beta \Gamma_{\alpha \mu}^\beta +e_{a,\mu}^\alpha e^b_{\alpha} \nonumber\\ \to e_b^{\gamma} \omega_{a \mu}^b&=e_b^{\gamma} e_a^\alpha e^b_\beta \Gamma_{\alpha \mu}^\beta +e_{a,\mu}^\alpha e^b_{\alpha} e_b^{\gamma} \nonumber\\ \to e_b^{\gamma} \omega_{a \mu}^b&= e_a^\alpha \Gamma_{\alpha \mu}^\gamma +e_{a,\mu}^\gamma \nonumber\\ \to e_{a,\mu}^\gamma &=e_b^{\gamma} \omega_{a \mu}^b- e_a^\alpha \Gamma_{\alpha \mu}^\gamma \label{eq:-frame,} \end{align}\] Since frame field takes value in \(GL\) group, the spin connection \(\omega_{a \mu}^b\) is \(\frak{gl}\)-value 1-form. Next, we will derive the relation between the tangent curvature \(R^\kappa_{\lambda\omega\sigma}\) and the spin curvature \(\mathscr{R}^a_{b\omega\sigma}\). The definition of these curvature are: \[\begin{align*} R^\kappa_{\lambda\omega\sigma} &= \Gamma^\kappa_{\lambda\sigma,\omega} - \Gamma^\kappa_{\lambda\omega,\sigma} + \Gamma^\kappa_{\nu\omega} \Gamma^\nu_{\lambda\sigma} - \Gamma^\kappa_{\nu\sigma} \Gamma^\nu_{\lambda\omega}\\ \mathscr{R}^a_{b\omega\sigma} &=\omega^a_{b\sigma,\omega} - \omega^a_{b\omega,\sigma} + \omega^a_{c\omega}\omega^c_{b\sigma} - \omega^a_{c\sigma} \omega^c_{b\omega} \end{align*}\] First, we take partial derivative on Eq.\eqref{eq:Gamma_omega}: \begin{adjustwidth}{-2cm}{-1.5cm} \[\begin{align*} \Gamma^\kappa_{\lambda\sigma,\omega} &=\left( e^\kappa_a e^b_\lambda \omega_{b \sigma}^a +e^\kappa_b e^b_{\lambda,\sigma} \right)_{,\omega}\\ &=e^\kappa_{a,\omega} e^b_\lambda \omega_{b \sigma}^a + e^\kappa_a e^b_{\lambda,\omega} \omega_{b \sigma}^a + e^\kappa_a e^b_\lambda \omega_{b \sigma,\omega}^a + e^\kappa_{b,\omega} e^b_{\lambda,\sigma}+ e^\kappa_b e^b_{\lambda,\sigma\omega}\\ &= \underbrace{\left(e_c^{\kappa} \omega_{a \omega}^c- e_a^\eta \Gamma_{\eta \omega}^\kappa \right)}_{\eqref{eq:-frame,}} e^b_\lambda \omega_{b \sigma}^a + e^\kappa_a \underbrace{\left(e^b_\eta\Gamma_{\lambda \omega}^\eta -e^c_\lambda\omega_{c \omega}^b \right)}_{\eqref{eq:frame,}} \omega_{b \sigma}^a + e^\kappa_a e^b_\lambda \omega_{b \sigma,\omega}^a + \underbrace{\left(e_c^{\kappa} \omega_{b \omega}^c- e_b^\eta \Gamma_{\eta \omega}^\kappa \right)}_{\eqref{eq:-frame,}} \underbrace{\left(e^b_\gamma \Gamma_{\lambda\sigma}^\gamma -e^d_\lambda\omega_{d \sigma}^b \right)}_{\eqref{eq:frame,}}+ e^\kappa_b e^b_{\lambda,\sigma\omega} \end{align*}\] \end{adjustwidth} Expand and rearrange: \[\begin{align*} \Gamma^\kappa_{\lambda\sigma,\omega} = \left(e_c^{\kappa} \omega_{a \omega}^c e^b_\lambda \omega_{b \sigma}^a- e_a^\eta \Gamma_{\eta \omega}^\kappa e^b_\lambda \omega_{b \sigma}^a \right) &+ \left( e^\kappa_a e^b_\eta\Gamma_{\lambda \omega}^\eta \omega_{b \sigma}^a-e^\kappa_a e^c_\lambda\omega_{c \omega}^b \omega_{b \sigma}^a \right) + e^\kappa_a e^b_\lambda \omega_{b \sigma,\omega}^a \\ &+\left( e_c^{\kappa} \omega_{b \omega}^ce^b_\gamma \Gamma_{\lambda\sigma}^\gamma -e_c^{\kappa} \omega_{b \omega}^c e^d_\lambda\omega_{d \sigma}^b -e_b^\eta \Gamma_{\eta \omega}^\kappa e^b_\gamma \Gamma_{\lambda\sigma}^\gamma +e_b^\eta \Gamma_{\eta \omega}^\kappa e^d_\lambda\omega_{d \sigma}^b \right)+ e^\kappa_b e^b_{\lambda,\sigma\omega}\\ = \left( \underbracket[0.4pt][0pt]{e_c^{\kappa} e^b_\lambda \omega_{a \omega}^c \omega_{b \sigma}^a }_{(*a)} - \underbracket[0.4pt][0pt]{e_a^\eta e^b_\lambda \Gamma_{\eta \omega}^\kappa \omega_{b \sigma}^a}_{(*b)} \right) &+ \left( e^\kappa_a e^b_\eta\Gamma_{\lambda \omega}^\eta \omega_{b \sigma}^a -e^\kappa_a e^{\textcolor{red}{b}}_\lambda \omega_{{\textcolor{red}{c}} \sigma}^a \omega_{{\textcolor{red}{b}} \omega}^{\textcolor{red}{c}}\right) +e^\kappa_a e^b_\lambda \omega_{b \sigma,\omega}^a \\ &+\left( e_c^{\kappa} e^b_\gamma \Gamma_{\lambda\sigma}^\gamma \omega_{b \omega}^c -\underbracket[0.4pt][0pt]{ e_c^{\kappa} e^{\textcolor{red}{b}}_\lambda \omega_{{\textcolor{red}{a}} \omega}^c \omega_{\textcolor{red}{b} \sigma}^{\textcolor{red}{a}} }_{(*a)} \underbracket[0.4pt][0pt]{-\Gamma_{\eta \omega}^\kappa \Gamma_{\lambda\sigma}^\eta}_{Mov\frac{1}{2\kappa} e\,to\,left} +\underbracket[0.4pt][0pt]{ e_{\textcolor{red}{a}}^\eta e^{\textcolor{red}{b}}_\lambda \Gamma_{\eta \omega}^\kappa \omega_{{\textcolor{red}{b}} \sigma}^{\textcolor{red}{a}} }_{(*b)} \right)+ e^\kappa_b e^b_{\lambda,\sigma\omega} \end{align*}\] Remove \((*a)\) and \((*b)\), we have: \[\begin{align} \label{eq:Gamma_sw} \Gamma^\kappa_{\lambda\sigma,\omega} +\Gamma_{\eta \omega}^\kappa \Gamma_{\lambda\sigma}^\eta = \left( \underbracket[0.4pt][0pt]{e^\kappa_a e^b_\eta\Gamma_{\lambda \omega}^\eta \omega_{b \sigma}^a}_{(\#a)} -e^\kappa_a e^b_\lambda \omega_{c \sigma}^a \omega_{b \omega}^c\right) +e^\kappa_a e^b_\lambda \omega_{b \sigma,\omega}^a +\left( \underbracket[0.4pt][0pt]{e_c^{\kappa} e^b_\gamma \Gamma_{\lambda\sigma}^\gamma \omega_{b \omega}^c}_{(\#b)} \right)+ \underbracket[0.4pt][0pt]{e^\kappa_b e^b_{\lambda,\sigma\omega}}_{(\#c)} \end{align}\] Similarly, swap \(\sigma \Leftrightarrow \omega \): \[\begin{align} \label{eq:Gamma_ws} \Gamma^\kappa_{\lambda\omega,\sigma} +\Gamma_{\eta \sigma}^\kappa \Gamma_{\lambda\omega}^\eta = \left( \underbracket[0.4pt][0pt]{ e^\kappa_a e^b_{\textcolor{red}{\gamma}}\Gamma_{\lambda \sigma}^{\textcolor{red}{\gamma}} \omega_{b \omega}^a}_{(\#b)} -e^\kappa_a e^b_\lambda \omega_{c \omega}^a \omega_{b \sigma}^c\right) +e^\kappa_a e^b_\lambda \omega_{b \omega,\sigma}^a +\left( \underbracket[0.4pt][0pt]{e_c^{\kappa} e^b_{\textcolor{red}{\eta}} \Gamma_{\lambda\omega}^{\textcolor{red}{\eta}} \omega_{b \sigma}^c}_{(\#a)} \right)+ \underbracket[0.4pt][0pt]{e^\kappa_b e^b_{\lambda,\omega\sigma}}_{(\#c)} \end{align}\] Eq.\eqref{eq:Gamma_sw} \(-\) Eq.\eqref{eq:Gamma_ws}: \[\begin{align*} \Gamma^\kappa_{\lambda\sigma,\omega} -\Gamma^\kappa_{\lambda\omega,\sigma} +\Gamma_{\eta \omega}^\kappa \Gamma_{\lambda\sigma}^\eta -\Gamma_{\eta \sigma}^\kappa \Gamma_{\lambda\omega}^\eta &=e^\kappa_a e^b_\lambda \left( \omega_{b \sigma,\omega}^a -\omega_{b \omega,\sigma}^a +\omega_{c \omega}^a \omega_{b \sigma}^c -\omega_{c \sigma}^a \omega_{b \omega}^c \right)\\ R^\kappa_{\lambda\omega\sigma} &= e^\kappa_a e^b_\lambda \mathscr{R}^a_{b\omega\sigma} \end{align*}\] \textbf{Eq.\eqref{eq:action_G}} \[\begin{align*} R&=g^{\kappa\sigma}\delta^\omega_\varepsilon R^\varepsilon_{\kappa\omega\sigma}\\ &=\left( \eta^{de}e^\kappa_d e^\sigma_e \right) \delta^\omega_\varepsilon \left(e^\varepsilon_c e^a_\kappa \mathscr{R}^c_{a\omega\sigma} \right)\\ &=\eta^{ae} e^\sigma_e e^\omega_c \mathscr{R}^c_{a\omega\sigma} \end{align*}\] \[\begin{align*} e_{,\gamma}=\left( \sqrt{-g} \right)_{,\gamma} &= \sqrt{-g}\Gamma_{\eta\gamma}^{\eta}\\ &=e\, \left(e^\eta_a e^b_\eta \omega_{b\gamma}^{a}-e^b_\eta e^\eta_{b,\gamma} \right) \end{align*}\] \subsection{GR-EMT for GR} \textbf{Eq.\eqref{eq:/omega,}} \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} &= \frac{\partial}{\partial \omega^b_{c\mu,\gamma}} \left(\frac{1}{2\kappa} e\, \eta^{ae}e^\sigma_e e^\omega_f \mathscr{R}^f_{a\omega\sigma} \right) \\ &= \frac{1}{2\kappa} e\, \eta^{ae}e^\sigma_e e^\omega_f \frac{\partial}{\partial \omega^b_{c\mu,\gamma}} \left( \omega ^f_{a\sigma,\omega}-\omega ^f_{a\omega,\sigma} +\omega ^f_{d\omega}\omega ^d_{a\sigma} -\omega ^f_{d\sigma}\omega ^d_{a\omega} \right)\\ &= \frac{1}{2\kappa} e\, \eta^{ae}e^\sigma_e e^\omega_f \left(\delta^f_b \delta^c_a \delta^\mu_\sigma \delta^\gamma_\omega - \delta^f_b \delta^c_a \delta^\mu_\omega \delta^\gamma_\sigma \right)\\ &= \frac{1}{2\kappa} e\, \eta^{ae} \delta^c_a \left(e^\sigma_e e^\omega_f \delta^f_b \delta^\mu_\sigma \delta^\gamma_\omega - e^\sigma_e e^\omega_f \delta^f_b \delta^\mu_\omega \delta^\gamma_\sigma \right)\\ &= \frac{1}{2\kappa} e\, \eta^{ce} \left(e^\mu_e e^\gamma_b - e^\gamma_e e^\mu_b \right) \end{align*}\] Similarly, \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma,\mu}}&= \frac{1}{2\kappa} e\, \eta^{ce} \left( e^\gamma_e e^\mu_b - e^\mu_e e^\gamma_b \right) =-\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \end{align*}\] Since \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\nu\mu,\gamma}^\alpha} &=\frac{1}{2\kappa} \sqrt{-g} \left( g^{\nu\mu} \delta_\alpha^\gamma - g^{\nu\gamma} \delta_\alpha^\mu \right)\\ &=\frac{1}{2\kappa} e \left( \eta^{ce} e^\nu_c e^\mu_e \delta_\alpha^\gamma - \eta^{ce} e^\nu_c e^\gamma_e \delta_\alpha^\mu \right)\\ &=\frac{1}{2\kappa} e \eta^{ce} \left(e^\nu_c e^\mu_e e^\gamma_b e^b_\alpha - e^\nu_c e^\gamma_e e^\mu_b e^b_\alpha\right)\\ &=e^\nu_c e^b_\alpha \frac{1}{2\kappa} e \eta^{ce} \left(e^\mu_e e^\gamma_b - e^\gamma_e e^\mu_b \right)\\ &=e^\nu_c e^b_\alpha \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \end{align*}\] Reversely, \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} &=e_\nu^c e_b^\alpha \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\nu\mu,\gamma}^\alpha} \end{align*}\] Evaluate \(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu}^\alpha}\): \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu}^\alpha} &=\frac{1}{2\kappa}\left( \sqrt{-g} \, g^{\kappa\sigma} \delta_\alpha^\nu \Gamma_{\kappa\sigma}^\mu +\sqrt{-g}\, g^{\mu\nu} \Gamma_{\alpha\omega}^\omega -\sqrt{-g}\, g^{\kappa\nu} \Gamma_{\kappa\alpha}^\mu -\sqrt{-g}\, g^{\mu\sigma} \Gamma_{\alpha\sigma}^\nu \right) \end{align*}\] \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\mu\nu,\gamma}^\alpha} &=\frac{1}{2\kappa} \sqrt{-g} \left( g^{\mu\nu} \delta_\alpha^\gamma - g^{\mu\gamma} \delta_\alpha^\nu \right) \end{align*}\] \begin{adjustwidth}{-2.2cm}{-1cm} \[\begin{align*} \left(\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\nu\gamma,\mu}^\alpha} \right)_{,\mu} &=\left(e^\nu_c e^b_\alpha \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma,\mu}}\right)_{,\mu}\\ &=e^\nu_c e^b_\alpha \left( \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma,\mu}}\right)_{,\mu} +\left( e^\nu_{c,\mu}e^b_\alpha +e^\nu_c e^b_{\alpha,\mu} \right) \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma,\mu}}\\ &=e^\nu_c e^b_\alpha \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma}} +\left( e^\nu_{c,\mu}e^b_\alpha +e^\nu_c e^b_{\alpha,\mu} \right) \frac{1}{2\kappa} e\, \eta^{ce} \left( e^\gamma_e e^\mu_b - e^\mu_e e^\gamma_b \right)\\ &=e^\nu_c e^b_\alpha \left\{ \frac{1}{2\kappa} e\,\eta^{ae} \left( e^{{\mu}}_e e^\gamma_b -e^\gamma_e e^{{\mu}}_b \right)\omega ^c_{a{{\mu}}} -\frac{1}{2\kappa} e\,\eta^{ce}\left(e^{{\mu}}_e e^\gamma_f - e^\gamma_e e^{{\mu}}_f \right)\omega ^f_{b{{\mu}}} \right\} +\left( e^\nu_{c,\mu}e^b_\alpha +e^\nu_c e^b_{\alpha,\mu} \right) \frac{1}{2\kappa} e\, \eta^{ce} \left( e^\gamma_e e^\mu_b - e^\mu_e e^\gamma_b \right)\\ &=\frac{1}{2\kappa} e \left\{ e^\nu_c e^b_\alpha\eta^{ae} \left( e^{{\mu}}_e e^\gamma_b -e^\gamma_e e^{{\mu}}_b \right)\omega ^c_{a{{\mu}}} -e^\nu_c e^b_\alpha\eta^{ce}\left(e^{{\mu}}_e e^\gamma_f - e^\gamma_e e^{{\mu}}_f \right)\omega ^f_{b{{\mu}}} +\left( e^\nu_{c,\mu}e^b_\alpha +e^\nu_c e^b_{\alpha,\mu} \right) \eta^{ce} \left( e^\gamma_e e^\mu_b - e^\mu_e e^\gamma_b \right) \right\}\\ &=\frac{1}{2\kappa} e \left\{ e^\nu_a e^b_\alpha\eta^{ce} \left( e^{{\mu}}_e e^\gamma_b -e^\gamma_e e^{{\mu}}_b \right)\omega ^a_{c{{\mu}}} -e^\nu_c e^f_\alpha\eta^{ce}\left(e^{{\mu}}_e e^\gamma_b - e^\gamma_e e^{{\mu}}_b \right)\omega ^b_{f{{\mu}}} -\left( e^\nu_{c,\mu}e^b_\alpha +e^\nu_c e^b_{\alpha,\mu} \right) \eta^{ce} \left( e^\mu_e e^\gamma_b -e^\gamma_e e^\mu_b \right) \right\}\\ &=\frac{1}{2\kappa} e \, \eta^{ce} \left( e^\mu_e e^\gamma_b -e^\gamma_e e^\mu_b \right) \left\{ e^\nu_a e^b_\alpha \omega ^a_{c{{\mu}}} -e^\nu_c e^f_\alpha\omega ^b_{f{{\mu}}} -\left( e^\nu_{c,\mu}e^b_\alpha +e^\nu_c e^b_{\alpha,\mu} \right) \right\}\\ &=\frac{1}{2\kappa} e \, \eta^{ce} \left( e^\mu_e e^\gamma_b -e^\gamma_e e^\mu_b \right) \left\{ e^b_\alpha \left( e^\nu_a \omega ^a_{c\mu}-e^\nu_{c,\mu}\right) -e^\nu_c\left( e^f_\alpha \omega^b_{f\mu} + e^b_{\alpha,\mu} \right) \right\}\\ &=\frac{1}{2\kappa} e \, \eta^{ce} \left( e^\mu_e e^\gamma_b -e^\gamma_e e^\mu_b \right) \left\{ e^b_\alpha e^\beta_c \Gamma^\nu_{\beta\mu} -e^\nu_c e^b_\beta \Gamma^\beta_{\alpha\mu} \right\} =\frac{1}{2\kappa} e \, \eta^{ce} \left\{ \left( e^\mu_e e^\gamma_b -e^\gamma_e e^\mu_b \right)e^b_\alpha e^\beta_c \Gamma^\nu_{\beta\mu} -\left( e^\mu_e e^\gamma_b -e^\gamma_e e^\mu_b \right)e^\nu_c e^b_\beta \Gamma^\beta_{\alpha\mu} \right\}\\ &=\frac{1}{2\kappa} e \, \eta^{ce} \left\{ e^\mu_e e^\gamma_be^b_\alpha e^\beta_c \Gamma^\nu_{\beta\mu} -e^\gamma_e e^\mu_b e^b_\alpha e^\beta_c \Gamma^\nu_{\beta\mu} -e^\mu_e e^\gamma_b e^\nu_c e^b_\beta \Gamma^\beta_{\alpha\mu}+e^\gamma_e e^\mu_b e^\nu_c e^b_\beta \Gamma^\beta_{\alpha\mu} \right\}\\ &=\frac{1}{2\kappa} e \, \eta^{ce} \left\{ e^\mu_e e^\beta_c \delta^\gamma_\alpha \Gamma^\nu_{\beta\mu} -e^\gamma_e e^\beta_c \delta^\mu_\alpha \Gamma^\nu_{\beta\mu} -e^\mu_e e^\nu_c \delta^\gamma_\beta \Gamma^\beta_{\alpha\mu} +e^\gamma_e e^\nu_c \delta^\mu_\beta \Gamma^\beta_{\alpha\mu} \right\}\\ &=\frac{1}{2\kappa} e \, \left\{ g^{\mu\beta} \delta^\gamma_\alpha \Gamma^\nu_{\beta\mu} -g^{\gamma\beta} \delta^\mu_\alpha \Gamma^\nu_{\beta\mu} -g^{\mu\nu} \delta^\gamma_\beta \Gamma^\beta_{\alpha\mu} + g^{\gamma\nu} \delta^\mu_\beta \Gamma^\beta_{\alpha\mu} \right\} =\frac{1}{2\kappa} \sqrt{-g} \, \left\{ g^{\mu\beta} \delta^\gamma_\alpha \Gamma^\nu_{\beta\mu} + g^{\gamma\nu} \Gamma^\mu_{\alpha\mu} -g^{\gamma\beta} \Gamma^\nu_{\beta\alpha} -g^{\mu\nu} \Gamma^\gamma_{\alpha\mu} \right\}\\ &=\frac{\partial \mathscr{L}_{GR}}{\partial \Gamma_{\nu\gamma}^\alpha} \end{align*}\] \end{adjustwidth} \[\begin{align*} \Gamma_{\alpha \mu}^\beta&=e^\beta_b e^a_\alpha\omega_{a \mu}^b +e^\beta_b e^b_{\alpha,\mu}\\ &=e^\beta_b e^a_\alpha\omega_{a \mu}^b - e^a_{\alpha} e^\beta_{a,\mu}\\ \\ e^b_\beta \Gamma_{\alpha \mu}^\beta&= e^a_\alpha\omega_{a \mu}^b + e^b_{\alpha,\mu}\\ \\ e^\alpha_a \Gamma_{\alpha \mu}^\beta&=e^\beta_b \omega_{a \mu}^b - e^\beta_{a,\mu} \end{align*}\] \textbf{Eq.\eqref{eq:/omega}} \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma}} &= \frac{\partial}{\partial \omega^b_{c\gamma}} \left( \frac{1}{2\kappa} e\, \eta^{ae}e^\sigma_e e^\omega_f \mathscr{R}^f_{a\omega\sigma} \right)\\ &=\frac{1}{2\kappa} e\, \eta^{ae}e^\sigma_e e^\omega_f\, \frac{\partial}{\partial \omega^b_{c\gamma}} \left( \omega ^f_{a\sigma,\omega}-\omega ^f_{a\omega,\sigma} +\omega ^f_{d\omega} \omega ^d_{a\sigma} -\omega ^f_{d\sigma} \omega ^d_{a\omega} \right)\\ &=\frac{1}{2\kappa} e\, \eta^{ae}e^\sigma_e e^\omega_f \left( \delta^{f}_{b}\delta^{c}_{d}\delta^{\gamma}_{\omega}\omega ^d_{a\sigma} +\omega ^f_{d\omega}\delta^{d}_{b}\delta^{c}_{a}\delta^{\gamma}_{\sigma} -\delta^{f}_{b}\delta^{c}_{d}\delta^{\gamma}_{\sigma}\omega ^d_{a\omega} -\omega ^f_{d\sigma}\delta^{d}_{b}\delta^{c}_{a}\delta^{\gamma}_{\omega} \right)\\ &=\frac{1}{2\kappa} e\, \eta^{ae}e^\sigma_e e^\omega_f \left( \delta^{f}_{b}\delta^{\gamma}_{\omega}\omega ^c_{a\sigma} +\omega ^f_{b\omega}\delta^{c}_{a}\delta^{\gamma}_{\sigma} -\delta^{f}_{b}\delta^{\gamma}_{\sigma}\omega ^c_{a\omega} -\omega ^f_{b\sigma}\delta^{c}_{a}\delta^{\gamma}_{\omega} \right)\\ &=\frac{1}{2\kappa} e\, \left( \eta^{ae}e^\sigma_e e^\omega_f \delta^{f}_{b}\delta^{\gamma}_{\omega}\omega ^c_{a\sigma} +\eta^{ae}e^\sigma_e e^\omega_f \omega ^f_{b\omega}\delta^{c}_{a}\delta^{\gamma}_{\sigma} -\eta^{ae}e^\sigma_e e^\omega_f \delta^{f}_{b}\delta^{\gamma}_{\sigma}\omega ^c_{a\omega} -\eta^{ae}e^\sigma_e e^\omega_f \omega ^f_{b\sigma}\delta^{c}_{a}\delta^{\gamma}_{\omega} \right)\\ &=\frac{1}{2\kappa} e\, \left( \eta^{ae}e^\sigma_e e^\gamma_b\omega ^c_{a\sigma} +\eta^{ce}e^\gamma_e e^\omega_f \omega ^f_{b\omega} -\eta^{ae}e^\gamma_e e^\omega_b \omega ^c_{a\omega} -\eta^{ce}e^\sigma_e e^\gamma_f \omega ^f_{b\sigma} \right)\\ &= \frac{1}{2\kappa} e\,\eta^{ae} e^{\textcolor{red}{\mu}}_e e^\gamma_b\omega ^c_{a{\textcolor{red}{\mu}}} +\frac{1}{2\kappa} e\,\eta^{ce}e^\gamma_e e^{\textcolor{red}{\mu}}_f \omega ^f_{b{\textcolor{red}{\mu}}} -\frac{1}{2\kappa} e\,\eta^{ae}e^\gamma_e e^{\textcolor{red}{\mu}}_b \omega ^c_{a{\textcolor{red}{\mu}}} -\frac{1}{2\kappa} e\,\eta^{ce}e^{\textcolor{red}{\mu}}_e e^\gamma_f \omega ^f_{b{\textcolor{red}{\mu}}}\\ &= \frac{1}{2\kappa} e\,\eta^{ae} \left( e^{{\mu}}_e e^\gamma_b -e^\gamma_e e^{{\mu}}_b \right)\omega ^c_{a{{\mu}}} -\frac{1}{2\kappa} e\,\eta^{ce}\left(e^{{\mu}}_e e^\gamma_f - e^\gamma_e e^{{\mu}}_f \right)\omega ^f_{b{{\mu}}} \end{align*}\] \textbf{Eq.\eqref{eq:/omega*/omega}} \[\begin{align*} \frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\gamma}}\omega^b_{c\varepsilon} &= \left[ \frac{1}{2\kappa} e\,\eta^{ae} \left( e^{{\mu}}_e e^\gamma_b -e^\gamma_e e^{{\mu}}_b \right) \omega ^c_{a{{\mu}}} -\frac{1}{2\kappa} e\,\eta^{ce} \left(e^{{\mu}}_e e^\gamma_f - e^\gamma_e e^{{\mu}}_f \right) \omega ^f_{b{{\mu}}} \right]\omega^b_{c\varepsilon}\\ &= \frac{1}{2\kappa} e\, \eta^{ae} \left( e^{{\mu}}_e e^\gamma_b -e^\gamma_e e^{{\mu}}_b \right) \omega^b_{c\varepsilon}\omega^c_{a\mu} -\frac{1}{2\kappa} e\, \eta^{ce} \left(e^{{\mu}}_e e^\gamma_f - e^\gamma_e e^{{\mu}}_f \right) \omega^f_{b\mu}\omega^b_{c\varepsilon}\\ &=\frac{1}{2\kappa} e\, \eta^{{\textcolor{red}{c}}e}\left({e^\mu_e e^\gamma_b-e^\gamma_e e^\mu_b} \right)\omega^b_{{\textcolor{red}{d}}\varepsilon}\omega^{\textcolor{red}{d}}_{{\textcolor{red}{c}}\mu} -\frac{1}{2\kappa} e\, \eta^{ce} \left(e^\mu_e e^\gamma_{\textcolor{red}{b}} -e^\gamma_e e^\mu_{\textcolor{red}{b}} \right)\omega^{\textcolor{red}{b}}_{{\textcolor{red}{d}}\mu}\omega^{\textcolor{red}{d}}_{c\varepsilon}\\ &=\frac{1}{2\kappa} e\, \eta^{ce}\left( e^\mu_e e^\gamma_b-e^\gamma_e e^\mu_b\right) \left( \omega^b_{{{d}}\varepsilon}\omega^{{d}}_{{{c}}\mu}- \omega^{{b}}_{{{d}}\mu}\omega^{{d}}_{c\varepsilon} \right)\\ &\underbrace{=}_{\eqref{eq:/omega,}}\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \left( \omega^b_{{{d}}\varepsilon}\omega^{{d}}_{{{c}}\mu}- \omega^{{b}}_{{{d}}\mu}\omega^{{d}}_{c\varepsilon} \right) \end{align*}\] \textbf{Eq.\eqref{eq:Noether_t}} \\ \[\begin{align*} t^\gamma_\varepsilon&=\frac{\partial \mathscr{L}_{GR}}{\partial \omega^b_{c\mu,\gamma}} \mathscr{R}^b_{c\varepsilon\mu} - \delta^\gamma_\varepsilon \mathscr{L}_{GR}\\ &=\frac{1}{2\kappa} e\,\eta^{ce}\left(e^\mu_e e^\gamma_b -e^\gamma_e e^\mu_b \right)\mathscr{R}^b_{c\varepsilon\mu}-\delta^\gamma_\varepsilon \mathscr{L}_{GR}\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\eta^{ce}\left(e^\mu_e e^\gamma_b -e^\gamma_e e^\mu_b\right)e^{b}_{\alpha}e^{\beta}_{c} R^\alpha_{\beta\varepsilon\mu}-\delta^\gamma_\varepsilon \mathscr{L}_{GR}\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\eta^{ce}\left(e^\mu_e e^\gamma_b e^{b}_{\alpha}e^{\beta}_{c} -e^\gamma_e e^\mu_b e^{b}_{\alpha}e^{\beta}_{c} \right)R^\alpha_{\beta\varepsilon\mu}-\delta^\gamma_\varepsilon \mathscr{L}_{GR}\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\eta^{ce}\left( e^\mu_e \delta^\gamma_{\alpha} e^{\beta}_{c} -e^\gamma_e \delta^\mu_{\alpha}e^{\beta}_{c} \right)R^\alpha_{\beta\varepsilon\mu}-\delta^\gamma_\varepsilon \mathscr{L}_{GR}\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\left( \eta^{ce}e^\mu_e e^{\beta}_{c} \delta^\gamma_{\alpha} R^\alpha_{\beta\varepsilon\mu} -\eta^{ce}e^\gamma_e e^{\beta}_{c} \delta^\mu_{\alpha} R^\alpha_{\beta\varepsilon\mu} \right)-\delta^\gamma_\varepsilon \mathscr{L}_{GR}\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\left( g^{\beta\mu} R^\gamma_{\beta\varepsilon\mu} - g^{\beta\gamma}\delta^\mu_{\alpha} R^\alpha_{\beta{\varepsilon\mu}} \right)-\delta^\gamma_\varepsilon \frac{1}{2\kappa} \sqrt{-g}\, R\\ &=\frac{1}{2\kappa} \sqrt{-g}\,\left( g^{\beta\mu} R^\gamma_{\beta\varepsilon\mu} \textcolor{red}{+} g^{\beta\gamma}\delta^\mu_{\alpha} R^\alpha_{\beta\textcolor{red}{\mu\varepsilon}} \right)-\delta^\gamma_\varepsilon \frac{1}{2\kappa} \sqrt{-g}\, R\\ &=\left(g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\frac{1}{2\kappa} \sqrt{-g} \end{align*}\] \[\begin{align*} g^{\beta\mu} R^\gamma_{\beta\varepsilon\mu} - g^{\beta\gamma}\delta^\mu_{\alpha} R^\alpha_{\beta{\varepsilon\mu}} &=g^{\beta\mu}\delta^\gamma_{\alpha} R^\alpha_{\beta\varepsilon\mu} - g^{\beta\gamma}\delta^\mu_{\alpha} R^\alpha_{\beta{\varepsilon\mu}}\\ \\ \\ \\ g^{\beta\mu}\delta^\gamma_{\alpha} - g^{\beta\gamma}\delta^\mu_{\alpha} =\eta^{ce}e^\beta_c e^\gamma_e-\eta^{ce}\\ \end{align*}\] \textbf{Eq.\eqref{eq:t_gravity}} \\ If metric compatible, the following is valid: \[\begin{align*} g_{\alpha\gamma}R^\gamma_{\beta\varepsilon\mu}=R_{\alpha\beta\varepsilon\mu}=-R_{\beta\alpha\varepsilon\mu}=-R_{\alpha\beta\mu\varepsilon}=R_{\varepsilon\mu\alpha\beta} \end{align*}\] \[\begin{align*} \left(g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\frac{1}{2\kappa} \sqrt{-g} &=\left(g^{\beta\mu}g^{\gamma\alpha}R_{\alpha\beta\varepsilon\mu}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(g^{\beta\mu}g^{\gamma\alpha}R_{\textcolor{red}{\beta\alpha}\textcolor{orange}{\mu\varepsilon}}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(g^{\gamma\alpha}R^\mu_{{\alpha}{\mu\varepsilon}}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(g^{\gamma\alpha}R_{{\alpha}{\varepsilon}}+g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(2g^{\beta\gamma}R_{\beta\varepsilon}-\delta^\gamma_\varepsilon R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(g^{\beta\gamma}R_{\beta\varepsilon}-\frac{1}{2}\delta^\gamma_\varepsilon R \right)\frac{1}{\kappa} \sqrt{-g}\\ \end{align*}\] \[\begin{align*} t_{\alpha\varepsilon}&=\left(g_{\alpha\gamma}g^{\beta\mu}R^\gamma_{\beta\varepsilon\mu} +R_{\alpha\varepsilon} -g_{\alpha\varepsilon} R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(g^{\beta\mu}R_{\alpha\beta\varepsilon\mu} +R_{\alpha\varepsilon} -g_{\alpha\varepsilon} R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(g^{\beta\mu}R_{\textcolor{red}{\beta\alpha}\textcolor{blue}{\mu\varepsilon}} +R_{\alpha\varepsilon} -g_{\alpha\varepsilon} R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(R^\mu_{{\alpha}{\mu\varepsilon}} +R_{\alpha\varepsilon} -g_{\alpha\varepsilon} R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(R_{{\alpha}{\varepsilon}} +R_{\alpha\varepsilon} -g_{\alpha\varepsilon} R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\left(2R_{{\alpha}{\varepsilon}} -g_{\alpha\varepsilon} R \right)\frac{1}{2\kappa} \sqrt{-g}\\ &=\frac{1}{\kappa} \mathbb{G}_{{\alpha}{\varepsilon}}\sqrt{-g} \end{align*}\]