Action Principle (Relativistic)

The foundations of classical field theory are electromagnetism and general relativity. Classical electromagnetism, due to its linear properties, is often used as a testing paradigm for many physical theories. It serves as the foundational demonstration for the relativistic principle of least action, and quantum field theory was initially pioneered by quantum electrodynamics.

On the other hand, general relativity successfully integrates gravitational effects into the curvature of spacetime through the equivalence principle and the generalized principle of relativity, providing new insights into the physical reality of spacetime. Similarly, Einstein's field equations can also be derived through the variation of the Hilbert-Einstein action.

The Hilbert-Einstein action, formulated by Hilbert in 1915 almost simultaneously with Einstein, derives Einstein's field equations. The core mathematical model of general relativity is (pseudo-)Riemannian geometry. In simple terms, it uses a manifold's tangent bundle with a point-wise variable metric \(g_{\mu\nu}\), endowing it with a new structure distinct from the flat Euclidean space. This constitutes an important branch of non-Euclidean geometry.

The Hilbert-Einstein action is expressed as: \[ S = \int \mathcal{L} \sqrt{-g} \, d^4x = \frac{c^4}{16\pi G} \int R \sqrt{-g} \, d^4x + \int \mathcal{L}_M \sqrt{-g} \, d^4x \] where: \[ \mathcal{L} = \left( \frac{c^4}{16\pi G} R + \mathcal{L}_M \right) \sqrt{-g} = \left( \frac{1}{2\kappa} R + \mathcal{L}_M \right) \sqrt{-g} \] \(R = g^{\mu\nu} R_{\mu\nu} = g^{\mu\nu} \delta^\beta_\alpha R^\alpha_{\mu\beta\nu}\) is the Ricci scalar, \(R_{\mu\nu}\) is the Ricci curvature tensor, \(R^\alpha_{\mu\beta\nu}\) is the Riemann curvature tensor, \(g = \det(g_{\mu\nu})\), and \(\mathcal{L}_M\) is the Lagrangian density of matter.

The curvature tensor \(R^\alpha_{\mu\beta\nu}\) is defined as: \[ R^\rho_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma} \] where \(\Gamma^\rho_{\nu\sigma}\) is the tangent connection, commonly referred to as the Christoffel symbol in introductory general relativity textbooks. It can be understood as describing the variation of basis vectors at different points in a non-flat coordinate system.

Interestingly, the tangent connection \(\Gamma^\rho_{\nu\sigma}\) and the curvature tensor \(R^\alpha_{\mu\beta\nu}\) hold a similar status in mathematical language to the electromagnetic field's \(A_\sigma\) and \(F_{\beta\mu}\). In fact, by "hiding" the first two indices of the tangent connection \(\Gamma^\rho_{\nu\sigma}\) and the curvature tensor \(R^\alpha_{\mu\beta\nu}\), we can express them as: \[ \Gamma^\rho_{\nu\sigma} = \left(\Gamma^\rho_\nu \right)_\sigma = \mathbf{\Gamma}_\sigma \leftrightarrow A_\sigma \] \[ R^\alpha_{\mu\beta\nu} = \left(R^\alpha_\mu \right)_{\beta\nu} = \mathbf{R}_{\beta\nu} \leftrightarrow F_{\beta\nu} \]

The first two indices are not vector indices but "Lie algebra indices." Readers who have the opportunity to study gauge theory (or Yang-Mills theory) can delve deeper into how electromagnetic interactions \(U(1)\), weak interactions \(SU(2)\), and strong interactions \(SU(3)\) are unified under the principal connection.

In fact, one can explore the similarities between general relativity and gauge theory or refer to Nobel laureate Chen-Ning Yang's seminal paper 10.1103/PhysRevD.12.3845, the popular science article "Chen-Ning Yang and Modern Mathematics," or Singer's article in "The Mathematical Heritage of Hermann Weyl (1988, American Mathematical Society)."

Originally written in Chinese by the author, these articles are translated into English to invite cross-language resonance.