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Chapter 13: Einstein Field Equations
Variational Derivation
The Einstein field equations can be derived from a variational principle, just like other fundamental equations in physics. The action principle provides the deepest understanding of why the equations take the form they do, and it is essential for quantization attempts and for coupling gravity to matter fields.
The Einstein-Hilbert Action
The gravitational action that yields the Einstein equations upon variation is the Einstein-Hilbert action:
$$S_{\text{EH}} = \frac{1}{16\pi G} \int d^4x\, \sqrt{-g}\, (R - 2\Lambda)$$
The simplest generally covariant action involving at most second derivatives of the metric
Here \( g = \det(g_{\mu\nu}) \) is the determinant of the metric tensor (negative for Lorentzian signature, hence \( \sqrt{-g} \)), \( R \) is the Ricci scalar, and\( \Lambda \) is the cosmological constant. The factor \( \sqrt{-g}\, d^4x \) is the invariant volume element.
The total action is the sum of the gravitational and matter parts:
$$S = S_{\text{EH}} + S_{\text{matter}} = \frac{1}{16\pi G} \int d^4x\, \sqrt{-g}\, (R - 2\Lambda) + S_{\text{matter}}[g_{\mu\nu}, \psi]$$
where \( \psi \) collectively denotes all matter fields. The stress-energy tensor is defined as the variational derivative of the matter action with respect to the metric.
Stress-Energy Tensor from the Action
The stress-energy tensor is defined as:
$$T_{\mu\nu} = -\frac{2}{\sqrt{-g}}\, \frac{\delta S_{\text{matter}}}{\delta g^{\mu\nu}}$$
This definition automatically produces a symmetric, covariantly conserved tensor. The factor of \( -2/\sqrt{-g} \) is chosen so that the definition agrees with the canonical stress-energy tensor for matter fields and gives the correct coupling in the Einstein equations.
For a matter Lagrangian density \( \mathcal{L}_{\text{matter}} \) with\( S_{\text{matter}} = \int d^4x\, \sqrt{-g}\, \mathcal{L}_{\text{matter}} \):
$$T_{\mu\nu} = -2\, \frac{\partial \mathcal{L}_{\text{matter}}}{\partial g^{\mu\nu}} + g_{\mu\nu}\, \mathcal{L}_{\text{matter}}$$
Variation of \( \sqrt{-g} \)
A key ingredient is the variation of the metric determinant. Using Jacobi's formula for the derivative of a determinant:
$$\delta g = g\, g^{\mu\nu}\, \delta g_{\mu\nu} = -g\, g_{\mu\nu}\, \delta g^{\mu\nu}$$
The second equality follows from differentiating the identity \( g^{\mu\alpha}g_{\alpha\nu} = \delta^\mu_\nu \), which gives \( \delta g_{\mu\nu} = -g_{\mu\alpha}g_{\nu\beta}\,\delta g^{\alpha\beta} \).
Therefore:
$$\delta\sqrt{-g} = -\frac{1}{2\sqrt{-g}}\,\delta g = -\frac{1}{2}\sqrt{-g}\, g_{\mu\nu}\, \delta g^{\mu\nu}$$
This formula appears repeatedly in variational calculations in general relativity. It tells us how the invariant volume element changes when the metric is varied.
Variation of the Ricci Scalar (Palatini Identity)
The variation of the Ricci scalar \( R = g^{\mu\nu}R_{\mu\nu} \) has two parts:
$$\delta R = R_{\mu\nu}\, \delta g^{\mu\nu} + g^{\mu\nu}\, \delta R_{\mu\nu}$$
The first term is straightforward. The second term requires the variation of the Ricci tensor, which is given by the Palatini identity:
$$\delta R_{\mu\nu} = \nabla_\rho\, \delta\Gamma^\rho_{\mu\nu} - \nabla_\nu\, \delta\Gamma^\rho_{\mu\rho}$$
The Palatini identity
This is remarkable: even though \( \Gamma^\rho_{\mu\nu} \) is not a tensor, the variation \( \delta\Gamma^\rho_{\mu\nu} \) is a tensor (the difference of two connections). Therefore the covariant derivatives in the Palatini identity are well-defined.
To derive this, note that \( R_{\mu\nu} = \partial_\rho\Gamma^\rho_{\mu\nu} - \partial_\nu\Gamma^\rho_{\mu\rho} + \Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho} \). Varying and using the fact that \( \delta\Gamma \) is a tensor, the partial derivatives can be promoted to covariant derivatives, and the cross terms cancel exactly, yielding the Palatini identity.
The second term in \( \delta R \) then becomes:
$$g^{\mu\nu}\, \delta R_{\mu\nu} = \nabla_\rho\left(g^{\mu\nu}\, \delta\Gamma^\rho_{\mu\nu} - g^{\mu\rho}\, \delta\Gamma^\sigma_{\mu\sigma}\right) \equiv \nabla_\rho v^\rho$$
This is a total covariant divergence! When integrated over spacetime with \( \sqrt{-g} \), it becomes a surface integral by the generalized Stokes theorem:\( \int d^4x\, \sqrt{-g}\, \nabla_\rho v^\rho = \oint d^3x\, \sqrt{|h|}\, n_\rho v^\rho \). This boundary term vanishes for compact variations (variations that vanish on the boundary), so it does not contribute to the field equations in the bulk.
Full Derivation: \( \delta S / \delta g^{\mu\nu} = 0 \)
Putting everything together, the variation of the total action is:
$$\delta S = \frac{1}{16\pi G}\int d^4x\left[\delta(\sqrt{-g})(R - 2\Lambda) + \sqrt{-g}\,\delta R\right] + \delta S_{\text{matter}}$$
Substituting our results for \( \delta\sqrt{-g} \) and \( \delta R \):
$$\delta S = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\left[-\frac{1}{2}g_{\mu\nu}(R - 2\Lambda)\,\delta g^{\mu\nu} + R_{\mu\nu}\,\delta g^{\mu\nu} + \nabla_\rho v^\rho\right]$$
$$\quad + \int d^4x\, \frac{\delta S_{\text{matter}}}{\delta g^{\mu\nu}}\,\delta g^{\mu\nu}$$
The total divergence \( \nabla_\rho v^\rho \) integrates to a boundary term that vanishes for variations with compact support. Using the definition\( T_{\mu\nu} = -(2/\sqrt{-g})\,\delta S_{\text{matter}}/\delta g^{\mu\nu} \):
$$\delta S = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\left(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} - 8\pi G\, T_{\mu\nu}\right)\delta g^{\mu\nu}$$
Requiring \( \delta S = 0 \) for all variations \( \delta g^{\mu\nu} \) gives:
$$\boxed{R_{\mu\nu} - \frac{1}{2}\, g_{\mu\nu}\, R + \Lambda\, g_{\mu\nu} = 8\pi G\, T_{\mu\nu}}$$
The Einstein Field Equations - derived from a variational principle
Gibbons-Hawking-York Boundary Term
There is a subtlety: the Ricci scalar contains second derivatives of the metric, so the variational principle requires that we fix both \( g_{\mu\nu} \) and its first derivatives on the boundary. To have a well-posed variational problem where only \( g_{\mu\nu} \) is fixed on the boundary (Dirichlet boundary conditions), we must add a boundary term:
$$S_{\text{GHY}} = \frac{1}{8\pi G} \oint_{\partial\mathcal{M}} d^3x\, \sqrt{|h|}\, K$$
The Gibbons-Hawking-York boundary term
Here \( h_{ab} \) is the induced metric on the boundary \( \partial\mathcal{M} \),\( h = \det(h_{ab}) \), and \( K = h^{ab}K_{ab} \) is the trace of the extrinsic curvature of the boundary:
$$K_{ab} = \nabla_a n_b$$
where \( n^\mu \) is the outward-pointing unit normal to the boundary
The complete gravitational action is then:
$$S_{\text{grav}} = \frac{1}{16\pi G}\int_\mathcal{M} d^4x\,\sqrt{-g}\,(R - 2\Lambda) + \frac{1}{8\pi G}\oint_{\partial\mathcal{M}} d^3x\,\sqrt{|h|}\, K$$
The GHY term does not affect the bulk equations of motion but is essential for: (1) a well-defined variational problem, (2) the correct thermodynamic properties of black holes (the Euclidean path integral gives the correct Bekenstein-Hawking entropy), and (3) the ADM formulation of general relativity where spacetime is decomposed into space + time.
Why the Action Principle is Fundamental
Uniqueness (Lovelock's Theorem)
In 4 dimensions, the Einstein-Hilbert action (with \( \Lambda \)) is the unique action that: (i) is built from the metric alone, (ii) is diffeomorphism invariant, and (iii) yields at most second-order field equations. This is Lovelock's theorem. In higher dimensions, additional terms (Gauss-Bonnet, Lovelock invariants) are possible.
Coupling to Matter
The action principle provides a systematic way to couple gravity to any matter field: simply replace the flat metric \( \eta_{\mu\nu} \) with \( g_{\mu\nu} \), partial derivatives with covariant derivatives, and the flat volume element with \( \sqrt{-g}\,d^4x \)in the matter action. This is the "minimal coupling" prescription.
Path to Quantization
The action is the starting point for quantum gravity. In the path integral approach, one sums over all metrics weighted by \( e^{iS/\hbar} \). While the full quantum theory of gravity remains unsolved, the action principle provides the foundation for perturbative quantum gravity, the Euclidean approach to black hole thermodynamics, and attempts at non-perturbative quantization (loop quantum gravity, causal dynamical triangulations).
Symmetries and Conservation Laws
Noether's theorem connects symmetries of the action to conservation laws. The diffeomorphism invariance of \( S_{\text{EH}} \) gives rise to the Bianchi identity\( \nabla^\mu G_{\mu\nu} = 0 \) and hence energy-momentum conservation. This is far more transparent in the action formulation than in the direct tensor approach.