Gravitationally Coupled Stress-Energy from Metric Variation

The stress–energy tensor that enters Einstein’s equation is the local response of the matter action to the inverse metric. This definition fixes its normalization, makes symmetry manifest, and remains meaningful on a curved background without relying on spacetime translations. Its covariant conservation is not part of the definition: it follows from diffeomorphism invariance when the matter equations hold and all dynamical fields exchanging energy and momentum are included.

Required background. Classical Fields and Their Actions provides action variation, Euler–Lagrange equations, and gauge symmetry; Flat-Background Hilbert Stress-Energy from Metric Variation provides the corresponding background response in special relativity; Affine Connections and Covariant Derivatives provides covariant differentiation and integration by parts.

Helpful background. Events, Intervals, and Minkowski Geometry reviews observer energy and Lorentz frames; Tensor Fields and Natural Operations reviews tensor contractions and pullbacks; Continuum Kinematics and Balance Laws supplies the meaning of local balance; Presymplectic Potentials and Currents prepares integrations by parts and boundary variations.

Metric response and its sign

For a matter action $S_{\rm m}[g,Ψ]$, where Ψ denotes all matter fields, the site convention is

$$\boxed{ T_{\mu\nu} =-\frac{2}{\sqrt{-g}} \frac{\delta S_{\rm m}}{\delta g^{\mu\nu}}.}$$

Equivalently, after moving derivatives off $\delta g^{\mu\nu}$ and imposing the boundary conditions of the variational problem,

$$\delta_g S_{\rm m} =-\frac12\int_{\mathcal M}d^4x\,\sqrt{-g}\, T_{\mu\nu}\,\delta g^{\mu\nu}.$$

Because $\delta g^{\mu\nu}$ is symmetric, only the symmetric part of its coefficient is probed. If instead the covariant metric is varied, $\delta g_{\alpha\beta}=-g_{\alpha\mu}g_{\beta\nu}\delta g^{\mu\nu}$ gives

$$T^{\mu\nu} =\frac{2}{\sqrt{-g}} \frac{\delta S_{\rm m}}{\delta g_{\mu\nu}}.$$

The functional derivative is taken with all independent matter variables fixed. This instruction matters when a displayed Lagrangian contains raised indices, a Hodge dual, covariant derivatives of nonscalar fields, or curvature: each of those objects carries hidden metric dependence. One must vary those dependencies, integrate derivatives of the metric variation by parts, and retain the boundary contribution appropriate to the stated variational problem. Merely differentiating the explicit $g^{\mu\nu}$ factors can miss entire terms. Actions related by use of a matter equation are also not automatically interchangeable off shell, because a functional derivative must be taken before imposing that equation.

Reall uses this latter form in Reall 2013, §10.3, pp. 119–121, PDF; translating the varied metric is essential before comparing signs.

Complete variation of a scalar action

Consider a minimally coupled real scalar,

$$S_{\rm m} =-\int d^4x\,\sqrt{-g} \left[ \frac12g^{\rho\sigma}\nabla_\rhoφ\nabla_\sigmaφ +V(φ) \right].$$

Hold φ fixed during the metric variation. A scalar’s first derivative is independent of the connection, so the two required identities are

$$\delta\sqrt{-g} =-\frac12\sqrt{-g}\,g_{\mu\nu}\delta g^{\mu\nu}, \qquad \delta\!\left(g^{\rho\sigma}\nabla_\rhoφ\nabla_\sigmaφ\right) =\nabla_\muφ\nabla_\nuφ\,\delta g^{\mu\nu}.$$

Keeping both contributions gives

$$\delta_g S_{\rm m} =-\frac12\int d^4x\,\sqrt{-g} \left\{ \nabla_\muφ\nabla_\nuφ -g_{\mu\nu} \left[ \frac12(\nablaφ)^2+V \right] \right\} \delta g^{\mu\nu},$$

and hence

$$T_{\mu\nu}^{(φ)} =\nabla_\muφ\nabla_\nuφ -g_{\mu\nu} \left[ \frac12(\nablaφ)^2+V(φ) \right].$$

In a local inertial frame, $T_{00}=\tfrac12\dotφ^2+\tfrac12|\boldsymbol\nablaφ|^2+V$. For a nonnegative potential this has the expected positive kinetic contribution, independently checking the determinant sign and the factor of two. The same variation is displayed in Reall 2013, §10.3, pp. 119–120, equations (10.27)–(10.34), PDF.

Why this tensor couples to gravity

With the gravitational normalization

$$S_{\rm grav} =\frac{1}{16\pi G} \int d^4x\,\sqrt{-g}\,(R-2\Lambda),$$

and the appropriate gravitational boundary term, its bulk inverse-metric variation is

$$\delta_g S_{\rm grav} =\frac{1}{16\pi G} \int d^4x\,\sqrt{-g}\, (G_{\mu\nu}+\Lambda g_{\mu\nu})\delta g^{\mu\nu}.$$

Stationarity of $S_{\rm grav}+S_{\rm m}$ for arbitrary interior variations therefore gives

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =8\pi G T_{\mu\nu}.$$

This calculation, not dimensional analogy alone, fixes the $8\pi G$ normalization and identifies the metric-response tensor as the gravitational source.

Diffeomorphism invariance supplies the associated identity. For the scalar tensor above, direct differentiation gives

$$\nabla_\mu T^\mu{}_{\nu} =\left(\Boxφ-V'(φ)\right)\nabla_\nuφ.$$

Thus $\nabla_\mu T^{\mu}{}_{\nu}=0$ on the scalar equation of motion. Off shell the divergence records the matter Euler–Lagrange expression; with a prescribed external field it instead records the force density transferred to that background. Reall derives this scalar diffeomorphism identity in Reall 2013, § 10.3, pp. 120–121, Eqs. (10.35)–(10.40), PDF. Tensor and gauge fields have additional Lie-derivative terms in the off-shell identity, although the total on-shell conservation conclusion remains the same.

Observer content and the canonical comparison

For any future-directed unit observer $u^\mu$, let $h_{\mu\nu}=g_{\mu\nu}+u_\mu u_\nu$. Define

$$\rho=T_{\mu\nu}u^\mu u^\nu, \qquad q_\mu=-h_\mu{}^\alpha T_{\alpha\beta}u^\beta, \qquad S_{\mu\nu}=h_\mu{}^\alpha h_\nu{}^\beta T_{\alpha\beta}.$$

Then $q_\mu u^\mu=0$, $S_{\mu\nu}u^\nu=0$, and

$$T_{\mu\nu} =\rho u_\mu u_\nu +2u_{(\mu}q_{\nu)} +S_{\mu\nu}.$$

These projections are measured energy density, momentum or energy flux, and spatial stress. Reall gives the corresponding local orthonormal-frame interpretation in Reall 2013, § 5.2, pp. 51–53, PDF.

In an observer-adapted orthonormal tetrad, these definitions read $T_{\hat0\hat0}=\rho$, $T_{\hat0\hat i}=-q_{\hat i}$, and $T_{\hat i\hat j}=S_{\hat i\hat j}$. They are local statements at one event. Comparing components at separated events requires a specified transport rule or a symmetry-generated conserved current; covariant conservation alone does not canonically identify tangent spaces or supply a global matter energy on an arbitrary spacetime.

The canonical Noether tensor of a flat-space Lagrangian instead arises from translation invariance. It need not be symmetric or gauge invariant, and it can be shifted by an identically conserved improvement. Under suitable hypotheses, Belinfante improvement relates it to the flat limit of the metric tensor; those hypotheses can fail for curvature couplings or higher derivatives. The decisive distinction for GR is operational: metric response is the tensor obtained when the dynamical metric is varied. Carroll compares the metric and canonical constructions in Carroll 1997, Chapter 4, pp. 116–117, PDF.

Exercise: constant potential as a sign check

Let the matter action contain only $S_V=-\int d^4x\sqrt{-g}\,V_0$, with $V_0$ constant. Find $T_{\mu\nu}$ and the energy density measured by any unit timelike observer.

Solution

Only the determinant varies:

$$\delta S_V =\frac12\int d^4x\,\sqrt{-g}\, V_0 g_{\mu\nu}\delta g^{\mu\nu}.$$

Comparison with $\delta S_{\rm m}=-\tfrac12\int\sqrt{-g}\,T_{\mu\nu}\delta g^{\mu\nu}$ gives

$$T_{\mu\nu}=-V_0g_{\mu\nu}.$$

Therefore $\rho=T_{\mu\nu}u^\mu u^\nu=V_0$, since $u^\mu u_\mu=-1$. A positive constant potential has positive observer energy and negative isotropic pressure; this catches the common error of losing the minus sign in $\delta\sqrt{-g}$.

References

• Carroll, Sean M. Lecture Notes on General Relativity. NSF-ITP/97-147, arXiv:gr-qc/9712019v1, 1997. arXiv. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. PDF.