Flat-Background Hilbert Stress-Energy from Metric Variation

Stress-energy can be defined as the local response of a matter action to a change in a prescribed background metric. This construction produces a symmetric tensor before one returns to Minkowski space, and it makes precise both how the Hilbert tensor differs from canonical energy-momentum and when the two agree up to improvements.

Required background. Classical Fields and Their Actions provides field actions and Euler–Lagrange equations, while Events, Intervals, and Minkowski Geometry fixes the spacetime conventions. Helpful background. Calculus and Linear Algebra for Relativity supplies variations and contractions; Presymplectic Potentials and Currents develops boundary variations and current ambiguities.

Metric response

Varying the metric measures how the matter action responds locally to changes in spacetime lengths and angles. With the global convention used throughout this site,

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

The metric is initially an arbitrary nondynamical background; setting $g_{\mu\nu}=\eta_{\mu\nu}$ afterward recovers flat-space stress-energy. This definition gives a symmetric tensor because $g^{\mu\nu}$ is symmetric.

The definition is equivalent, after all integrations by parts required by the functional derivative, to

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

If $g^{\mu\nu}=\eta^{\mu\nu}+k^{\mu\nu}$, the leading coupling is $-\tfrac12\int T_{\mu\nu}k^{\mu\nu}$. Equivalently, for $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, inversion gives $k^{\mu\nu}=-h^{\mu\nu}$ and the coupling is $+\tfrac12\int T^{\mu\nu}h_{\mu\nu}$. This sign pair is a useful check against confusing variations of the metric and inverse metric.

Tong, General Relativity, §§4.5.2–4.5.3, printed pp. 183–186, Eqs. (4.46)–(4.50) derives the same response definition and its flat-space translation interpretation. The response is local: changing a component of the background metric probes the conjugate energy, momentum, or stress density.

Because the metric is dimensionless in ordinary coordinate conventions, $\delta S/\delta g^{\mu\nu}$ has the dimensions of an action density. Dividing by $\sqrt{-g}$ therefore leaves energy per spatial volume, as required. This dimensional check does not impose positivity: positivity is a further property of the matter theory and observer projection.

Scalar derivation

Consider minimally coupled real scalar matter,

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

Hold $\phi$ fixed while varying the inverse metric. Because $\nabla_\mu\phi=\partial_\mu\phi$ for a scalar, no connection variation enters this example. Jacobi’s determinant identity and $g_{\mu\rho}g^{\rho\nu}=\delta_\mu{}^\nu$ give

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

Substitution gives

$$\boxed{ T_{\mu\nu} =\nabla_\mu\phi\nabla_\nu\phi -g_{\mu\nu} \left[ \frac12\nabla_\rho\phi\nabla^\rho\phi+V \right].}$$

In flat space, $T_{00}=\tfrac12\dot\phi^2+\tfrac12|\boldsymbol\nabla\phi|^2+V$. The positive kinetic energy checks both the determinant variation and the $(-,+,+,+)$ signature. This calculation is worked explicitly in Tong, §4.5.3, printed pp. 184–186, Eq. (4.51).

Conservation and comparison

For the scalar tensor just derived, direct differentiation gives the stronger diagnostic

$$\boxed{ \nabla_\mu T^{\mu\nu} =\left(\Box\phi-V'(\phi)\right)\nabla^\nu\phi.}$$

The crossed second-derivative terms cancel because covariant derivatives commute on a scalar. Thus the scalar equation $\Box\phi-V'=0$ implies covariant conservation, and the identity shows exactly what remains off shell.

More generally, for a diffeomorphism-covariant matter action, a compactly supported infinitesimal diffeomorphism varies both matter fields and metric. After integration by parts, invariance gives

$$\nabla_\mu T^{\mu\nu}=0$$

when the matter equations hold and no nondynamical external field supplies momentum. Off shell, the divergence is instead tied to the matter Euler–Lagrange expressions. If a prescribed electromagnetic or other background is held fixed, matter stress-energy can have a force-density source; the combined modeled system must be used for conservation.

The scalar also gives a direct canonical comparison. Returning to flat space,

$$T_{\rm H}^{\mu\nu} =\partial^\mu\phi\,\partial^\nu\phi +\eta^{\mu\nu}\mathcal L =T_{\rm can}^{\mu\nu}.$$

No improvement or field equation is needed for this standard first-derivative scalar. In the general bosonic tensor theory, however, the logical chain is

$$T_{\rm B}=T_{\rm can}+\partial B, \qquad T_{\rm H}=T_{\rm B}+\text{terms proportional to the matter equations},$$

for a specified minimal covariantization, so Hilbert and Belinfante agree on shell rather than by definition. Curvature terms that vanish after setting $g=\eta$ can still vary nontrivially and produce symmetric, identically conserved flat-space improvements. Pons, §§4.2–4.4 and the Appendix separates these equation-of-motion and covariantization ambiguities. The source-free Maxwell calculation on Canonical Energy-Momentum and Improvement provides the vector-field comparison; the metric variation itself is independently derived in Lorentz Force and Electromagnetic Stress-Energy.

Metric response is defined off shell, whereas its covariant conservation is generally an on-shell statement. Keeping those facts separate matters when deriving Ward identities, coupling to external backgrounds, or comparing actions related only after use of field equations.

Exercise: homogeneous scalar matter

Let $\phi=\phi(t)$ in Minkowski space. Compute the energy density and isotropic pressure, then take the constant-field limit.

Solution

For a comoving observer $u^\mu=(1,0,0,0)$,

$$\rho=T_{\mu\nu}u^\mu u^\nu =\frac12\dot\phi^2+V(\phi).$$

The spatial diagonal components are

$$T_{ij} =\left(\frac12\dot\phi^2-V\right)\delta_{ij}, \qquad p=\frac13\delta^{ij}T_{ij} =\frac12\dot\phi^2-V.$$

If $\dot\phi=0$ at a constant configuration, then

$$T_{\mu\nu}=-Vg_{\mu\nu}, \qquad p=-\rho=-V.$$

The observer energy remains $\rho=T_{\mu\nu}u^\mu u^\nu=V$ because $u^\mu u_\mu=-1$. A field that stays constant in time is an actual solution only if $V'(\phi)=0$; a turning point with $\dot\phi=0$ can instead have nonzero acceleration and only instantaneously share this stress tensor. This limit distinguishes vacuum-like stress from pressureless matter even when both are spatially homogeneous.

Limits and onward use

Spinor actions require a tetrad or coframe variation before translating the result to a metric tensor. Higher-derivative and nonminimal actions require boundary terms and integrations by parts; varying only explicit metric factors is then wrong. The definition above concerns matter, not a generally covariant local gravitational energy tensor.

Local Balance Laws and Flux Integrals turns $\nabla_\mu T^{\mu\nu}=0$ and its sourced variants into finite-region measurements with explicit orientation signs.

References

• Josep M. Pons, “Noether Symmetries, Energy-Momentum Tensors and Conformal Invariance in Classical Field Theory,” Journal of Mathematical Physics 52, 012904 (2011), doi:10.1063/1.3532941, Open PDF, §§4.2–4.4, printed pp. 14–18, and Appendix, printed pp. 27–29.
• David Tong, General Relativity, University of Cambridge Part III lecture notes (2019), official PDF, §§4.5.2–4.5.3, printed pp. 183–186, Eqs. (4.46)–(4.52).