Why Gravitational Energy Is Not Generally Local

Matter stress-energy is a local covariant tensor because it is obtained by varying the matter action with respect to the metric. Gravitational energy cannot generally be packaged into another universal local tensor of the metric with the same role. The equivalence principle allows first metric derivatives to vanish at an event, while gravitational energy in useful formulations is often built from those derivatives. Curvature, quasi-local quantities, and asymptotic charges remain meaningful; localization changes its form rather than disappearing.

Required background. We use spacetime symmetries, Einstein’s equation, and the gravitational action with boundary terms.

Helpful background. The argument draws on special-relativistic energy, tensor fields, and presymplectic boundary charges.

Why matter has a local tensor

For a covariant matter action,

$$T_{ab}=-\frac{2}{\sqrt{-g}} \frac{\delta S_{\mathrm m}}{\delta g^{ab}}.$$

This object is symmetric under the usual metric assumptions, transforms tensorially, and gives local energy density $T_{ab}u^au^b$ for an observer. On the matter equations it satisfies $\nabla^aT_{ab}=0$. It appears on the source side of Einstein’s equation.

Moving the Einstein tensor to the right does not define a second stress tensor: on solutions it would merely cancel the matter source, and it contains the geometry being solved for. A gravitational energy expression should also distinguish radiative, binding, and boundary energy in appropriate regimes; no unique local tensor built solely from $g_{ab}$ meets all these demands.

The local-inertial obstruction

Many canonical gravitational energy densities are quadratic in connection coefficients or first derivatives of the metric. At any chosen event $p$, Riemann normal coordinates make

$$\Gamma^a{}_{bc}(p)=0, \qquad \partial_cg_{ab}(p)=0,$$

even when $R^a{}_{bcd}(p)\ne0$. Thus a connection-quadratic expression can be made zero at an arbitrary event by coordinates. If it were a nonzero tensor, vanishing in one chart would force vanishing in all charts; therefore it is a pseudotensor or coordinate-dependent density.

MIT’s discussion of gravitational stress constructions emphasizes this coordinate dependence and the separate status of total fluxes Guth–Turner 2017, Lectures 24–25, pp. 104–113, PDF.

This argument is bounded. It rules out interpreting connection-based expressions as universal local tensors. It does not say that curvature is unobservable or that no curvature tensor can be formed. The Bel–Robinson tensor, for example, is covariant and quadratic in Weyl curvature, but has different dimensions and algebraic meaning from an ordinary material stress tensor.

Pseudotensors and superpotentials

In a chosen coordinate or background framework one can write a combined balance law

$$\partial_\mu\left[\sqrt{-g} (T^\mu{}_\nu+t^\mu{}_\nu)\right]=0,$$

where $t^\mu{}_\nu$ is a gravitational pseudotensor. Different rearrangements of Einstein’s nonlinear terms and different superpotentials give different local densities. Their coordinate dependence is not automatically a defect when the background, gauge, or asymptotic frame is part of the problem.

Integrating a divergence can turn differences between pseudotensors into surface terms. Under common falloff and coordinate conditions, distinct prescriptions may agree on total asymptotic energy even while disagreeing pointwise. The invariant content resides in the controlled boundary limit, not in an unqualified local density.

Quasi-local and asymptotic alternatives

Quasi-local energy associates a quantity with a closed two-surface and the geometry it bounds. Different constructions impose different reference embeddings, causal conditions, or Hamiltonian boundary terms; there is no single universally preferred value for every surface. Szabados reviews why gravitational energy naturally moves from pointwise densities to two-surface and boundary constructions Szabados 2009, §§ 3–4.

At spatial or null infinity, asymptotic structure selects symmetry generators and normalizations. Surface charges can then define total energy, momentum, angular momentum, and radiative balance. In a stationary spacetime, a Killing field provides a further normalization. These quantities can be nonzero in vacuum, where $T_{ab}=0$, because they measure the gravitational field through boundary data.

The necessary qualifications are part of every result: background or reference term, boundary surface, falloff, asymptotic frame, symmetry generator, and orientation. Calling any one of them “the local gravitational stress tensor” erases precisely the structure that makes it meaningful.

A two-coordinate check

Compute a connection-quadratic pseudodensity in a generic chart at $p$, then transform to Riemann normal coordinates. It can change or vanish while the curvature and a small-loop holonomy remain. Next integrate the associated superpotential over a large surface under allowed asymptotic transformations. The surface charge may remain unchanged. This contrast demonstrates coordinate-dependent localization together with invariant total information.

Exercises

Why does the ability to set a gravitational pseudotensor to zero at one event not imply that an isolated spacetime has zero total mass?

Solution

The pseudotensor is a coordinate-dependent local expression built from connection coefficients, which can vanish in normal coordinates at one event. Total mass is a boundary charge determined by the geometry and normalization on an asymptotic surface. A local coordinate change cannot generally alter those prescribed asymptotic data, so the boundary integral can remain nonzero even though the chosen pointwise density vanishes.

References

• Guth, Alan H., lectures with notes by Andrew P. Turner. 8.962 General Relativity, Spring 2017. MIT (2017). PDF.
• Szabados, László B. “Quasi-Local Energy-Momentum and Angular Momentum in General Relativity.” Living Reviews in Relativity 12, 4 (2009). DOI.