Gravitational Energy and Pseudo-Tensors

Core Idea

Defining gravitational energy in general relativity is fundamentally problematic because the equivalence principle allows gravity to be locally eliminated — there is no local, covariant expression for gravitational energy density. Pseudo-tensors (such as the Einstein, Landau-Lifshitz, and Moller prescriptions) provide coordinate-dependent expressions for gravitational energy-momentum that, combined with the matter stress-energy tensor, yield a conserved total energy-momentum: ∂_μ(√(-g)(T^μν + t^μν_LL)) = 0 (ordinary, not covariant, divergence). These are not tensors — they transform inhomogeneously and can be made to vanish at any point by choice of coordinates. Meaningful definitions of gravitational energy exist only in special circumstances: total (ADM) mass for asymptotically flat spacetimes, Bondi mass at null infinity (accounting for energy radiated as gravitational waves), and quasi-local energy constructions for bounded regions. The non-localizability of gravitational energy is one of the deepest conceptual features of GR.

Explainer

In Newtonian gravity and in electromagnetism, energy density is a well-defined local quantity: you can point to a region of space and unambiguously say how much energy is stored in the gravitational or electromagnetic field there. In general relativity, this is impossible for gravitational energy. The obstacle is the equivalence principle: at any single point, you can choose coordinates (a local freely falling frame) in which the gravitational field and all its associated effects vanish. If gravitational energy were described by a tensor, it would have to be nonzero in every coordinate system if nonzero in any — but the equivalence principle requires it to vanish in the freely falling frame. This contradiction means no covariant, local expression for gravitational energy density exists.

Pseudo-tensors circumvent this by abandoning covariance. The Landau-Lifshitz pseudo-tensor t^μν_LL, for example, is defined so that ∂_μ((-g)(T^μν + t^μν_LL)) = 0 — an ordinary (partial) divergence, not a covariant one. This is a genuine conservation law that can be integrated over a spatial volume using Gauss's theorem to give a conserved total energy. However, t^μν_LL depends on the coordinate system: in one set of coordinates, the gravitational energy density might be large and positive at a point; in another, it could be zero or negative. Different pseudo-tensor prescriptions (Einstein, Moller, Weinberg, Bergmann-Thomson) give different local distributions but agree on total energy when integrated over all space for asymptotically flat spacetimes.

Meaningful, coordinate-independent gravitational energy is defined only in special geometric situations. For asymptotically flat spacetimes (isolated systems in otherwise empty space), the ADM mass provides the total energy measured at spatial infinity. It equals the sum of all matter energy plus gravitational binding energy and is conserved in time. At future null infinity, the Bondi mass provides the total energy remaining after accounting for energy radiated as gravitational waves. The difference M_ADM - M_Bondi is the total energy carried away by gravitational radiation, and the Bondi mass is monotonically non-increasing — gravitational waves carry positive energy. The positive energy theorem (Schoen-Yau, 1979; Witten, 1981) proves that the ADM mass is non-negative for physically reasonable matter, a deep result that confirms the stability of Minkowski spacetime.

For finite regions (not extending to infinity), quasi-local energy constructions (Brown-York, Wang-Yau, Hawking) attempt to define the gravitational energy within a bounded 2-surface. These are gauge-independent but depend on the choice of reference (what you compare the geometry against) and have various technical subtleties. The lack of a universal, local, covariant definition of gravitational energy is not a deficiency of the theory but a reflection of a deep physical truth: gravity is geometry, and "gravitational energy" is inseparable from the structure of spacetime itself. This non-localizability has profound implications for quantum gravity, where the standard techniques for quantizing field energies (which assume a well-defined local energy density) must be fundamentally reconsidered.