Coordinate-independent local energy density of the gravitational field in general relativity

P. A. M. Dirac's 1959 analysis of the energy of the gravitational field (Phys. Rev. Lett. 2, 368) proposed, in the second-order weak-field Hamiltonian formulation of general relativity, the energy density $w+\mathcal{K}_{ML}$ and showed that it changes by terms of the same order as the energy itself under coordinate transformations of order $\gamma$ (his Eqs. (14), (15)), the ambiguities vanishing only for unidirectional gravitational waves and for retarded-potential far fields; he concluded that there does not seem to be any general definition of energy density independent of the coordinate system. The problem is whether a generally applicable, coordinate- and gauge-invariant local energy density for the gravitational field exists (within general relativity, possibly using alternative variables or an empirically equivalent reformulation), or whether its non-existence can be rigorously established for precisely formalized classes of candidates. Dirac's analysis itself was carried out only to second order in the weak-field expansion; the problem concerns a generally applicable definition in general relativity. The literature status is open: the mainstream holds gravitational energy to be non-localizable, but the strongest no-go result (Curiel's theorem) excludes only tensorial concomitants of the metric (quantities constructed naturally from the metric and finitely many of its derivatives) under natural conditions; total energies are invariant only at infinity (ADM, Bondi) or quasi-locally (Penrose, Bartnik, Brown–York, Wang–Yau — several inequivalent definitions without consensus); a covariant wave-energy density exists only in the high-frequency averaged limit (Isaacson); and localizability claims in teleparallel gravity remain frame-dependent, with new strictly local proposals appearing as recently as 2026.

Background

General relativity (GR) describes gravity as spacetime curvature, and Einstein's field equations hold in any coordinate system: the metric $g_{\mu\nu}$ determines the geometry, but the freedom to relabel coordinates (diffeomorphism invariance) makes it hard to separate physical effects from coordinate artifacts. For matter, the stress-energy tensor $T^{\mu\nu}$ — the local density of energy and momentum — is a true tensor and is covariantly conserved, $\nabla_\mu T^{\mu\nu}=0$. For the gravitational field itself no such tensorial object is known: the classical device is the Einstein stress pseudotensor $t^\mu{}_\nu$, a coordinate-dependent expression built from first derivatives of the metric whose divergence cancels that of $T^{\mu\nu}$ in a fixed coordinate system, so that total energy-momentum is 'conserved' only in that system; its component $t^0{}_0$ is the usual candidate energy density, and it depends strongly on the coordinates. Two neighboring lines set the stage for the 1959 analysis. Møller (1958) proposed a new expression intended to localize the energy of a physical system better than the pseudotensor. Dirac (1958) had recast GR in Hamiltonian (canonical) form: the state of the gravitational field at a given time is described by the six spatial metric components $g_{rs}$ ($r,s=1,2,3$) and their conjugate momenta $p^{rs}$, while the four components $g_{\mu 0}$ are not state variables but only connect states at neighboring times. Dirac therefore required that a physically acceptable energy at a given time depend only on $g_{rs}$, $p^{rs}$ and the matter variables — not on the $g_{\mu 0}$; both $t^0{}_0$ and Møller's expression fail this test. In the 1959 Physical Review Letters paper, Dirac worked in the weak-field approximation, where the metric deviates from its special-relativistic (Minkowski) values by quantities of order $\gamma$ (the gravitational constant, used as the small parameter), keeping terms to second order — the lowest order at which field energy, quadratic in first derivatives, appears. He split the Hamiltonian into a main part $H_{\mathrm{main}}=\int(w+\mathcal{K}_{ML})\,d^3x$ (his Eq. (5)) — whose integrand he interpreted as the energy density, $w$ (his Eq. (6)) being the quadratic gravitational part built from $p^{rs}$ and first spatial derivatives of $g_{rs}$ — and correction terms $H_{\mathrm{cor}}$ (his Eq. (7)) involving the $g_{\mu 0}$. The constraints of the canonical formalism force the full Hamiltonian to vanish, but $H_{\mathrm{main}}$ is a nontrivial constant of the motion (the discrepancy being a surface term at infinity) and is identified with the total energy; when convergent it equals a surface integral at infinity (his Eq. (16)) agreeing with the pseudotensor expression provided the $g_{\mu 0}$ approach their background values fast enough at large distances. Under special coordinate conditions he verified a local conservation law with an identifiable energy flux. The essential negative result is that under a coordinate change $x^\mu\to x^\mu+\xi^\mu$ of order $\gamma$, and under a normal shift of the time hypersurface, the energy density $w+\mathcal{K}_{ML}$ changes by two terms (his Eqs. (14) and (15)) of the same order as the energy itself. The ambiguities vanish in two special cases: for gravitational waves all moving in a single direction ('the energy density of gravitational waves moving in a single direction is well-defined, independent of the coordinate system. It is only the interference energy density of waves moving in different directions that is subject to uncertainties'), and for retarded-potential solutions (fields determined by the past positions of the sources, i.e., containing outgoing but no incoming waves) describing accelerating masses that emit waves, so that 'the uncertainties in the energy density ... do not affect calculations of the emission of energy by gravitational waves'. His overall conclusion: 'On account of the uncertainties (14), (15), there does not seem to be any general definition for the energy density independent of the coordinate system.' The later literature built three pillars around this question. For total energy, the ADM canonical program (Arnowitt–Deser–Misner, 1960–1962), developing Dirac's Hamiltonian variables, expressed the total energy of an isolated system as a surface integral at spatial infinity of the metric — the ADM energy — invariant under coordinate transformations at infinity and valid in the full nonlinear (strong-field) theory; Bondi, van der Burg and Metzner (1962) did the same at null infinity (the far-field limit along outgoing light rays), obtaining the Bondi mass and its mass-loss law for radiating systems. For wave energy, Isaacson (1968) showed that in the short-wavelength (high-frequency) limit, averaging over a wavelength yields a gauge-invariant effective stress-energy tensor for gravitational waves — a covariant local wave-energy density, but only in that limit and with an external averaging scale. For finite regions, Penrose (1982), Bartnik (1989), Brown–York (1993) and Wang–Yau (2009) associated energy to closed two-dimensional surfaces rather than to volume densities — quasi-local energy; several inequivalent definitions coexist without consensus (reviewed by Szabados). On the negative side, Curiel (preprint 2009, published 2019) proved — under natural conditions on how a tensor may depend on the metric, formulated via jet bundles (the machinery describing how geometric objects depend on the derivatives of fields) — that no tensor can represent gravitational stress-energy, so gravitational energy in GR is necessarily non-local, formalizing the long-standing equivalence-principle folklore (in normal coordinates at a point, the connection and the first derivatives of the metric can be made to vanish). A further line keeps the localizability program alive in the teleparallel equivalent of GR (TEGR), a tetrad reformulation (a tetrad is a field of four orthonormal vector fields serving as a local reference frame) in which the gravitational field strength is the torsion (the antisymmetric part of the connection) of the Weitzenböck connection (the flat connection naturally determined by a tetrad) instead of curvature; there the gravitational energy-momentum density is a true tensor, but it depends on the choice of tetrad frame (the local Lorentz gauge).

Problem Statement

Does there exist a generally applicable local energy density for the gravitational field — an expression defined at each spacetime point, invariant under arbitrary coordinate (diffeomorphism) transformations, and valid for general field configurations rather than only for the unidirectional waves and far-field retarded-potential solutions in which Dirac's second-order weak-field analysis showed the transformation ambiguities of $w+\mathcal{K}_{ML}$ to vanish — or, failing that, can the non-existence of such a quantity be rigorously established for precisely formalized classes of definitions? Dirac's own analysis was carried out only to second order in the weak-field expansion; the existence question above concerns a generally applicable definition in general relativity.

A proposed construction must specify its precise framework: standard metric general relativity, possibly using alternative variables or an empirically equivalent reformulation (such as the teleparallel tetrad formulation, in which case the density must be invariant under the full gauge freedom of that formulation, including the local Lorentz/frame freedom). A valid construction must satisfy jointly the following adequacy conditions: the energy must be a useful integral of the equations of motion (a nontrivial conserved quantity); it must depend only on the canonical state variables (in Dirac's Hamiltonian formulation, the spatial metric components $g_{rs}$ and their conjugate momenta $p^{rs}$) and not on the remaining components $g_{\mu 0}$ (which in that formulation connect states at neighboring times rather than describing the state at a given time); it must obey a local conservation law with an identifiable energy flux; and it must agree, when convergent, with the surface-integral expression for the total energy at infinity (his Eq. (16)). Consistency with the two special cases Dirac verified — the unidirectional-wave energy density and the retarded-potential emission results — serves as a consistency check rather than as an added completion condition. Full coordinate invariance and dependence solely on the canonical hypersurface data are required jointly: Dirac applied both tests (coordinate transformations of order $\gamma$, and a normal shift of the time hypersurface — his Eqs. (14) and (15)) to a density built from the canonical variables and treated both resulting changes as ambiguities a general definition should avoid; reconciling the two requirements is the crux of the open question, and no precedence between them is stipulated. For context, the ADM energy (1960–1962) and the Bondi energy (1962) postdate the source and settled the total-energy component of the question at spatial and null infinity; compatibility with them and with the Newtonian weak-field limit is background context rather than a mandatory acceptance condition.

A proposed impossibility result must formalize the class of admissible candidate densities precisely: the class must be fixed by general structural conditions on how a candidate density may depend on the metric, its derivatives and any auxiliary structure, not by a definition chosen so as to make the non-existence conclusion true. To count as a resolution, the result must cover a class of candidates that Curiel's theorem does not already exclude. Curiel's theorem — the exclusion of tensorial concomitants of the metric under natural conditions — is background delimiting what is already known; a no-go result that only restates it does not resolve the problem.

References

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