Energy Momentum PseudotensorEdit
The energy-momentum pseudotensor is a classical tool in the study of gravity within the framework of general relativity. In a theory where gravity is geometry rather than a force field in a fixed background, there is no unique, coordinate-independent local energy density for the gravitational field. The energy-momentum pseudotensor provides a way to package the energy and momentum of matter together with a representative gravitational contribution inside a chosen coordinate system. This construction rests on the idea that, while gravity cannot be localized in a tensorial sense, one can still write down conservation-like relations in a fixed coordinate chart that resemble the familiar energy and momentum balances of nonrelativistic physics. The concept appears in several versions, the most famous being the Einstein energy-momentum complex, and has played a central role in teaching, calculation, and historical development of the subject general relativity.
In practice, the pseudotensor is not a genuine tensor. Its components transform in a coordinate-dependent way, which means different observers (using different coordinate systems) can attribute different local gravitational energies to the same region of spacetime. This property is intimately tied to the equivalence principle, which implies that gravitational effects can be transformed away at a point by a freely falling frame. Nevertheless, integrated quantities built from the pseudotensor—such as the total energy or energy flux across a closed surface in the appropriate asymptotic regime—can be meaningful and agree with other global notions of energy in situations like isolated systems. The historical appeal of this approach is that it offers a concrete locus for energy accounting in gravity, much as one uses energy and momentum densities in other physical theories, provided one remains mindful of the coordinate dependence. See for example discussions of the Einstein field equations and their implications for energy conservation in a curved spacetime.
Formalism
Einstein energy-momentum complex: In the standard formulation, the Einstein equations G^{μν} = 8πG T^{μν} can be rearranged into a conservation law involving a combined quantity T^{μν} + t_E^{μν}, where t_E^{μν} is the Einstein pseudotensor. The expression is constructed so that ∂_ν [√−g (T^{μν} + t_E^{μν})] = 0 in a chosen coordinate system. Because t_E^{μν} is not a true tensor, its numerical values depend on the coordinates, even though the total energy and momentum of an isolated system (defined with appropriate boundary conditions) can be well defined in asymptotically flat spacetimes. See Noether's theorem in the context of diffeomorphism symmetry and the way conserved currents arise in this setting.
Other prescriptions: Several alternative energy-momentum complexes have been proposed to address coordinate issues or to emphasize symmetry properties. The Landau-Lifshitz pseudotensor, for example, is constructed to form part of a symmetric energy-momentum complex and is often used in calculations involving gravitational waves and radiative spacetimes. The Møller pseudotensor was introduced to offer a formulation that can be applied more freely in a variety of coordinate systems. Each of these prescriptions yields the same total energy in certain limits, even though their local densities may disagree, underscoring the coordinate-sensitive nature of gravitational energy in GR. See Landau-Lifshitz pseudotensor and Møller energy-momentum complex.
Relationship to quasi-local and global energy: In modern practice, many physicists prefer quasi-local definitions of energy that assign energy to a finite region via boundary data, or global notions such as the ADM energy for spacetimes that are asymptotically flat, and the Bondi energy at null infinity. These concepts avoid some of the ambiguities of local densities while still capturing the physically meaningful content of gravitational energy. Important entries in this broader view include ADM energy, Bondi energy, and Brown-York quasi-local energy.
Applications to gravitational radiation: The pseudotensor framework has been used to estimate the energy flux carried by gravitational waves across surfaces and to analyze the energy balance of radiating systems. In the wave zone, the pseudotensor provides a way to compute the energy flux that propagates away from sources, consistent with the predictions of GR for energy loss in systems such as binary mergers. See discussions of energy flux in the context of gravitational waves.
Controversies and debates
A central point of debate is whether a local gravitational energy density should be expected to exist at all. The equivalence principle implies that locally one can always transform away gravity, which argues against a unique, observer-independent local energy density for the gravitational field. This has led many theorists to view the energy-momentum pseudotensor as a coordinate artifact rather than a fundamental physical quantity. Critics emphasize the non-tensorial, coordinate-dependent character of the object and point to modern quasi-local constructs as a more faithful description of gravitational energetics. See discussions linked to General relativity and the general issue of energy localization.
Proponents of the pseudotensor reply that:
In a fixed coordinate chart, the pseudotensor provides a workable and historically productive way to express energy conservation laws for matter plus gravity, and to compute total energy and radiative flux in many situations of physical interest. The global content extracted by integrating the appropriate combinations over boundaries agrees with other well-defined notions in the right limits, such as ADM or Bondi energy.
The framework is not claiming a universal local energy density in the same sense as in nonrelativistic physics; rather, it encodes energy accounting in a way that is consistent within a chosen coordinate system and yields correct global results in asymptotically flat spacetimes.
Modern formulations of gravity energy—such as the Brown-York quasi-local energy or the Hamiltonian approach to GR—complement the pseudotensor view and help resolve ambiguities about localization by focusing on finite regions and boundary data rather than a pointwise density. See Brown-York quasi-local energy and Hamiltonian formulation of general relativity.
From a certain traditional or pragmatic viewpoint, the pseudotensor remains a useful bridge between old calculations and new ideas: it preserves a familiar conservation structure in a curved setting, supports straightforward calculations in many coordinate systems, and helps illuminate how energy and momentum flow through spacetime in processes like gravitational radiation. Critics who insist on strict tensorial, coordinate-independent quantities might see it as an artifact, but its utility in teaching, calculation, and the historical development of the subject is widely acknowledged. See also discussions surrounding Noether's theorem and the broader question of energy in GR.