Energy In General RelativityEdit
Energy in General Relativity
Introduction
General relativity reframes energy as a property that sits closely with matter and non-gravitational fields, rather than as a universal clockwork quantity attached to spacetime itself. The theory’s central equation, the Einstein field equations, ties spacetime curvature to the distribution of energy and momentum through the energy-momentum tensor. Yet gravity—the geometry of spacetime—does not admit a simple, coordinate-independent local energy density. This leads to a nuanced view: at the level of matter and fields, energy and momentum obey familiar conservation laws, but the gravitational field’s energy is most meaningfully discussed only in certain limits or via global charges defined at boundaries. The result is a coherent framework in which energy concepts are robust and operative in practical settings, even as they resist naive localization.
From a traditional perspective, the emphasis is on well-defined, observable quantities and on limits where the spacetime behaves in a controlled way (for example, being asymptotically flat). In such regimes, one can assign global energies to entire systems, understand how energy flows, and connect theoretical constructs with measurements. This approach has yielded concrete and testable notions such as the total energy of an isolated system, the energy carried by gravitational radiation, and the role of cosmological energy density in expanding universes. The subject also contains deep theoretical debates about what counts as “energy” in a gravitational field, and how best to describe it without undermining the core principles of the theory.
Core concepts
- Energy-momentum tensor and conservation laws
The energy-momentum tensor, denoted energy-momentum tensor, describes the density and flux of energy and momentum of matter and non-gravitational fields. Its covariant divergence vanishes, ∇_μ T^{μν} = 0, once the matter fields satisfy their equations of motion. This expresses a generalized form of energy-momentum conservation within curved spacetime, consistent with the diffeomorphism invariance of General relativity. In flat spacetime, Noether’s theorem also yields conserved quantities associated with translations, but in curved spacetime the interpretation is subtler because there is no global translation symmetry. The gravitational field itself does not contribute to T^{μν} in a way that yields a universally defined local energy density.
- Local versus global energy
In GR, a local, coordinate-invariant energy density for gravity does not exist in the same sense as for matter. Attempts to assign a local gravitational energy density lead to objects such as energy-momentum pseudotensors, which depend on the choice of coordinates and are not generally covariant. This has been a point of debate among physicists who seek a robust local gravitational energy concept. A pragmatic stance is to recognize that energy and momentum of the non-gravitational fields are well-defined locally, while gravitational energy becomes meaningful primarily through global charges or quasi-local constructions that depend on boundary data.
- Global energy and boundary charges
For spacetimes that approach a simple reference geometry at infinity, one can define total energy and momentum as boundary charges. The archetypal examples are: - the ADM energy and ADM momentum in asymptotically flat spacetimes, which quantify the total energy and linear momentum of a system as measured from spatial infinity, - the Bondi energy (or Bondi mass) and Bondi-Sachs formalism for radiating systems, which track the energy content as carried away by gravitational waves and describe a mass-loss formula.
These global notions provide physically meaningful accounts of energy in regions where the gravitational field can be controlled at large distances from sources. They are essential tools for understanding astrophysical processes such as binary mergers and the emission of gravitational radiation.
- Stationary and quasi-local energies
In stationary spacetimes, the Komar energy provides another way to assign a mass associated with a timelike Killing vector. For finite regions, various quasi-local energy proposals aim to assign an energy to a bounded domain by combining information about its boundary with reference geometries. These include definitions such as the Hawking energy, Brown–York energy, and other quasi-local constructions. While useful, no single quasi-local notion has achieved universal acceptance, reflecting the fundamental difficulty of localizing gravitational energy in a coordinate-independent manner.
Gravitational radiation and energy transfer
- Gravitational waves carry energy
A central empirical result is that gravitational waves transport energy away from radiating systems. In the high-frequency, short-wavelength regime, the Isaacson effective stress-energy tensor provides a way to describe the average energy and momentum carried by these waves. Observationally, this energy flux is manifested in the orbital evolution of compact binaries and in the observed decrease of the Bondi mass as waves propagate to infinity.
- Observational and practical implications
The detection of gravitational waves by interferometers such as LIGO and Virgo confirms that gravitational radiation is real and energy-bearing. The observed waveforms are consistent with the energy radiated by systems like binary black holes or neutron star mergers, and the energy balance inferred from the data aligns with the predictions of the global energy definitions in GR. This concrete link between energy transfer and measurable signals underpins confidence in both the theory and the practical methods used to analyze gravitational-wave sources.
Debates and viewpoints in the field
- Local energy localization versus global charges
The core debate centers on whether a local, coordinate-invariant gravitational energy density can be defined. The conservative position emphasizes global charges (ADM, Bondi) and quasi-local constructs as the physically meaningful way to account for energy in GR. Critics of local energy density argue that any such attempt inevitably introduces coordinate dependence and breaks the geometric coherence of the theory.
- The role of boundary conditions
The global energy definitions rely on asymptotic conditions and reference backgrounds. In realistic cosmological settings, spacetime is not asymptotically flat, and defining a global energy becomes more nuanced. Proponents of a pragmatic approach emphasize that, as a matter of physics, one should rely on the most well-defined charges available in the given context and be transparent about the limitations when boundaries depart from idealized cases.
- Cosmology and the energy budget of the universe
In cosmology, the absence of a global timelike Killing vector in an expanding universe complicates a single universal energy conservation law. The cosmological constant (or dark energy) contribution is often treated as part of the overall energy-momentum content, yet its global accounting interacts with the dynamics of expansion. The conservative stance is to work with well-defined local conservation laws and, where possible, with well-mounded global quantities in appropriate limits, while recognizing that “total energy of the universe” is not defined in the same way as for isolated systems.
Energy concepts in practice
- Numerical relativity and modeling
In simulations of strong-field gravity, energy accounting is typically done using well-behaved global or quasi-local measures. ADM-like quantities are used for initial data and for global checks, while gravitational-wave fluxes through large spheres are used to monitor energy emission. These practices reflect a pragmatic balance between theoretical rigor and computational tractability.
- Experimental implications
Real-world tests of GR and astrophysical modeling rely on energy concepts that are robust and testable. For instance, the observed orbital decay of compact binaries matches the predicted energy loss through gravitational radiation. The energy balance is a key consistency check that supports the underlying GR description and the associated energy definitions in the relevant spacetime regime.
- Conceptual clarity and pedagogy
A coherent pedagogy emphasizes that energy in GR is best understood through the interplay of local matter content and global boundary charges, with a clear caveat about the nonexistence of a universal local gravitational energy density. This clarity helps students and researchers compare GR with Newtonian intuition and with alternative theories of gravity, while preserving the unique geometric nature of energy in curved spacetime.