Quasi Local EnergyEdit
Quasi-local energy is a concept in the framework of general relativity that seeks to assign an energy value to a finite region of spacetime bounded by a closed two-surface. In Einstein’s theory, gravity is not a field that sits on top of spacetime like electromagnetism; it is the geometry of spacetime itself. This makes the notion of “gravitational energy density” at a point ill-defined, due to the equivalence principle and diffeomorphism invariance. As a result, physicists have pursued quasi-local definitions that attribute energy to a bounded region by examining the geometry of its boundary and its embedding in a reference spacetime. These approaches aim to provide a practical accounting of energy for systems such as black holes, neutron stars, and gravitational-wave emitting configurations, while remaining faithful to the mathematical structure of General relativitygeneral relativity.
Over the past few decades, several competing proposals have been developed to formalize quasi-local energy. The core idea is to compare the physical geometry of a boundary surface with a chosen reference geometry, and to extract a scalar quantity that behaves like energy for the enclosed region. This program traces its roots to the recognition that global energy notions (such as the total energy in asymptotically flat spacetimes) do not directly transfer to a finite region. Classic global notions include the ADM energy ADM energy and the Bondi energy Bondi energy, which apply at spatial and null infinity, respectively. Quasi-local definitions fill the gap by attaching energy to a finite, closed surface rather than to the entire spacetime.
Notable formulations
Brown-York quasi-local energy: This approach derives energy from the Hamiltonian formulation of general relativity, using the extrinsic curvature of the boundary surface. It quantifies the energy as the difference between the physical boundary term and the corresponding term in a reference spacetime. It has attractive features, such as a clear geometric interpretation and good behavior in many physically relevant situations; however, it relies on a choice of reference and an embedding of the boundary, which can be ambiguous in practice. See Brown-York quasi-local energy and its relationship to ADM energy in the appropriate limits.
Hawking-Hayward quasi-local mass: This is a broader mass concept built from the geometry of lightlike or null surfaces. It provides a definition of energy content that is more conservative about the choice of reference geometry, but it is not always straightforward to compute in generic spacetimes. For foundational details, see Hawking-Hayward quasi-local mass.
Liu-Yau quasi-local mass: The Liu-Yau construction emphasizes positivity under reasonable energy conditions and uses the geometry of the boundary to define a mass-like quantity. It is often discussed in concert with other “quasi-local mass” measures, especially when addressing positivity properties on physically meaningful surfaces. See Liu-Yau quasi-local mass.
Wang-Yau quasi-local energy: The Wang-Yau proposal aims to be broadly applicable and physically robust by solving a variational problem that identifies an optimal reference embedding for the boundary. It has become a widely discussed contender in the quasi-local energy landscape, with arguments about its universality and numerical practicality. See Wang-Yau quasi-local energy.
In practice, all quasi-local definitions face common challenges: they are generally frame-dependent, require a choice of reference spacetime, and can yield different results for the same physical situation depending on the chosen construction. They often agree with the well-known global energies in the appropriate limits (for example, they recover ADM energy at spatial infinity for suitable asymptotically flat spacetimes), but outside of those limits the numbers can vary. See discussions of how these formulations relate to Komar mass and other energy measures.
Applications and implications
Numerical relativity and simulations: Quasi-local energy measures provide practical diagnostics for tracking the energy content of a simulated system on finite grids or within specific bounded regions. This is particularly useful in studies of binary mergers, gravitational-wave emission, and energy exchange between matter and geometry. See applications in numerical relativity and the study of gravitational waves.
Black hole physics and horizons: The boundary-focused character of quasi-local energy makes it a natural tool for analyzing energy fluxes across horizons and for characterizing the energetic content of dynamical spacetimes around compact objects. Related notions appear in discussions of black hole thermodynamics and horizon mechanics.
Astrophysical and cosmological contexts: In bounded regions such as stellar interiors, accretion disks, or finite cosmological patches, quasi-local energy concepts help formalize energy accounting in strong-field regimes where global energy notions are not directly applicable.
Conceptual clarity and measurement: By tying energy to a boundary, quasi-local definitions align with the intuition that energy content should be associated with the region’s boundary interactions with its surroundings, rather than with a local, pointwise gravitational density. This resonates with conservative, property-rights–oriented thinking about measurement and accountability in complex systems.
Controversies and debates
Non-uniqueness and observer-dependence: A central point of contention is that no single quasi-local energy definition has achieved universal acceptance. Different proposals offer different advantages and are suited to different problems. Critics argue that this non-uniqueness complicates physical interpretation, while proponents emphasize that multiple consistent definitions illuminate distinct facets of gravitational energy.
Reference subtraction and embedding ambiguities: Most quasi-local constructions require subtracting a reference term that encodes the energy of a chosen background spacetime. The choice of reference surface embedding is often not unique, leading to ambiguities in the resulting energy value. Proponents counter that a careful, context-appropriate reference captures the physically meaningful contrast between the actual spacetime and a sensible baseline.
Localizability and the equivalence principle: A perennial objection is that gravity, as geometry, resists localization in the same way as other fields, so any attempt to assign a pointwise gravitational energy density is suspect. Quasi-local programs circumvent this by focusing on finite regions, but critics ask how far boundary-based energy really tells us about internal dynamics.
Relationship to global energy concepts: Proponents of quasi-local energy argue that these definitions provide a bridge between local geometric data and global energetic statements, especially in dynamical settings with gravitational radiation. Detractors might point out that, for many practical purposes, the existing global notions already capture the essential energetics at infinity; thus, the added complexity of quasi-local definitions must be justified by clear physical payoff.
Philosophical and practical tradeoffs: In a field driven by both rigorous mathematics and computational physics, the tradeoffs between mathematical elegance, computational tractability, and physical interpretability shape ongoing debate. Advocates for a conservative, economy-minded approach emphasize models and calculations that offer robust predictions with minimal dependence on arbitrary choices, while others push for broader formulations that capture a wider class of spacetimes.
History and context
The problem of assigning energy in gravity has a long history, beginning with attempts to define gravitational mass and energy in the early 20th century and evolving through the development of conserved quantities in asymptotically flat spacetimes. The ADM and Bondi energies established important asymptotic benchmarks, but they proved insufficient for finite, bounded regions. The modern quasi-local program develops from the insight that meaningful energy statements can be tied to the geometry of a boundary, with several concrete proposals offering different ways to implement that idea. Researchers often compare quasi-local energies to classical concepts like Komar mass in stationary spacetimes and to the way energy fluxes are defined for gravitational radiation through a closed surface.