Hawking mass

Hawking mass
Name	Hawking mass
Field	Mathematical physics; General relativity
Introduced	1968
Introduced by	Stephen Hawking
Related	Brown–York energy; Bartnik mass; ADM mass; Bondi mass; Geroch energy; Penrose inequality

Hawking mass The Hawking mass is a quasi-local measure of energy associated with a closed two-dimensional spacelike surface in a spacetime of General relativity. It was introduced by Stephen Hawking and plays a role in studies linking geometric analysis, black hole physics, and global properties of spacetimes such as the Penrose inequality, ADM mass, and Bondi mass. The quantity appears in proofs and conjectures involving curvature flows, energy inequalities, and horizon properties in the work of researchers connected to Christodoulou, Bartnik, Geroch, and York.

Definition and mathematical formulation

The Hawking mass of a closed orientable spacelike two-surface S embedded in a three-dimensional Riemannian slice or four-dimensional Lorentzian manifold is defined using the area |S| and mean curvature H of S. For a surface S in a Riemannian hypersurface, the Hawking mass m_H(S) is given by a formula involving |S| and the integral of H^2 over S, which ties into notions appearing in Gauss–Bonnet theorem, Yau's work on minimal surfaces, and techniques used by Schoen and Yau in positivity arguments. The definition uses geometric quantities similar to those in the definitions of the Brown–York energy and the Geroch energy, and it is sensitive to the embedding of S as studied by Bartnik and Bray.

Properties and basic results

The Hawking mass satisfies monotonicity properties under certain flows, notably the inverse mean curvature flow studied by Huisken and Ilmanen, which was instrumental in a proof of the Riemannian Penrose inequality where relations among Hawking mass, ADM mass, and minimal surfaces are central. For round spheres in asymptotically flat slices of known metrics such as Schwarzschild metric and Reissner–Nordström metric, the Hawking mass recovers expected values that match global masses computed by Arnowitt–Deser–Misner and others. Rigidity statements tie to uniqueness theorems associated with Israel and Bunting–Masood-ul-Alam for static black holes, and stability properties connect to spectral estimates used by Lichnerowicz and Obata.

Applications in general relativity

In black hole physics, the Hawking mass is used to formulate and test quasi-local energy bounds for apparent horizons appearing in the analysis of gravitational collapse studied by Penrose, Christodoulou, and Wald. It appears in considerations of horizon area theorems by Hawking and energy flux inequalities studied by Bardeen, Carter, and Hawking in the context of stationary solutions like Kerr metric and Kerr–Newman metric. Numerical relativity groups, including those influenced by work at Caltech, Max Planck Institute for Gravitational Physics, and AEI, use Hawking mass evaluations in simulations related to binary mergers studied by collaborations such as LIGO Scientific Collaboration and Virgo Collaboration to track quasi-local energy during evolution.

Comparison with other quasi-local masses

The Hawking mass is one among several quasi-local mass proposals alongside the Brown–York energy, Bartnik mass, Geroch energy, and definitions by Penrose and Wang–Yau. Each compares differently under limits to global invariants like the ADM mass and Bondi mass; for instance, the Brown–York energy employs boundary terms inspired by York and Arnowitt–Deser–Misner formalism, while the Bartnik mass is variational and relates to filling problems studied by Bartnik and Schoen–Yau. The Wang–Yau mass, influenced by methods of Wang and Yau, uses isometric embeddings into Minkowski space and provides positivity results akin to those proven by Witten in the spinorial approach. Comparisons are often mediated by geometric flows investigated by Huisken, Ilmanen, and Bray.

Examples and explicit computations

For the round coordinate spheres in the Schwarzschild metric of mass M, the Hawking mass evaluates to M for spheres that are the geometric round horizons and approaches M at spatial infinity, consistent with the ADM mass computed by Arnowitt–Deser–Misner and the asymptotic analysis used by Regge–Teitelboim. In spherically symmetric spacetimes like Reissner–Nordström metric and Vaidya-like models studied by Vaidya and Israel for radiating solutions, explicit formulas relate Hawking mass to Misner–Sharp mass and to charged or radiative parameters appearing in the work of Bondi and Sachs. Examples in cosmological contexts reference metrics investigated by Friedmann, Lemaître, and Tolman–Bondi where quasi-local masses illuminate inhomogeneity effects.

Extensions and generalizations

Generalizations of the Hawking mass include modified definitions incorporating shear and null expansions used in the study of marginally outer trapped surfaces (MOTS) analyzed by Andersson, Mars, and Simon. Spinor-based and Hamiltonian approaches by Witten, Brown–York, and Szabados inspire alternative quasi-local constructions connected to positivity results established by Schoen and Yau. Extensions to asymptotically anti-de Sitter contexts bring in methods related to Henneaux and Teitelboim, and geometric-analytic generalizations interface with research programs at institutions like Princeton University, Cambridge University, and University of California, Berkeley where interplay among geometric flows, scalar curvature problems, and quasi-local mass remains active.

Category:Mathematical physics Category:General relativity Category:Geometric analysis