Komar mass

Komar mass The Komar mass is a conserved quantity defined for stationary spacetimes in general relativity that measures the total gravitating mass as seen by asymptotic observers. It provides a coordinate-invariant evaluation of energy associated with time-translation symmetry and connects to notions used in the study of black holes, cosmological models, and compact objects. The Komar construction plays a central role in comparisons between different mass definitions and in proofs of uniqueness and positivity theorems.

Definition and physical interpretation

The Komar mass is defined for a spacetime admitting a timelike Killing vector field, and it captures the contribution of matter and gravitational fields to the total energy measured at spatial infinity in contexts such as the Schwarzschild solution, the Kerr solution, and stationary stars. Its interpretation parallels energy notions in analyses of the Schwarzschild metric, the Kerr metric, and the Reissner–Nordström metric, and it enters prominently in the study of the ADM mass and the Bondi mass when stationarity or asymptotic structure is imposed. In black hole thermodynamics related to the Bekenstein–Hawking entropy and the Smarr formula, the Komar mass supplies the gravitational energy term paired with angular momentum and surface gravity. Historically, the Komar mass complements results by Arnowitt–Deser–Misner, Bondi et al., and others who developed mass concepts in relativistic gravitation.

Mathematical formulation

Mathematically, the Komar mass is expressed as an integral of the exterior derivative of the timelike Killing one-form over a two-sphere at infinity in asymptotically flat spacetimes. The construction uses differential-geometric tools common to treatments of the Einstein field equations and exploits identities related to the Killing vector field and the Ricci tensor appearing in the contracted Bianchi identities. In a stationary axisymmetric spacetime such as the Kerr–Newman metric, one combines Komar integrals for the timelike and axial Killing vectors to derive relations that appear in proofs by contributors like Smarr and in uniqueness theorems by Carter and Robinson. When applied to perfect-fluid models studied by Tolman and Oppenheimer–Volkoff, the Komar expression yields integrals involving the stress–energy tensor that correspond to conserved charges in geometric formulations used by Noether-based approaches and variational derivations.

Relation to other mass definitions

The Komar mass relates to and sometimes differs from the ADM mass and the Bondi mass depending on stationarity and asymptotic conditions. For strictly stationary, asymptotically flat solutions satisfying suitable energy conditions, the Komar mass equals the ADM mass; in radiating spacetimes described by the work of Bondi and Sachs, the Bondi mass captures energy loss by gravitational waves and can differ from Komar values. In electrovac solutions studied by Carter and Newman and Penrose, gauge fields modify Komar integrals and require inclusion of electromagnetic contributions akin to treatments by Wald and Israel. Comparisons with quasi-local masses developed by Brown and York, Hawking mass, Bartnik mass, and constructions from the Penrose inequality highlight regimes where Komar measures coincide or deviate, informing results in gravitational collapse analyzed by Penrose and numerical relativity investigations by groups at Caltech and Max Planck Institute for Gravitational Physics.

Applications and examples

Komar mass is used to compute the mass of stationary black holes like Schwarzschild black hole, Kerr black hole, and electrically charged Reissner–Nordström black hole, and it enters derivations of the Smarr relation and first law of black hole mechanics as explored by Bardeen, Carter, and Hawking. In models of rotating neutron stars studied at institutions such as Kavli Institute for Theoretical Physics and in work by Hartle and Thorne, Komar integrals provide checks against numerical ADM computations. In cosmological models with asymptotic structures examined by Friedmann, Lemaître, and de Sitter, the Komar approach clarifies limitations when timelike Killing fields are absent; nevertheless, modified Komar-like constructions appear in studies by Gibbons and Henneaux for spacetimes with a cosmological constant. The Komar concept also aids in analyzing energy conditions employed in singularity theorems by Hawking and Penrose and in verifying mass parameters in analytic solutions cataloged by Kramer et al..

Limitations and conditions for validity

The Komar mass requires the existence of a global timelike Killing vector and appropriate asymptotic flatness or specified boundary conditions; therefore it is inapplicable in dynamical, nonstationary spacetimes treated in the radiative frameworks developed by Bondi and Sachs. The construction assumes energy conditions such as the dominant or weak energy condition often invoked in works by Hawking, Penrose, and Wald, and electromagnetic or other gauge fields demand extra boundary terms as shown in analyses by Israel and Wald. In asymptotically anti-de Sitter spacetimes addressed by Maldacena and researchers in the AdS/CFT correspondence, Komar integrals must be adapted or replaced by background-subtracted conserved charges developed by Abbott and Deser and holographic renormalization programs by Henningson and Skenderis. Finally, quasi-local mass proposals by Brown and York or Bartnik provide alternative tools when Komar assumptions fail.

Category:General relativity