Komar MassEdit
Komar mass is a geometric way to assign a total mass-energy content to certain spacetimes in general relativity. It arises when a spacetime possesses a timelike symmetry, encoded in a Killing vector field, and it yields a coordinate-independent surface integral that can be evaluated at large distances in stationary, asymptotically flat settings. The concept bears the name of the mathematician and physicist who introduced it as part of the broader effort to understand how energy and mass can be defined in a theory where gravity itself embodies geometry rather than a traditional field in a fixed background.
In practical terms, Komar mass provides a conserved quantity associated with the time-translation symmetry of a stationary spacetime. It is most transparent for black holes and isolated gravitating systems where the spacetime settles to a stable configuration at infinity. It is closely related to other global mass notions such as the ADM mass and the Bondi mass, and it plays a role in the Smarr relation that connects a black hole’s mass to its surface gravity, horizon area, angular momentum, and charge. The concept is a cornerstone of the way relativists connect the mathematics of spacetime to something that behaves like total energy as measured by observers far from gravitating sources.
Definition and formalism
The key ingredients are a stationary spacetime and a timelike Killing vector field. If the spacetime admits a Killing vector field ξ^a that asymptotically approaches the time translation at infinity, the Komar mass M_K is defined by a two-surface integral
- M_K = - (1/8π) ∮S ∇^a ξ^b dS{ab},
where ∇^a is the Levi-Civita connection, and dS_{ab} is the surface element of a closed two-surface S, oriented with the outward normal. The integral is typically evaluated on a sphere at spatial infinity in an asymptotically flat region, in which case M_K yields a total mass parameter characterizing the gravitating system.
Several variations and related quantities appear in the same formalism. For axisymmetric spacetimes, the angular momentum is defined by a similar integral with the rotational Killing vector χ^a:
- J = (1/8π) ∮S ∇^a χ^b dS{ab}.
These expressions reflect how symmetries of the spacetime translate into conserved currents via a generalization of Gauss’s law to curved spacetime. In familiar solutions such as the Schwarzschild or Kerr spacetimes, the Komar mass reduces to the standard mass parameter that one reads off from the metric, and in stationary, asymptotically flat spacetimes it aligns with the ADM mass.
The mathematical underpinning rests on the Komar current constructed from the Killing vector field, and the identity ∇[a ξ{b]} = - ∇{[b} ξ{a]} captures the antisymmetric nature that makes the two-surface integral well-defined. Because the integral relies on asymptotic structure, its meaning and value are most robust in spacetimes that are stationary far from the sources.
Linking concepts: the idea of Komar mass is part of the broader dialogue among related quantities such as ADM mass, Bondi mass, and the study of Killing vector fields. It also connects to the Schwarzschild solution Schwarzschild metric and the Kerr solution Kerr metric in concrete examples, and to the notion of quasi-local energy that appears in discussions of objects like the Brown-York quasi-local energy.
Examples and applications
Schwarzschild spacetime: In the simplest static, spherically symmetric case, the Komar mass equals the parameter M that appears in the metric, and thus M_K = M, matching the ADM mass for this spacetime.
Kerr spacetime: For a rotating black hole, the Komar mass again corresponds to the mass parameter M of the solution, with the angular momentum captured by the axial Killing vector. The separation of mass and angular momentum via the two Killing vectors is central to the geometric interpretation of the Kerr geometry.
Smarr relation: In stationary black-hole spacetimes, the Komar mass participates in the Smarr formula, which relates the mass to horizon properties such as surface gravity κ, horizon area A, angular velocity Ω_H, and angular momentum J, among other quantities. This relation depends on the same symmetries that underpin the Komar constructions.
Relation to other mass notions: In asymptotically flat, stationary spacetimes, M_K often coincides with the ADM mass, reinforcing its physical interpretation as the total energy content as seen from infinity. In radiating or non-stationary contexts, the connection becomes more subtle, and other mass notions (like Bondi mass) become more appropriate.
Limitations and debates
Stationarity requirement: The Komar mass is defined via a timelike Killing vector, so it is intrinsically tied to stationary spacetimes. In dynamical situations—such as during mergers or gravitational radiation—the symmetry is absent or broken over time, and the Komar construction either fails to apply or loses its invariant meaning.
Global versus local: Like many gravitational energy measures in general relativity, Komar mass is not a local energy density. It encodes global information about the spacetime’s asymptotic structure rather than a pointwise energy density, reflecting the general relativistic principle that gravitational energy cannot be localized in a coordinate-independent way.
Cosmological settings: In spacetimes with a cosmological constant or non-asymptotically flat behavior (for example, de Sitter or anti-de Sitter backgrounds), the standard Komar integral is not straightforwardly defined or lacks the same physical interpretation. In such contexts researchers turn to alternative definitions or modified constructions.
Comparative landscape of mass concepts: The physics community uses a variety of mass notions to suit different problems. ADM mass is a global measure for spatial infinity in stationary, asymptotically flat spacetimes, Bondi mass tracks energy carried by radiation at null infinity, and Brown–York-type quasi-local energies aim to assign energy to finite regions. The choice among them depends on the physical situation and the symmetries present.
Controversies and critiques: A persistent theme in the broader discussion is the nonlocalizability of gravitational energy and the proliferation of competing quasi-local definitions. Critics caution against overinterpreting any single integral as a universal measure of “gravitational energy,” while proponents emphasize the practical utility and clean geometric origin of Komar-type expressions in appropriate symmetric settings.
See, for example, discussions comparing Komar mass with ADM mass, Bondi mass, and quasi-local approaches such as Brown-York quasi-local energy and related concepts in the study of energy in general relativity.