The Hawking energy or Hawking mass is one of the possible definitions of mass in general relativity. It is a measure of the bending of ingoing and outgoing rays of light that are orthogonal to a 2-sphere surrounding the region of space whose mass is to be defined.
Let be a 3-dimensional sub-manifold of a relativistic spacetime, and let be a closed 2-surface. Then the Hawking mass of is defined to be
where is the mean curvature of .
In the Schwarzschild metric, the Hawking mass of any sphere about the central mass is equal to the value of the central mass.
A result of Geroch implies that Hawking mass satisfies an important monotonicity condition. Namely, if has nonnegative scalar curvature, then the Hawking mass of is non-decreasing as the surface flows outward at a speed equal to the inverse of the mean curvature. In particular, if is a family of connected surfaces evolving according to
where is the mean curvature of and is the unit vector opposite of the mean curvature direction, then
- Page 21 of Schoen, Richard, 2005, "Mean Curvature in Reimannian Geometry and General Relativity," in Global Theory of Minimal Surfaces: Proceedings of the Clay Mathematics Institute 2001 Summer School, David Hoffman (Ed.), p.113-136.
- Geroch, Robert. 1973. "Energy Extraction." doi:10.1111/j.1749-6632.1973.tb41445.x.
- Lemma 9.6 of Schoen (2005).
- Section 4 of Yuguang Shi, Guofang Wang and Jie Wu (2008), "On the behavior of quasi-local mass at the infinity along nearly round surfaces".
- Section 2 of Shing Tung Yau (2002), "Some progress in classical general relativity," Geometry and Nonlinear Partial Differential Equations, Volume 29.
- Section 6.1 in Szabados, László B. (2004), "Quasi-Local Energy-Momentum and Angular Momentum in GR", Living Rev. Relativity 7, retrieved 2007-08-23