One of the possible definitions of mass in general relativity
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.
Definition
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[1] to be
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[2] 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
Hawking mass is not necessarily positive. However, it is asymptotic to the
ADM[4] or the
Bondi mass, depending on whether the surface is asymptotic to spatial infinity or null infinity.[5]
^Page 21 of Schoen, Richard, 2005, "Mean Curvature in Riemannian Geometry and General Relativity," in Global Theory of Minimal Surfaces: Proceedings of the Clay Mathematics Institute 2001 Summer School, David Hoffman (Ed.), pp. 113–136.