Hawking energy

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 formula_1 be a 3-dimensional sub-manifold of a relativistic spacetime, and let formula_2 be a closed 2-surface. Then the Hawking mass formula_3 of formula_4 is defined to be
where formula_6 is the mean curvature of formula_4.
Properties.
In the Schwarzschild metric, the Hawking mass of any sphere formula_8 about the central mass is equal to the value formula_9 of the central mass.
A result of Geroch implies that Hawking mass satisfies an important monotonicity condition. Namely, if formula_10 has nonnegative scalar curvature, then the Hawking mass of formula_4 is non-decreasing as the surface formula_4 flows outward at a speed equal to the inverse of the mean curvature. In particular, if formula_13 is a family of connected surfaces evolving according to
where formula_6 is the mean curvature of formula_13 and formula_17 is the unit vector opposite of the mean curvature direction, then
Said otherwise, Hawking mass is increasing for the inverse mean curvature flow.
Hawking mass is not necessarily positive. However, it is asymptotic to the ADM or the Bondi mass, depending on whether the surface is asymptotic to spatial infinity or null infinity.