# 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 ${\displaystyle ({\mathcal {M}}^{3},g_{ab})}$  be a 3-dimensional sub-manifold of a relativistic spacetime, and let ${\displaystyle \Sigma \subset {\mathcal {M}}^{3}}$  be a closed 2-surface. Then the Hawking mass ${\displaystyle m_{H}(\Sigma )}$  of ${\displaystyle \Sigma }$  is defined[1] to be

${\displaystyle m_{H}(\Sigma ):={\sqrt {\frac {{\text{Area}}\,\Sigma }{16\pi }}}\left(1-{\frac {1}{16\pi }}\int _{\Sigma }H^{2}da\right),}$

where ${\displaystyle H}$  is the mean curvature of ${\displaystyle \Sigma }$ .

## Properties

In the Schwarzschild metric, the Hawking mass of any sphere ${\displaystyle S_{r}}$  about the central mass is equal to the value ${\displaystyle m}$  of the central mass.

A result of Geroch[2] implies that Hawking mass satisfies an important monotonicity condition. Namely, if ${\displaystyle {\mathcal {M}}^{3}}$  has nonnegative scalar curvature, then the Hawking mass of ${\displaystyle \Sigma }$  is non-decreasing as the surface ${\displaystyle \Sigma }$  flows outward at a speed equal to the inverse of the mean curvature. In particular, if ${\displaystyle \Sigma _{t}}$  is a family of connected surfaces evolving according to

${\displaystyle {\frac {dx}{dt}}={\frac {1}{H}}\nu (x),}$

where ${\displaystyle H}$  is the mean curvature of ${\displaystyle \Sigma _{t}}$  and ${\displaystyle \nu }$  is the unit vector opposite of the mean curvature direction, then

${\displaystyle {\frac {d}{dt}}m_{H}(\Sigma _{t})\geq 0.}$

Said otherwise, Hawking mass is increasing for the inverse mean curvature flow.[3]

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]