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

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

where $H$ is the mean curvature of $\Sigma$.

## Properties

In the Schwarzschild metric, the Hawking mass of any sphere $S_r$ about the central mass is equal to the value $m$ of the central mass.

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

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

where $H$ is the mean curvature of $\Sigma_t$ and $\nu$ is the unit vector opposite of the mean curvature direction, then

$\frac{d}{dt}m_H(\Sigma_t) \geq 0.$

Hawking mass is also monotonically 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]