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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

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

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]