Hawking energy

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 ${\displaystyle \Sigma }$ .

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]

• Mass in general relativity
• Inverse mean curvature flow

• Section 6.1 in Szabados, László B. (2004), "Quasi-Local Energy-Momentum and Angular Momentum in GR", Living Rev. Relativ., 7 (1): 4, Bibcode:2004LRR.....7....4S, doi:10.12942/lrr-2004-4, PMC 5255888, PMID 28179865, S2CID 40602589, retrieved 2007-08-23