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.

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]

References

^ Hoffman 2005, p. 21
^ Geroch, Robert (December 1973). "Energy extraction*". Annals of the New York Academy of Sciences. 224 (1): 108–117. Bibcode:1973NYASA.224..108G. doi:10.1111/j.1749-6632.1973.tb41445.x. ISSN 0077-8923. S2CID 222086296.{{cite journal}}: CS1 maint: date and year (link)
^ Hoffman 2005, Lemma 9.6
^ Section 4 of Shi, Yuguang; Wang, Guofang; Wu, Jie (2008). On the behavior of quasi-local mass at the infinity along nearly round surfaces. arXiv:0806.0678.
^ Section 2 of Finster, Felix; Smoller, Joel; Yau, Shing-Tung (2000-06-01). "Some Recent Progress in Classical General Relativity". Journal of Mathematical Physics. 41 (6): 3943–3963. arXiv:gr-qc/0001064. Bibcode:2000JMP....41.3943F. doi:10.1063/1.533332. S2CID 18904339.

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