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]

See also

• Mass in general relativity
• Inverse mean curvature flow

References

1. Hoffman 2005, p. 21
2. 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.
3. Hoffman 2005, Lemma 9.6
4. 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.
5. 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.

Further reading

• Section 6.1 in Szabados, László B. (December 2004). Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article. Vol. 7. p. 4. Bibcode:2004LRR.....7....4S. doi:10.12942/lrr-2004-4. ISSN 2367-3613. PMC 5255888. PMID 28179865. S2CID 40602589.
• Hoffman, David A., ed. (2005). Global theory of minimal surfaces: proceedings of the Clay Mathematics Institute 2001 Summer School, Mathematical Sciences Research Institute, Berkeley, California, June 25-July 27, 2001. Clay mathematics proceedings. Providence, RI : Cambridge, MA: American Mathematical Society ; Clay Mathematics Institute. ISBN 978-0-8218-3587-6.