Physics: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 [math]\displaystyle{ (\mathcal{M}^3, g_{ab}) }[/math] be a 3-dimensional sub-manifold of a relativistic spacetime, and let [math]\displaystyle{ \Sigma \subset \mathcal{M}^3 }[/math] be a closed 2-surface. Then the Hawking mass [math]\displaystyle{ m_H(\Sigma) }[/math] of [math]\displaystyle{ \Sigma }[/math] is defined[1] to be
- [math]\displaystyle{ m_H(\Sigma) := \sqrt{\frac{\text{Area}\,\Sigma}{16\pi}}\left( 1 - \frac{1}{16\pi}\int_\Sigma H^2 da \right), }[/math]
where [math]\displaystyle{ H }[/math] is the mean curvature of [math]\displaystyle{ \Sigma }[/math].
Properties
In the Schwarzschild metric, the Hawking mass of any sphere [math]\displaystyle{ S_r }[/math] about the central mass is equal to the value [math]\displaystyle{ m }[/math] of the central mass.
A result of Geroch[2] implies that Hawking mass satisfies an important monotonicity condition. Namely, if [math]\displaystyle{ \mathcal{M}^3 }[/math] has nonnegative scalar curvature, then the Hawking mass of [math]\displaystyle{ \Sigma }[/math] is non-decreasing as the surface [math]\displaystyle{ \Sigma }[/math] flows outward at a speed equal to the inverse of the mean curvature. In particular, if [math]\displaystyle{ \Sigma_t }[/math] is a family of connected surfaces evolving according to
- [math]\displaystyle{ \frac{dx}{dt} = \frac{1}{H}\nu(x), }[/math]
where [math]\displaystyle{ H }[/math] is the mean curvature of [math]\displaystyle{ \Sigma_t }[/math] and [math]\displaystyle{ \nu }[/math] is the unit vector opposite of the mean curvature direction, then
- [math]\displaystyle{ \frac{d}{dt}m_H(\Sigma_t) \geq 0. }[/math]
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
References
- ↑ Page 21 of Schoen, Richard, 2005, "Mean Curvature in Riemannian Geometry and General Relativity," in Global Theory of Minimal Surfaces: Proceedings of the Clay Mathematics Institute 2001 Summer School, David Hoffman (Ed.), pp. 113–136.
- ↑ Geroch, Robert (1973). "Energy Extraction". Annals of the New York Academy of Sciences 224: 108–117. doi:10.1111/j.1749-6632.1973.tb41445.x. Bibcode: 1973NYASA.224..108G.
- ↑ Lemma 9.6 of Schoen (2005).
- ↑ Section 4 of Yuguang Shi, Guofang Wang and Jie Wu (2008), "On the behavior of quasi-local mass at the infinity along nearly round surfaces".
- ↑ Section 2 of Finster, Felix; Smoller, Joel; Yau, Shing-Tung (2000). "Some recent progress in classical general relativity". Journal of Mathematical Physics 41 (6): 3943–3963. doi:10.1063/1.533332. Bibcode: 2000JMP....41.3943F.
Further reading
- Section 6.1 in Szabados, László B. (2004), "Quasi-Local Energy-Momentum and Angular Momentum in GR", Living Rev. Relativ. 7 (1): 4, doi:10.12942/lrr-2004-4, PMID 28179865, PMC 5255888, Bibcode: 2004LRR.....7....4S, http://www.livingreviews.org/lrr-2004-4, retrieved 2007-08-23
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