Английская Википедия:Hawking energy
Шаблон:Short description 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>(\mathcal{M}^3, g_{ab})</math> be a 3-dimensional sub-manifold of a relativistic spacetime, and let <math>\Sigma \subset \mathcal{M}^3</math> be a closed 2-surface. Then the Hawking mass <math>m_H(\Sigma)</math> of <math>\Sigma</math> is defined[1] to be
- <math>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>H</math> is the mean curvature of <math>\Sigma</math>.
Properties
In the Schwarzschild metric, the Hawking mass of any sphere <math>S_r</math> about the central mass is equal to the value <math>m</math> of the central mass.
A result of Geroch[2] implies that Hawking mass satisfies an important monotonicity condition. Namely, if <math>\mathcal{M}^3</math> has nonnegative scalar curvature, then the Hawking mass of <math>\Sigma</math> is non-decreasing as the surface <math>\Sigma</math> flows outward at a speed equal to the inverse of the mean curvature. In particular, if <math>\Sigma_t</math> is a family of connected surfaces evolving according to
- <math>\frac{dx}{dt} = \frac{1}{H}\nu(x),</math>
where <math>H</math> is the mean curvature of <math>\Sigma_t</math> and <math>\nu</math> is the unit vector opposite of the mean curvature direction, then
- <math>\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
Further reading
- Section 6.1 in Шаблон:Citation
- ↑ 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.
- ↑ Шаблон:Cite journal
- ↑ 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 Шаблон:Cite journal
