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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]

Mass in general relativity
Inverse mean curvature flow

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.), p.113-136.
Geroch, Robert. 1973. "Energy Extraction." doi:10.1111/j.1749-6632.1973.tb41445.x.
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 Shing Tung Yau (2002), "Some progress in classical general relativity," Geometry and Nonlinear Partial Differential Equations, Volume 29.

Section 6.1 in Szabados, László B. (2004), "Quasi-Local Energy-Momentum and Angular Momentum in GR", Living Rev. Relativity 7, retrieved 2007-08-23

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