Hawking energy

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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[edit]

Let be a 3-dimensional sub-manifold of a relativistic spacetime, and let be a closed 2-surface. Then the Hawking mass of is defined[1] to be

where is the mean curvature of .

Properties[edit]

In the Schwarzschild metric, the Hawking mass of any sphere about the central mass is equal to the value of the central mass.

A result of Geroch[2] implies that Hawking mass satisfies an important monotonicity condition. Namely, if has nonnegative scalar curvature, then the Hawking mass of is non-decreasing as the surface flows outward at a speed equal to the inverse of the mean curvature. In particular, if is a family of connected surfaces evolving according to

where is the mean curvature of and is the unit vector opposite of the mean curvature direction, then

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[edit]

References[edit]

  1. ^ 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.
  2. ^ Geroch, Robert. 1973. "Energy Extraction." doi:10.1111/j.1749-6632.1973.tb41445.x.
  3. ^ Lemma 9.6 of Schoen (2005).
  4. ^ 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".
  5. ^ Section 2 of Shing Tung Yau (2002), "Some progress in classical general relativity," Geometry and Nonlinear Partial Differential Equations, Volume 29.