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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


  • 1 Definition
  • 2 Properties
  • 3 See also
  • 4 References


Let M 3 , g a b }^,g_} be a 3-dimensional sub-manifold of a relativistic spacetime, and let Σ ⊂ M 3 }^} be a closed 2-surface Then the Hawking mass m H Σ \Sigma } of Σ is defined to be

m H Σ := Area Σ 16 π 1 − 1 16 π ∫ Σ H 2 d a , \Sigma :=}\,\Sigma }}}\left1-}\int _H^da\right,}

where H is the mean curvature of Σ


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 implies that Hawking mass satisfies an important monotonicity condition Namely, if M 3 }^} has nonnegative scalar curvature, then the Hawking mass of Σ is non-decreasing as the surface Σ flows outward at a speed equal to the inverse of the mean curvature In particular, if Σ t } is a family of connected surfaces evolving according to

d x d t = 1 H ν x , }=}\nu x,}

where H is the mean curvature of Σ t } and ν is the unit vector opposite of the mean curvature direction, then

d d t m H Σ t ≥ 0 }m_\Sigma _\geq 0}

Said otherwise, Hawking mass is increasing for the inverse mean curvature flow

Hawking mass is not necessarily positive However, it is asymptotic to the ADM or the Bondi mass, depending on whether the surface is asymptotic to spatial infinity or null infinity

See also

  • Mass in general relativity
  • Inverse mean curvature flow


  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, p113-136
  2. ^ Geroch, Robert 1973 "Energy Extraction" doi:101111/j1749-66321973tb41445x
  3. ^ Lemma 96 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
  • Section 61 in Szabados, László B 2004, "Quasi-Local Energy-Momentum and Angular Momentum in GR", Living Rev Relativ, 7, retrieved 2007-08-23 

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