Hawking energy

One of the possible definitions of mass in general relativity

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 ( M 3 , g a b ) {\displaystyle ({\mathcal {M}}^{3},g_{ab})} be a 3-dimensional sub-manifold of a relativistic spacetime, and let Σ ⊂ M 3 {\displaystyle \Sigma \subset {\mathcal {M}}^{3}} be a closed 2-surface. Then the Hawking mass m H ( Σ ) {\displaystyle m_{H}(\Sigma )} of Σ {\displaystyle \Sigma } is defined[1] to be

m H ( Σ ) := Area Σ 16 π ( 1 − 1 16 π ∫ Σ H 2 d a ) , {\displaystyle m_{H}(\Sigma ):={\sqrt {\frac {{\text{Area}}\,\Sigma }{16\pi }}}\left(1-{\frac {1}{16\pi }}\int _{\Sigma }H^{2}da\right),}

where H {\displaystyle H} is the mean curvature of Σ {\displaystyle \Sigma } .

Properties

In the Schwarzschild metric, the Hawking mass of any sphere S r {\displaystyle S_{r}} about the central mass is equal to the value m {\displaystyle m} of the central mass.

A result of Geroch[2] implies that Hawking mass satisfies an important monotonicity condition. Namely, if M 3 {\displaystyle {\mathcal {M}}^{3}} has nonnegative scalar curvature, then the Hawking mass of Σ {\displaystyle \Sigma } is non-decreasing as the surface Σ {\displaystyle \Sigma } flows outward at a speed equal to the inverse of the mean curvature. In particular, if Σ t {\displaystyle \Sigma _{t}} is a family of connected surfaces evolving according to

d x d t = 1 H ν ( x ) , {\displaystyle {\frac {dx}{dt}}={\frac {1}{H}}\nu (x),}

where H {\displaystyle H} is the mean curvature of Σ t {\displaystyle \Sigma _{t}} and ν {\displaystyle \nu } is the unit vector opposite of the mean curvature direction, then

d d t m H ( Σ t ) ≥ 0. {\displaystyle {\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]

See also

References

  1. ^ Hoffman 2005, p. 21
  2. ^ Geroch, Robert (December 1973). "Energy extraction*". Annals of the New York Academy of Sciences. 224 (1): 108–117. Bibcode:1973NYASA.224..108G. doi:10.1111/j.1749-6632.1973.tb41445.x. ISSN 0077-8923. S2CID 222086296.{{cite journal}}: CS1 maint: date and year (link)
  3. ^ Hoffman 2005, Lemma 9.6
  4. ^ Section 4 of Shi, Yuguang; Wang, Guofang; Wu, Jie (2008). On the behavior of quasi-local mass at the infinity along nearly round surfaces. arXiv:0806.0678.
  5. ^ Section 2 of Finster, Felix; Smoller, Joel; Yau, Shing-Tung (2000-06-01). "Some Recent Progress in Classical General Relativity". Journal of Mathematical Physics. 41 (6): 3943–3963. arXiv:gr-qc/0001064. Bibcode:2000JMP....41.3943F. doi:10.1063/1.533332. S2CID 18904339.

Further reading

  • Section 6.1 in Szabados, László B. (December 2004). Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article. Vol. 7. p. 4. Bibcode:2004LRR.....7....4S. doi:10.12942/lrr-2004-4. ISSN 2367-3613. PMC 5255888. PMID 28179865. S2CID 40602589.{{cite book}}: CS1 maint: date and year (link)
  • Hoffman, David A., ed. (2005). Global theory of minimal surfaces: proceedings of the Clay Mathematics Institute 2001 Summer School, Mathematical Sciences Research Institute, Berkeley, California, June 25-July 27, 2001. Clay mathematics proceedings. Providence, RI : Cambridge, MA: American Mathematical Society ; Clay Mathematics Institute. ISBN 978-0-8218-3587-6.