Hawking energy explained

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

( l{M} ^{3,} g _{ab} ) be a 3-dimensional sub-manifold of a relativistic spacetime, and let \Sigma \subset l{M} ^{3} be a closed 2-surface. Then the Hawking mass m _{H(\Sigma)} of \Sigma is defined to bem _{H(\Sigma)} : = \sqrt{ (Area \Sigma)/(16 \pi) }\left(1 - \frac\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^[1] implies that Hawking mass satisfies an important monotonicity condition. Namely, if

l{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(dx)/(dt) = (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(d)/(dt) m _{H(\Sigma} _{t)} \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^[2] or the Bondi mass, depending on whether the surface is asymptotic to spatial infinity or null infinity.^[3]

See also

• Mass in general relativity
• Inverse mean curvature flow

Further reading

• Section 6.1 in Book: Szabados, László B. . Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article . December 2004 . 2004 . 7 . 4 . en . 2004LRR.....7....4S . 10.12942/lrr-2004-4 . 2367-3613 . 5255888 . 28179865 . free . 1 . 40602589.
• Book: 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 . 2005 . American Mathematical Society ; Clay Mathematics Institute . 978-0-8218-3587-6 . Hoffman . David A. . Clay mathematics proceedings . Providence, RI : Cambridge, MA.

Notes and References

1. Geroch . Robert . December 1973 . 1973 . Energy extraction* . Annals of the New York Academy of Sciences . en . 224 . 1 . 108–117 . 1973NYASA.224..108G . 10.1111/j.1749-6632.1973.tb41445.x . 0077-8923 . 222086296.
2. Section 4 of Book: Shi . Yuguang . On the behavior of quasi-local mass at the infinity along nearly round surfaces . Wang . Guofang . Wu . Jie . 2008 . 0806.0678 .
3. Section 2 of Finster . Felix . Smoller . Joel . Yau . Shing-Tung . 2000-06-01 . Some Recent Progress in Classical General Relativity . en . 41 . 6 . 3943–3963 . gr-qc/0001064 . 2000JMP....41.3943F . 10.1063/1.533332 . 18904339 . Journal of Mathematical Physics.