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
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
}\left(1 - \frac\int_\Sigma H^2 da \right),
where
is the
mean curvature of
.
Properties
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
Further reading
Notes and References
- https://books.google.com/books?id=HRr3jdBDXIgC&pg=PA122 Page 21
- 10.1111/j.1749-6632.1973.tb41445.x. Energy Extraction. 1973. Geroch. Robert. Annals of the New York Academy of Sciences. 224. 108–117. 1973NYASA.224..108G. 222086296.
- https://books.google.com/books?id=HRr3jdBDXIgC&pg=PA124 Lemma 9.6
- 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 10.1063/1.533332. gr-qc/0001064. Some recent progress in classical general relativity. 2000. Finster. Felix. Smoller. Joel. Yau. Shing-Tung. Journal of Mathematical Physics. 41. 6. 3943–3963. 2000JMP....41.3943F. 18904339.
