4.2 Reduction of the Einstein-Hilbert 4 Stationary Space-Times4 Stationary Space-Times

4.1 Killing Horizons 

The black hole region of an asymptotically flat spacetime tex2html_wrap_inline3665 is the part of M which is not contained in the causal past of future null infinity. Popup Footnote Hence, the event horizon, being defined as the boundary of the black hole region, is a global concept. Of crucial importance to the theory of black holes is the strong rigidity theorem, which implies that the event horizon of a stationary spacetime is a Killing horizon. Popup Footnote The definition of the latter is of purely local nature: Consider a Killing field tex2html_wrap_inline3669, say, and the set of points where tex2html_wrap_inline3669 is null, tex2html_wrap_inline3673 . A connected component of this set which is a null hyper-surface, tex2html_wrap_inline3675, is called a Killing horizon, tex2html_wrap_inline3677 . Killing horizons possess a variety of interesting properties: Popup Footnote

It is an interesting fact that the surface gravity plays a similar role in the theory of stationary black holes as the temperature does in ordinary thermodynamics. Since the latter is constant for a body in thermal equilibrium, the result

  equation423

is usually called the zeroth law of black hole physics [3Jump To The Next Citation Point In The Article]. The zeroth law can be established by different means: Each of the following alternatives is sufficient to prove that tex2html_wrap_inline3689 is uniform over the Killing horizon generated by tex2html_wrap_inline3669 .

(i) The original proof of the zeroth law rests on the first assumption [3]. The reasoning is as follows: First, Einstein's equations and the fact that tex2html_wrap_inline3735 vanishes on the horizon (see above), imply that tex2html_wrap_inline3737 on tex2html_wrap_inline3677 . Hence, the one-form tex2html_wrap_inline3741 Popup Footnote is perpendicular to tex2html_wrap_inline3669 and, therefore, space-like or null on tex2html_wrap_inline3677 . On the other hand, the dominant energy condition requires that tex2html_wrap_inline3741 is time-like or null. Thus, tex2html_wrap_inline3741 is null on the horizon. Since two orthogonal null vectors are proportional, one has, using Einstein's equations again, tex2html_wrap_inline3755 on tex2html_wrap_inline3677 . The result that tex2html_wrap_inline3689 is uniform over the horizon now follows from the general property Popup Footnote

  equation437

(ii) By virtue of Eq. (6Popup Equation) and the general Killing field identity tex2html_wrap_inline3763, the zeroth law follows if one can show that the twist one-form is closed on the horizon [147Jump To The Next Citation Point In The Article]:

  equation443

While the original proof (i) takes advantage of Einstein's equations and the dominant energy condition to conclude that the twist is closed, one may also achieve this by requiring that tex2html_wrap_inline3699 vanishes identically, Popup Footnote which then proves the second version of the first zeroth law. Popup Footnote

(iii) The third version of the zeroth law, due to Kay and Wald [105], is obtained for bifurcate Killing horizons. Computing the derivative of the surface gravity in a direction tangent to the bifurcation surface shows that tex2html_wrap_inline3689 cannot vary between the null-generators. (It is clear that tex2html_wrap_inline3689 is constant along the generators.) The bifurcate horizon version of the zeroth law is actually the most general one: First, it involves no assumptions concerning the matter fields. Second, the work of Rácz and Wald strongly suggests that all physically relevant Killing horizons are either of bifurcate type or degenerate [146], [147].



4.2 Reduction of the Einstein-Hilbert 4 Stationary Space-Times4 Stationary Space-Times

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
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