4.1 Killing horizons

The black hole region of an asymptotically flat spacetime (M, g ) is the part of M which is not contained in the causal past of future null infinity.30 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.31 The definition of the latter is of purely local nature: Consider a Killing field ξ, say, and the set of points where ξ is null, N ≡ (ξ, ξ) = 0. A connected component of this set which is a null hyper-surface, (dN , dN ) = 0, is called a Killing horizon, H [ξ]. Killing horizons possess a variety of interesting properties:32

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

κ = constant on H [ξ] (5 )
is usually called the zeroth law of black hole physics [3Jump To The Next Citation Point]. The zeroth law can be established by different means: Each of the following alternatives is sufficient to prove that κ is uniform over the Killing horizon generated by ξ.

(i) Einstein’s equations are fulfilled with matter satisfying the dominant energy condition.

(ii) The domain of outer communications is either static or circular.

(iii) H [ξ] is a bifurcate Killing horizon.

(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 R (ξ,ξ) vanishes on the horizon (see above), imply that T (ξ,ξ) = 0 on H [ξ]. Hence, the one-form T (ξ)35 is perpendicular to ξ and, therefore, space-like or null on H [ξ]. On the other hand, the dominant energy condition requires that T(ξ) is time-like or null. Thus, T(ξ) is null on the horizon. Since two orthogonal null vectors are proportional, one has, using Einstein’s equations again, ξ ∧ R (ξ ) = 0 on H [ξ]. The result that κ is uniform over the horizon now follows from the general property36

ξ ∧ dκ = − ξ ∧ R (ξ) on H [ξ]. (6 )

(ii) By virtue of Eq. (6View Equation) and the general Killing field identity dω = ∗[ξ ∧ R (ξ)], the zeroth law follows if one can show that the twist one-form is closed on the horizon [147Jump To The Next Citation Point]:

[d ω]H[ξ] = 0 = ⇒ κ = constant on H [ξ]. (7 )
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 ω vanishes identically,37 which then proves the second version of the first zeroth law.38

(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 κ cannot vary between the null-generators. (It is clear that κ 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 [146147].

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