4.1 Killing horizons

The black hole region of an asymptotically flat spacetime is the part of which is not contained in the causal past of future null infinity. 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. The definition of the latter is of purely local nature: Consider a Killing field , say, and the set of points where is null, . A connected component of this set which is a null hyper-surface, , is called a Killing horizon, . Killing horizons possess a variety of interesting properties:
• An immediate consequence of the above definition is the fact that and are proportional on . (Note that , since , and that two orthogonal null vectors are proportional.) This suggests the following definition of the surface gravity, ,
Since the Killing equation implies , the above definition shows that the surface gravity measures the extent to which the parametrization of the geodesic congruence generated by is not affine.
• A theorem due to Vishveshwara [172] gives a characterization of the Killing horizon in terms of the twist of : The surface is a Killing horizon if and only if
• Using general identities for Killing fields one can derive the following explicit expressions for :
Introducing the four velocity for a time-like , the first expression shows that the surface gravity is the limiting value of the force applied at infinity to keep a unit mass at in place: , where (see, e.g. [177]).
• Of crucial importance to the zeroth law of black hole physics (to be discussed below) is the fact that the -component of the Ricci tensor vanishes on the horizon,
This follows from the above expressions for and the general Killing field identity .

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

is usually called the zeroth law of black hole physics [3]. The zeroth law can be established by different means: Each of the following alternatives is sufficient to prove that is uniform over the Killing horizon generated by .

(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) 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 vanishes on the horizon (see above), imply that on . Hence, the one-form is perpendicular to and, therefore, space-like or null on . On the other hand, the dominant energy condition requires that is time-like or null. Thus, is null on the horizon. Since two orthogonal null vectors are proportional, one has, using Einstein’s equations again, on . The result that is uniform over the horizon now follows from the general property

(ii) By virtue of Eq. (6) and the general Killing field identity , the zeroth law follows if one can show that the twist one-form is closed on the horizon [147]:

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 vanishes identically, which then proves the second version of the first zeroth law.

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