The Schwarzschild black hole has an event horizon with a specific structure, which is captured by the following definition: A set is called a bifurcate Killing horizon if it is the union of a a smooth spacelike submanifold of co-dimension two, called the bifurcation surface, on which a Killing vector field vanishes, and of four smooth null embedded hypersurfaces obtained by following null geodesics in the four distinct null directions normal to .
For example, the Killing vector in Minkowski spacetime has a bifurcate Killing horizon, with the bifurcation surface . As already mentioned, another example is given by the set in Schwarzschild–Kruskal–Szekeres spacetime with positive mass parameter .
In the spirit of the previous definition, we will refer to the union of two null hypersurfaces, which intersect transversally on a 2-dimensional spacelike surface as a bifurcate null surface.
The reader is warned that a bifurcate Killing horizon is not a Killing horizon, as defined in Section 2.5, since the Killing vector vanishes on . If one thinks of as not being part of the bifurcate Killing horizon, then the resulting set is again not a Killing horizon, since it has more than one component.
One of the key steps of the uniqueness theory, as described in Section 3, forces one to consider “horizon candidates” with local properties similar to those of a proper event horizon, but with global behavior possibly worse: A connected, not necessarily embedded, null hypersurface to which is tangent is called a Killing prehorizon. In this terminology, a Killing horizon is a Killing prehorizon, which forms a embedded hypersurface, which coincides with a connected component of . The Minkowskian Killing vector provides an example where is not a hypersurface, with every hyperplane being a prehorizon.
The Killing vector on , equipped with the flat metric, where is an -dimensional torus, and where is a unit Killing vector on with dense orbits, admits prehorizons, which are not embedded. This last example is globally hyperbolic, which shows that causality conditions are not sufficient to eliminate this kind of behavior.
Of crucial importance to the zeroth law of black-hole physics (to be discussed shortly) is the fact that the -component of the Ricci tensor vanishes on horizons or prehorizons,
The following two properties of Killing horizons and prehorizons play a role in the theory of stationary black holes:
An immediate consequence of the definition of a Killing horizon or prehorizon is the proportionality of and on , where
The Killing equation implies ; we see that the surface gravity measures the extent to which the parametrization of the geodesic congruence generated by is not affine.
A fundamental property is that the surface gravity is constant over horizons or prehorizons in several situations of interest. This leads to the intriguing 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.
The constancy of holds in vacuum, or for matter fields satisfying the dominant-energy condition, see, e.g., [151, Theorem 7.1]. The original proof of the zeroth law  proceeds as follows: First, Einstein’s equations and the fact that vanishes on the horizon imply that on . Hence, the vector field is perpendicular to and, therefore, space-like (possibly zero) or null on . On the other hand, the dominant energy condition requires that is zero, time-like or null. Thus, vanishes or is null on the horizon. Since two orthogonal null vectors are proportional, one has, using Einstein’s equations again, on , where . The result that is constant over each horizon follows now from the general property (see, e.g., )
The proof of (2.16) given in  generalizes to all spacetime dimensions ; the result also follows in all dimensions from the analysis in  when the horizon has compact spacelike sections.
By virtue of Eq. (2.17) and the identity , the zeroth law follows if one can show that the twist one-form is closed on the horizon :
Yet another situation of interest is a spacetime with two commuting Killing vector fields and , with a Killing horizon associated to a Killing vector . Such a spacetime is said to be circular if the distribution of planes spanned by and is hypersurface-orthogonal. Equivalently, the metric can be locally written in a 2+2 block-diagonal form, with one of the blocks defined by the orbits of and . In the circular case one shows that implies on the horizon generated by ; see , Chapter 7 for details.
A significant observation is that of Kay and Wald , that must be constant on bifurcate Killing horizons, regardless of the matter content. This is proven by showing that the derivative of the surface gravity in directions tangent to the bifurcation surface vanishes. Hence, cannot vary between the null-generators. But it is clear that is constant along the generators.
Summarizing, each of the following hypotheses is sufficient to prove that is constant over a Killing horizon defined by :
See  for some further observations concerning (2.16).
A Killing horizon is called degenerate if vanishes, and non-degenerate otherwise.
As an example, in Minkowski spacetime, consider the Killing vector . We have
A key theorem of Rácz and Wald  asserts that non-degenerate horizons (with a compact cross section and constant surface gravity) are “essentially bifurcate”, in the following sense: Given a spacetime with such a non-degenerate Killing horizon, one can find another spacetime, which is locally isometric to the original one in a one-sided neighborhood of a subset of the horizon, and which contains a bifurcate Killing horizon. The result can be made global under suitable conditions.
The notion of average surface gravity can be defined for null hypersurfaces, which are not necessarily Killing horizons: Following , near a smooth null hypersurface one can introduce Gaussian null coordinates, in which the metric takes the formaverage surface gravity is defined as
A smooth null hypersurface, not necessarily a Killing horizon, with a smooth compact cross-section such that is said to be mean non-degenerate.
Using general identities for Killing fields (see, e.g., , Chapter 2) one can derive the following explicit expressions for :).
Living Rev. Relativity 15, (2012), 7
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