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 [9] 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., [314
])
The proof of (2.16) given in [314
] generalizes to all spacetime dimensions
; the result also
follows in all dimensions from the analysis in [165
] 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 [270
]:
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 [151
], Chapter 7 for details.
A significant observation is that of Kay and Wald [184], 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 [270] 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 [270] 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 [238], near a smooth null hypersurface
one can introduce Gaussian null
coordinates, in which the metric takes the form
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., [151], Chapter 2) one can derive the following
explicit expressions for
:
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Living Rev. Relativity 15, (2012), 7
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