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.
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. [178]).
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) 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 [146], [147].
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Stationary Black Holes: Uniqueness and Beyond
Markus Heusler http://www.livingreviews.org/lrr-1998-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |