horizon generators, and
k
a
is the tangent vector to the horizon generators with respect to
λ
. The Killing
vector
ξ
a
is given on the horizon by
ξ
a
=
κλk
a
(with a certain choice for the origin of
λ
). Using the Einstein
equation we thus have
∆
M
=
(
κ/
8
πG
)
Z
R
ab
k
a
k
b
λ dλdA
(2.3)
=
(
κ/
8
πG
)
Z
dρ
dλ
λ dλdA
(2.4)
=
(
κ/
8
πG
)
Z
(

ρ
)
dλdA
(2.5)
=
(
κ/
8
πG
) ∆
A.
(2.6)
The second equality uses the focusing equation neglecting the quadratic terms
ρ
2
and
σ
2
, the third uses
integration by parts with the boundary term dropped since the black hole is initially and finally stationary,
and the last equality follows directly from the definition of
ρ
.
2.1.4
Second and Third Laws
Continuing with the analogy, the
Second Law
is of course Hawking’s area theorem, stating that the horizon
area can never decrease assuming Cosmic Censorship and a positive energy condition. The
Third Law
also
has an analog in black hole physics, namely, the surface gravity of the horizon cannot be reduced to zero in
a finite number of steps. Validity of this law has been suggested by investigations of the orbits of charged
test particles around a charged rotating black hole. A precise formulation of this Third Law has been given
and proved under some assumptions by Israel.
Significance of the Third Law
An idea of the significance of the Third Law can be gleaned by thinking about how one might try to violate
it. First, for a nonrotating neutral black hole,
κ
is decreased when mass is added to the hole. (So the hole
has negative specific heat.) But it would take an infinite amount of mass to reduce
κ
to zero. A general
rotating, charged black hole with angular momentum
J
and charge
Q
has a surface gravity and horizon area
given by
κ
= 4
πμ/A,
A
= 4
π
[2
M
(
M
+
μ
)

Q
2
]
(2.7)
with
μ
= (
M
2

Q
2

J
2
/M
2
)
1
/
2
.
(2.8)
An
extremal
black hole is one for which
μ
= 0. For an extremal black hole,
κ
vanishes and
A
= 4
π
(2
M
2

Q
2
).
Thus, an extremal black hole has zero “temperature”, but nonzero “entropy”. (Thus the Planck form of the
Third law does not hold for black holes. Also it should be remarked that if the extremal state is “eternal”
rather than being reached from a nonextremal one, the entropy that enters a proper variational form of
the first law is not the area and, in fact, vanishes.) If
M
2
< Q
2
+
J
2
/M
2
then the spacetime has a naked
singularity and is not a black hole at all. Thus if the surface gravity could actually be reduced to zero, one
would be only infinitesimally far from creating a naked singularity, violating Cosmic Censorship.
To reduce the surface gravity to zero you might thus try to inject a sufficient amount of charge or angular
momentum into the hole.
Suppose you try to drop a charge
q
with mass
m
into a nonrotating charged
black hole of mass
M
and charge
Q < M
, trying to make
Q
+
q
=
M
+
m
. In order for the gravitational
attraction to be stronger than the electrostatic repulsion you must choose
mM > qQ
, so
q/m < M/Q
. But
this inequality insures that
Q
+
q < M
+
m
. Similarly if you try to inject enough orbital angular momentum
to a spinning black hole you find that the particle simply misses the hole.