Entropy of black holes
281
Suppose that the convergence p of neighbouring generators has a positive value
PO > 0 at some point q on a generator of j-(I+). then p increases to infinity
within a finite affine distance .6 v :::: 1/ PO to the future of q. The point r at
which p becomes infinite is a focal point at which neighbouring null geodesics
intersect. In other words, if the generators of j-(I+) ever start converging.
they are destined to have future endpoints within a finite affine distance. This
contradicts the previously stated property that such generators have no future
endpoints. It follows [I] that p :::: 0 everywhere on the event horizon and.
therefore. from (10.17). that the area of the two-dimensional cross section cannot
decrease with time I. As already noted. this prompted the observation that this
area is analogous to the entropy of a thermodynamic system.
10.3 Entropy of black holes
In fact. the area of the two-dimensional section of the event horizon will remain
constant only if the black hole is in a stationary state. If the black hole interacts
with anything else the area always increases. In this respect. the area behaves
similarly to the entropy of a thermodynamic system. In favourable circumstances.
one can arrange that the increase in area can be made arbitrarily small. which
corresponds to nearly reversible transformations in thermodynamics.
During black-hole formation in the collapse of a star. the metric is strongly
time-dependent and a complete classification of all solutions has not been
found. However. the possible final stationary states have been identified. An
asymptotically flat metric is called stationary if there exists a Killing vector2 k
that is timelike near infinity (where it may be normalized such that k 2 = I). In
other words. outside of the horizon k = f,. where t is a time coordinate. In these
coordinates. the general stationary metric has a line element of the form
ds 2 = goo(x) dt 2 + 2g0i(X) dt d,xi + gij (x) d,xi d,xi.
(10.29)
A stationary metric is called static if it is also invariant under time-reversal. at
least near infinity. Thus. the general static metric has gOi = 0 and the line element
takes the form
ds 2 = goo(x) dt 2 + gij (x) d,xi d,xj.
(10.30)
I It would be possible to escape this conclusion if the generators were prevented from reaching the
finite affine distance to the endpoint because of an intervening singularity. However, Hawking [5] has
shown, using the general requirements of asymptotic predictability, that this does not occur.
2 A general coordinate transformation x ~ x' is called an 'isometry' if the transformed metric
8~v(X') is the same function of its argument x'''' a~ the original metric 8",v(X) was of its argument x"'.
TIle generators of such transformations may be found by considering an infinitesimal transformation
in which x'''' = x'" + £~'" with £ « 1. This is an isometry if ~ satisfies ~"';\I + ~v;'" = O. and
any vector satisfying this is called a 'Killing' vector. We may equivalently write the Killing vector as
~ = ~jLJr.
281
Suppose that the convergence p of neighbouring generators has a positive value
PO > 0 at some point q on a generator of j-(I+). then p increases to infinity
within a finite affine distance .6 v :::: 1/ PO to the future of q. The point r at
which p becomes infinite is a focal point at which neighbouring null geodesics
intersect. In other words, if the generators of j-(I+) ever start converging.
they are destined to have future endpoints within a finite affine distance. This
contradicts the previously stated property that such generators have no future
endpoints. It follows [I] that p :::: 0 everywhere on the event horizon and.
therefore. from (10.17). that the area of the two-dimensional cross section cannot
decrease with time I. As already noted. this prompted the observation that this
area is analogous to the entropy of a thermodynamic system.
10.3 Entropy of black holes
In fact. the area of the two-dimensional section of the event horizon will remain
constant only if the black hole is in a stationary state. If the black hole interacts
with anything else the area always increases. In this respect. the area behaves
similarly to the entropy of a thermodynamic system. In favourable circumstances.
one can arrange that the increase in area can be made arbitrarily small. which
corresponds to nearly reversible transformations in thermodynamics.
During black-hole formation in the collapse of a star. the metric is strongly
time-dependent and a complete classification of all solutions has not been
found. However. the possible final stationary states have been identified. An
asymptotically flat metric is called stationary if there exists a Killing vector2 k
that is timelike near infinity (where it may be normalized such that k 2 = I). In
other words. outside of the horizon k = f,. where t is a time coordinate. In these
coordinates. the general stationary metric has a line element of the form
ds 2 = goo(x) dt 2 + 2g0i(X) dt d,xi + gij (x) d,xi d,xi.
(10.29)
A stationary metric is called static if it is also invariant under time-reversal. at
least near infinity. Thus. the general static metric has gOi = 0 and the line element
takes the form
ds 2 = goo(x) dt 2 + gij (x) d,xi d,xj.
(10.30)
I It would be possible to escape this conclusion if the generators were prevented from reaching the
finite affine distance to the endpoint because of an intervening singularity. However, Hawking [5] has
shown, using the general requirements of asymptotic predictability, that this does not occur.
2 A general coordinate transformation x ~ x' is called an 'isometry' if the transformed metric
8~v(X') is the same function of its argument x'''' a~ the original metric 8",v(X) was of its argument x"'.
TIle generators of such transformations may be found by considering an infinitesimal transformation
in which x'''' = x'" + £~'" with £ « 1. This is an isometry if ~ satisfies ~"';\I + ~v;'" = O. and
any vector satisfying this is called a 'Killing' vector. We may equivalently write the Killing vector as
~ = ~jLJr.