Gravity and Strings

7.1 Schwarzschild’s solution 197
(b) Since the gravitational collapse of many different systems always gives rise
to the same BHs, characterized by a very small number of parameters, it is
natural to wonder what has happened to all the information about the original
state. This is essentially the BH information problem, which can be stated
more precisely in quantum-mechanical language. Furthermore, it is also natural
to attribute to the BHs a very big entropy that we should be able to compute
by standard statistical methods if we knew all the BH microstates that a BH
characterized by M, Q, and J can be in. This is the essence of the BH entropy
problem. Tosolve these two problems, we need a theory of quantum gravity.
12. The event horizons of stationary BHs are usually Killing horizons,hypersurfaces that
are invariant under one isometry wherein the modulus of the corresponding Killing
vector k µ of the metric vanishes, k2 = 0. In the Schwarzschild case, k µ = δ µt and
generates translations in time: k2 r=RS = gtt| r=RS = 0. Furthermore, the horizon hypersurface
r = RS is, as a whole, time-translation-invariant.
Killing horizons (and, hence, event horizons) are null hypersurfaces. 19 Furthermore,
for each value of t, the Killing horizon is a two-sphere of radius RS. This is the
only topology allowed according to the topological-censorship theorems [407, 507].
Like many other important results in GR, these theorems depend heavily on energypositivity
conditions and, thus, it is not surprising that they break down in the presence
of a negative cosmological constant and then it is possible to find topological
black holes [42, 156, 186, 202, 203, 572, 628, 629, 648, 649, 650, 683, 684, 859, 921]
whose event horizons can have the topology of any compact Riemann surface. In particular,
generalizations of the asymptotically anti-de Sitter Schwarzschild BH with
horizons with the topology of Riemann surfaces of arbitrary genus were given in
[921].
13. The area of the event horizon is
A =
r=RS
d 2 r 2 = 4π R 2 S . (7.20)
Hawking proved in [512] that the Einstein equations imply that the area A of the
event horizon of a BH never decreases with time. On top of this, if two BHs coalesce
to form a new BH, the area of the horizon of this final BH is larger than the sum of
the areas of the horizons of the initial BHs. (This result holds for more general kinds
of BHs having electric charge and angular momentum.)
There is a clear analogy between the area A of a BH event horizon and the entropy
of a thermodynamical system as never decreasing quantities [95, 97, 98, 241], which
deserves to be investigated further.
19 That is, the vector field normal to a Killing horizon is a null vector field. This vector field, due to the
Lorentzian signature, always belongs to the tangent space of the null hypersurface.