11.1 Black holes in general relativity

In order to introduce the reader to some basic notions of black-hole physics,

let us begin with the simplest black-hole solutions of general relativity in

four dimensions, which are the Schwarzschild and Reissner“Nordstr¨m black

o

holes. The latter black hole is a generalization of the Schwarzschild solution

that is electrically charged. Another generalization, known as the Kerr

black hole, is a black hole with angular momentum. Certain black holes

with angular momentum are considered in Section 11.3.

Schwarzschild black hole

The Schwarzschild solution in spherical coordinates

For a spherically symmetric mass distribution of mass M in four space-time

dimensions, there is a unique solution to the vacuum Einstein™s equations

Rµν = 0, (11.3)

that describes the geometry outside of the mass distribution.4 In four

dimensions it is given by the Schwarzschild black-hole metric, which in

Schwarzschild coordinates (t, r, θ, φ) is

rH rH ’1

ds2 = gµν dxµ dxν = ’ 1 ’ dt2 + 1 ’ dr2 + r2 d„¦2 , (11.4)

2

r r

where

rH = 2G4 M. (11.5)

3 The LHC is the Large Hadron Collider at CERN, which is scheduled to start operating in 2007.

4 The statement that the Schwarzschild black hole is the unique vacuum solution of Einstein™s

equations in four dimensions with spherical symmetry. Its time independence is known as

Birkho¬™s theorem.

11.1 Black holes in general relativity 553

Here rH is known as the Schwarzschild radius, and G4 is Newton™s constant.

The metric describing the unit two-sphere is

d„¦2 = dθ2 + sin2 θdφ2 . (11.6)

2

The Schwarzschild metric only depends on the total mass M (which is

both inertial and gravitational), and it reduces to the Minkowski metric

as M ’ 0. Note that t is a time-like coordinate for r > rH and a space-like

coordinate for r < rH , while the reverse is true for r. The surface r = rH ,

called the event horizon, separates the previous two regions. This metric is

stationary in the sense that the metric components are independent of the

Schwarzschild time coordinate t, so that ‚/‚t is a Killing vector. This Killing

vector is time-like outside the horizon, null on the horizon, and space-like

inside the horizon.

It becomes clear that M has the interpretation of a mass by considering

the weak ¬eld limit, that is, the asymptotic r ’ ∞ behavior of Eq. (11.4). In

this limit we should recover Newtonian gravity.5 The Newtonian potential

¦ in these stationary coordinates can be read o¬ from the tt component of

the metric

gtt ∼ ’ (1 + 2¦) . (11.7)

As a result, in the case of the Schwarzschild black hole,

M G4

¦=’ , (11.8)

r

so that it becomes clear that the parameter M is the black-hole mass.

Schwarzschild black hole in D dimensions

The four-dimensional Schwarzschild metric (11.4) can be generalized to D

dimensions, where it takes the form

ds2 = ’hdt2 + h’1 dr2 + r2 d„¦2 , (11.9)

D’2

with

rH D’3

h=1’ (11.10)

r

and

16πM GD

D’3

rH = . (11.11)

(D ’ 2)„¦D’2

5 This is nicely illustrated by considering a massive test particle moving in the curved background.

This is a homework problem.

554 Black holes in string theory

Here „¦n is the volume of a unit n-sphere, namely6

2π (n+1)/2

„¦n = . (11.12)

“ n+12

For large r, this again determines the Newton potential and therefore the

black-hole mass M .

The singularities

As can be seen from Eq. (11.4), the coe¬cients of the metric become sin-

gular at r = 0 and also at the Schwarzschild radius r = rH . In general,

a singularity in a metric component could be a coordinate-dependent phe-

nomenon. In order to determine whether a physical singularity is present,

coordinate-independent quantities, that is, scalars, should be analyzed. Such

a scalar quantity should involve the Riemann tensor. For example, the D = 4

Schwarzschild solution yields, after a straightforward calculation,

2

12rH

µνρσ

R Rµνρσ = 6 . (11.13)

r

This is evidence that the singularity at the horizon r = rH is only a coor-

dinate singularity, as we will prove shortly, while it proves that a physical

singularity is located at r = 0.

For objects that are not black holes, the behavior of the solution at the

point r = 0 is of no physical relevance, since these objects have a mass

distribution of ¬nite size, and there is no horizon or singularity. The metric

describing the sun, for example, is perfectly well de¬ned at r = 0. If,

however, the mass is concentrated inside the Schwarzschild radius, then the

singularity at r = 0 becomes relevant, and the resulting solution is called a

Schwarzschild black hole.

In general relativity, it is common practice to set Newton™s constant equal

to unity, G4 = 1, as a choice of length scale. We prefer not to do so,

both because we are interested in Newton™s constant in various space-time

dimensions, and because the string scale, rather than Newton™s constant, is

the natural length scale in string theory. G4 , and more generally GD , are

related to the string scale, the string coupling, and a (10 ’ D)-dimensional

compacti¬cation volume V by GD = G10 /V and G10 = 8π 6 gs 8 . 2

s

Schwarzschild solution in Kruskal“Szekeres coordinates

There are other coordinate systems in which the Schwarzschild solution does

not even have a coordinate singularity at the horizon. One such coordinate

R

6 This can be derived by computing exp(’r 2 ) dn+1 x in spherical coordinates and comparing

to the answer computed in Cartesian coordinates.

11.1 Black holes in general relativity 555

system, called the Kruskal“Szekeres coordinate system, is related to the

Schwarzschild coordinates previously introduced by

1/2

r t

er/2rH cosh

’1

u= , (11.14)

rH 2rH

1/2

r t

er/2rH sinh

’1

v= . (11.15)

rH 2rH

In these coordinates the metric takes the form

3

4rH ’r/rH

2

’dv 2 + du2 + r2 d„¦2 .

ds = e (11.16)