String Theory and M-Theory

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
dΩ 2 2 = dθ 2 + sin 2 θdφ 2 . (11.6)
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 field limit, that is, the asymptotic r → ∞ behavior of Eq. (11.4). In
this limit we should recover Newtonian gravity. 5 The Newtonian potential
Φ in these stationary coordinates can be read off from the tt component of
the metric
gtt ∼ − (1 + 2Φ) . (11.7)
As a result, in the case of the Schwarzschild black hole,
Φ = − MG4
, (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
with
and
ds 2 = −hdt 2 + h −1 dr 2 + r 2 dΩ 2 D−2, (11.9)
h = 1 −
r D−3
H
rH
D−3
r
(11.10)
16πMGD
= . (11.11)
(D − 2)ΩD−2
5 This is nicely illustrated by considering a massive test particle moving in the curved background.
This is a homework problem.