String Theory Demystified

242 String Theory Demystifi ed
We will ignore that here, we only mention it for information purposes. Continuing,
the simplest case we can imagine is a black hole of mass m which is static (i.e.,
nonrotating) and spherically symmetric. The metric which describes the space-time
outside a black hole of this form is called the Schwarzschild metric. The full solution
to the vacuum fi eld equations used to arrive at this metric can be found in Chap. 10
of Relativity Demystifi ed. We simply state the metric here:
2 mG4
2 mG4
ds 1 dt
r
r
2
1 2
⎛ ⎞ ⎛ ⎞
=− +
⎝
⎜
⎠
⎟ + +
⎝
⎜
⎠
⎟
−1
2 2
dr + r dΩ 2 (14.6)
2 2 2 2
where dΩ= dθ + sin θdφ . The point rH G m = 2 4 is called the horizon. This
appears to be a singular point because setting r = 2G4mcauses the coeffi cient of dr
to blow up. It can be shown, however, that this is not a real singularity—this singular
behavior is just an artifact of the coordinate system. To see this we can calculate a
scalar which is an invariant, which gives us insight into the true nature of the
horizon. One such invariant is
2
µνρσ 12rH R Rµνρσ
= (14.7)
6
r
This expression tells us that there is a true singularity at r = 0.
Although rH G m = 2 4 is not a singularity, it is still an important location. This
location as we have already indicated denotes the event horizon. This is a boundary,
in the case of (3 + 1)-dimensional space-time the surface of a sphere which divides
space-time into the external world and a point of no return. Nothing that crosses the
event horizon can ever return to the rest of the universe, not even light. This is why
black holes are black, because light cannot escape from inside the horizon.
It will be of interest to study black holes in arbitrary space-time dimension D.
With that in mind, before moving on to our next black hole let’s defi ne some basic
quantities. The fi rst item to note is the volume of a unit sphere in d dimensions. This
is given by
Ω
d
( d+
1)/ 2
2π
=
⎛ d + 1⎞
Γ
⎝
⎜
2 ⎠
⎟
(14.8)
where Γ is the gamma factorial function. The radius of the horizon in D-dimensional
space-time is given by
r
16πmG
=
( D − 2)Ω
D−3 D
H
D−2
(14.9)