Mathematical Principles of Theoretical Physics

434 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

and case (b) is the embedding

~r−=

{

~rext forr>Rs,

~r−int forr<Rs.

The base space marked asR^3 in (a) and (b) are taken as the coordinate space (i.e. the projec-
tive space), and the surfaces marked byMrepresent the real world which are separated into
two closed parts by the spherical surface of radiusRs: the black hole(r<Rs)and the exterior
world(r>Rs).

Rs R^3

M

Rs R^3

M

(a) (b)

Figure 7.4:Mis the real world with the metric (7.3.5) forr>Rsand the metric (7.3.7)
forr<Rs, and in the base spaceR^3 the coordinate system is taken as spherical coordinates
(r,θ,φ).

In particular, the geometric realization of (7.3.7) for a black hole clearly manifests that the
real world in the black hole is a hemisphere with radiusRsembedded inR^4 ; see Figure7.4(a):

x^21 +x^22 +x^23 +x^24 =R^2 s for 0≤ |x 4 | ≤Rs,

where

(x 1 ,x 2 ,x 3 ,x 4 ) =

(

rsinθcosφ,rsinθsinφ,rcosθ,±

√

R^2 s−r^2

)

.

We remark that the singularity ofMatr=Rs, where the tangent space ofMis per-
pendicular to the coordinate spaceR^3 , is essential, and cannot be removed by any coordi-
nate transformations. The coordinate transformations such as those given by Eddington and
Kruskal possess the singularity as well, and, consequently, cannot be used as the coordinate
systems for the metrics (7.3.5) and (7.3.7).

7.3.2 Blackhole theorem

The main objective of this section is to prove the following blackhole theorem.

Theorem 7.15(Blackhole Theorem).Assume the validity of the Einstein theory of general
relativity, then the following assertions hold true:

1) black holes are closed: matters can neither enter nor leave their interiors;