7.3. BLACK HOLES 433
The coordinate system(t,x)withx= (r,θ,φ)represents the projection of the real world to
the coordinate space. Therefore the radial motion speeddr/dtin the projected world differs
from the proper speedvrby a factor
√
−g 00 /g 11 , i.e.dr
dt=
√
−g 00 /g 11 vr.Hence, the singularity (7.3.3) and (7.3.4) means that for an object moving toward to the
boundary of a black hole, its projection speed vanishes:
dr
dt= 0 atr=Rs.This implies that any object in the exterior of the black holecannot pass through its boundary
and enter into the interior. In the next subsection we shall rigorously prove that a black hole
is a closed and innate system.
Mathematically, a Riemannian manifold(M,gij)is called a geometric realization (i.e.
isometric embedding) inRN, if there exists a one to one mapping
~r:M→RN,such that
gij=
d~r
dxi·
d~r
dxj.
The geometric realization provides a “visual” diagram ofM, the real world of our Universe.
In the following we present the geometric realization of a 3Dmetric space of a black hole
near its boundary. By (7.3.2), the space metric of a black hole is given by
(7.3.5) ds^2 =
(
1 −
Rs
r)− 1
dr^2 +r^2 (dθ^2 +sin^2 θdφ^2 ) forr>Rs=2 MG
c^2.
It is easy to check that a geometric realization of (7.3.5) is given by~r:M→R^4 :(7.3.6) ~rext=
{
rsinθcosφ,rsinθsinφ,rcosθ, 2√
Rs(r−Rs)}
forr>Rs.In the interior of a black hole, the Riemannian metric near the boundary is given by the
TOV solution (7.1.42), and its space metric is in the form
(7.3.7) ds^2 =
(
1 −
r^2
R^2 s)− 1
dr^2 +r^2 (dθ^2 +sin^2 θdφ^2 ) forr<Rs,A geometrical realization of (7.3.7) is
(7.3.8) ~r±int=
{
rsinθcosφ,rsinθsinφ,rcosθ,±√
R^2 s−r^2}
.
The diagrams of (7.3.6) and (7.3.8) are as shown in Figure7.4, where case (a) is the embed-
ding
~r+={
~rext forr>Rs,
~r+int forr<Rs,