Mathematical Principles of Theoretical Physics

432 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

7.3 Black Holes

7.3.1 Geometric realization of black holes

The concept of black holes was originated from the Einstein general theory of relativity.
Based on the Einstein gravitational field equations, K. Schwarzschild derived in 1916 an
exact exterior solution for a spherically symmetrical matter field, and Tolman-Oppenheimer-
Volkoff derived in 1939 an interior solution; see Section7.1.2. In both solutions if the radius
Rof the matter field with massMis less than or equal to a critical radiusRs, called the
Schwarzschild radius:

(7.3.1) R≤Rs=

2 MG

c^2

,

then the matter field generates a singular spherical surfacewith radiusRs, where time stops
and the spatial metric blows-up; see Figure7.3. The spherical region with radiusRsis called
the black hole.

R

Rs Black hole

Figure 7.3: The spherical region enclosing a matter field with massMand radiusRsatisfying
(7.3.1) is called black hole.

We recall again the Schwarzschild metric in the exterior of ablack hole written as

(7.3.2)

ds^2 =g 00 c^2 dt^2 +g 11 dr^2 +r^2 (dθ^2 +sin^2 θdφ^2 ),

g 00 =−

(

1 −

2 MG

c^2 r

)

, g 11 =

(

1 −

2 MG

c^2 r

)− 1

,

wherer>Rswhen the condition (7.3.1) is satisfied.
In (7.3.2) we see that atr=Rs, the time interval is zero, and the spatial metric blows up:

√

−g 00 dt=

(

1 −

Rs

r

) 1 / 2

(7.3.3) dt= 0 atr=Rs,

√

g 11 dr=

(

1 −

Rs

r

)− 1 / 2

(7.3.4) dr=∞ atr=Rs.

Physically, the proper time and distance for (7.3.2) are

proper time=

√

−g 00 t,

proper distance=

√

g 11 dr^2 +r^2 dθ^2 +r^2 sinθdφ^2.