Mathematical Principles of Theoretical Physics

402 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

We are now in position to discuss the solutions of problem (7.1.28)-(7.1.33). Let

(7.1.34) M(r) =
c^2 r
2 G

( 1 −e−v).

Then the equation (7.1.28) can be rewritten as

1

r^2

dM

dr

= 4 π ρ,

whose solution is given by

(7.1.35) M(r) =

∫r

0

4 πr^2 ρdr for 0<r<R.

By (7.1.35), we see thatM(r)is the mass, contained in the ballBr. It follows from (7.1.34)
that

(7.1.36) e−v= 1 −

2 GM(r)

c^2 r

.

Inserting (7.1.36) in (7.1.29) we obtain

(7.1.37) u′=

1

r(c^2 r− 2 MG)

[

8 πG

c^2

pr^3 + 2 GM(r)

]

.

Putting (7.1.37) into (7.1.31) we get

(7.1.38) p′=−
(p+c^2 ρ)
2 r(c^2 r− 2 MG)

[

8 πG

c^2

pr^3 + 2 GM(r)

]

.

Thus, it suffices for us to derive the solutionp,Mandρfrom (7.1.32)-(7.1.34) and
(7.1.38), and thenvanduwill follow from (7.1.36)-(7.1.37) and (7.1.33).
The equation (7.1.38) is called the TOV equation, which was derived to describe the
structure of neutron stars.
We note that (7.1.36) is the interior metric of a blackhole provided that 2GM(r)/c^2 r=

1. Thus the TOV solution (7.1.36) gives a rigorous proof of the following theorem for the

7.3 Black Holes.

Theorem 7.3.If the matter field in a ball BRof radius R is spherically symmetric, and the
mass MRand the radius R satisfy
2 GMR
c^2 R

= 1 ,

then the ball must be a blackhole.

An idealized model is that the density is homogeneous, i.e. (7.1.32) is given by

ρ=ρ 0 a constant.