400 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
Comparing (7.1.25) with (7.1.15), we can obtainb=2 mG
c^2.
Thus, we get the solution of (7.1.17) as
g 00 =−(
1 −
2 mG
c^2 r)
,
g 11 =(
1 −
2 mG
c^2 r)− 1
,
and the metric reads
(7.1.26) ds^2 =−
(
1 −
2 mG
c^2 r)
c^2 dt^2 +(
1 −
2 mG
c^2 r)− 1
dr^2 +r^2 dθ^2 +r^2 sin^2 θdφ^2 ,We have in particular
e−v=eu= 1 −
2 mG
c^2 r.
which is called the Schwarzschild solution or metric.
TOV metric
The Schwarzschild metric (7.1.26) describes the exterior gravitational fields of a centrally
symmetric ball. For the interior gravitational fields, the metric is given by the TOV solution.
Letmbe the mass of a centrally symmetric ball, andRbe the radius of this ball. In the
interior of the ball, the variablersatisfies 0≤r<R.Let the ball be a static liquid sphere
consisting of idealized fluid, an approximation of stars. The energy-momentum tensor of an
idealized fluid is in the form
Tμ ν= (ρ+p)uμuν+pgμ ν,wherepis the pressure,ρis the density, anduμis the 4-velocity. For a static fluid,uμis
given by
uμ=1
√
−g 00( 1 , 0 , 0 , 0 ).
Hence, the (1,1)-type of the energy-momentum tensor is in the form
Tμν=
−c^2 ρ 0 0 0
0 p 0 0
0 0 p 0
0 0 0 p
.
The Einstein gravitational field equations of an interior ball read(7.1.27)
Rνμ−1
2
δμνR=−8 πG
c^4
Tμν,DνTμν= 0.