Mathematical Principles of Theoretical Physics

398 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

where(r,θ,φ)is the spherical coordinate system, andu=u(r,t)andv=v(r,t)are functions
ofrandt, which are determined by the gravitational field equations.
In the exterior of a ball, Schwarzschild first obtained an exact solution of the Einstein
field equations in 1916, which describes the gravitational fields for the external vacuum state
of a static spherically symmetric matter field.
Letmbe the total mass of a centrally symmetric ball. Then the classical Newtonian
gravitational potential of the ball reads

(7.1.13) φ=−

mG

r

.

Based on the Einstein general theory of relativity, the time-componentg 00 of gravitational
potentialgμ νand the Newton potentialφhave the following relation

(7.1.14) g 00 =−

(

1 +

2

c^2

φ

)

Hence, by (7.1.13) and (7.1.14) for the ball we have

(7.1.15) g 00 =− 1 +

2 mG

c^2 r

.

Now we consider the gravitational field equations in the exterior of the ball. In the vacuum
state,

(7.1.16) Tμ ν= 0.

On the other hand, byR=gμ νRμ ν, the Einstein gravitational field equations

Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν

can be equivalently written as

Rμ ν=−

8 πG

c^4

(Tμ ν−

1

2

gμ νT), T=gklTkl.

Thus, by (7.1.16) the Einstein field equations become

(7.1.17) Rμ ν= 0

The nonzero components of the metric (7.1.12)gμ νare

(7.1.18) g 00 =−eu, g 11 =ev, g 22 =r^2 , g 33 =r^2 sin^2 θ.

Since the gravitational resource is static,uandvonly depend onr. By (7.1.18), all
nonzero components of the Levi-Civita connection (7.1.5) are given by

(7.1.19)

Γ^100 =

1

2

eu−vu′, Γ^111 =

1

2

v′, Γ^122 =−re−v,

Γ^133 =−re−vsin^2 θ, Γ^010 =

1

2

u′, Γ^212 =

1

r

,

Γ^233 =−sinθcosθ, Γ^313 =

1

r

, Γ^323 =

cosθ

sinθ

.