Mathematical Principles of Theoretical Physics

7.1.5 Astrophysical Fluid Dynamics Equations

Thus, by (7.1.4) and (7.1.19) we getRμ νas

(7.1.20)

R 00 =eu−v

[

−

1

2

u′′−

1

r

u′+

1

r

u′(v′−u′)

]

,

R 11 =

1

2

u′′−

1

r

v′−

1

4

u′(v′−u′),

R 22 =e−v

[

1 −ev+

r

2

(u′−v′)

]

,

R 33 =R 22 sin^2 θ,
Rμ ν= 0 ∀μ 6 =ν.
Therefore, the vacuum Einstein field equations (7.1.17) become the following system of
ordinary differential equations

1

2

u′′+

1

r

u′−

1

4

(7.1.21) (v′−u′)u′= 0 ,

1

2

u′′−

1

r

v′−

1

4

(7.1.22) (v′−u′)u′= 0 ,

r

2

(7.1.23) (u′−v′)−ev+ 1 = 0.

By the Bianchi identity, only two equations of (7.1.22)-(7.1.23) are independent. The
difference of (7.1.21) and (7.1.22) leads to

u′+v′= 0 ,

which implies that

(7.1.24) u+v=β (constant),

and (7.1.23) becomes
ev+rv′− 1 = 0.

Namely
d
dr

(re−v) = 1.

It follows that

e−v= 1 −

b

r

,

wherebis a to-be-determined constant.
Then it follows from (7.1.24) that

eu=eβe−v=eβ( 1 −

b

r

).

By scaling timet, we can takeeβ=1. Hence the solution of the Einstein field equations
(7.1.17) is given by

(7.1.25)

g 00 =−eu=−

(

1 −

b

r

)

,

g 11 =ev=

(

1 −

b

r

)− 1

.