Mathematical Principles of Theoretical Physics

7.1. ASTROPHYSICAL FLUID DYNAMICS 401

By (7.1.20) we have

R=gμ νRμ ν=−e−uR 00 +e−νR 11 +

2

r^2

R 22.

Then, by (7.1.20) andRνμ=gν αRα μwe get

R^00 −

1

2

R=−e−v

(

1

r^2

−

v′

r

)

+

1

r^2

,

R^11 −

1

2

R=−e−v

(

u′

r

+

1

r^2

)

+

1

r^2

,

R^22 −

1

2

R=−

1

2

e−v

(

u′′−

1

2

u′v′+

1

2

u′^2 +

1

r

u′−

1

r

v′

)

In addition, we know that

DαTμα=D 0 Tμ^0 +D 1 Tμ^1 +D 2 Tμ^2 +D 3 Tμ^3 ,

DαTμβ=

∂Tμβ

∂xα

+Γβα νTμν−Γνα μTνβ.

Thus, by (7.1.19) we have

DαT 1 α=

dp

dr

+

1

2

(p+c^2 ρ)u′,

DαTμα= 0 forμ 6 = 1.

Hence, the field equations (7.1.27) can be written as

e−v

(

1

r^2

−

v′

r

)

−

1

r^2

=−

8 πG

c^2

(7.1.28) ρ,

e−v

(

1

r^2

+

u′

r

)

−

1

r^2

=

8 πG

c^4

(7.1.29) p,

e−v(u′′−

1

2

u′v′+

1

2

u′^2 +

1

r

u′−

1

r

v′) =

16 πG

c^2

(7.1.30) p,

p′+

1

2

(7.1.31) (p+c^2 ρ)u′= 0.

By the Bianchi identity, only three equations of (7.1.28)-(7.1.31) are independent. Here
we also regardpandρas unknown functions. Therefore, for the four unknown functions
u,v,p,ρ, we have to add an equation of state to the system of (7.1.28)-(7.1.31):

(7.1.32) ρ=f(p),

and the functionfwill be given according to physical conditions.
On the surfacer=Rof the ball,p=0 anduandvare given in terms of the Schwarzschild
solution:

(7.1.33) p(R) = 0 , u(R) =−v(R) =ln

(

1 −

2 Gm

Rc^2

)

.