426 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
the nonzero components of the Levi-civita connections are
Γ^000 =
1
2 cut, Γ^011 =1
2 cev−u(kt+vt), Γ^010 =1
2
ur,Γ^022 =
r^2
2 ce−ukt, Γ^033 =r^2
2 ce−uktsin^2 θ, Γ^100 =1
2
eu−vurΓ^111 =
1
2
vr, Γ^110 =1
2 c
(kt+vt), Γ^122 =−re−v,Γ^133 =−re−vsin^2 θ, Γ^221 =1
r, Γ^233 =sinθcosθ,Γ^331 =1
r, Γ^332 =
cosθ
sinθ,
and the nonzero components of the Ricci curvature tensor read
R 00 =
1
2 c^2[
3 ktt+3
2
kt^2 +vtt+1
2
vt^2 +ktvt−ut(kt+vt)]
−
1
2
eu−k−v[
urr+1
2
u^2 r−1
2
urvr+2
rur]
,
R 11 =−
ek+v−u
2 c^2[
ktt+3
2
kt^2 +vtt+v^2 t+ 3 ktvt−1
2
ut(kt+vt)]
+
1
2
[
urr+1
2
u^2 r−1
2
urvr−2
r
vr]
,
R 22 =−
r^2 ek−u
2 c^2[
ktt+3
2
kt^2 +1
2
kt(vt−ut)]
−e−v[
ev+r
2
(kr+vr−ur)− 1]
,
R 33 =R 22 sin^2 θ,R 10 =−1
cr[
( 1 +
r
2ur)kt+vr]
.
The energy-momentum tensor is in the form
Tμ ν=
ρ g 00 g 11 Prc 0 0
g 00 g 11 Prc g 11 p 0 0
0 0 g 22 p 0
0 0 0 g 33 p
,
whereρis the energy density,Pris the radial component of the momentum density. Then
direct computations imply that
T=gμ νTμ ν=−ρ+ 3 p, T 00 −1
2
g 00 T=1
2
(ρ+ 3 p),T 11 −1
2
g 11 T=1
2
ek+v(ρ−p), T 22 −1
2
g 22 T=1
2
ekr^2 (ρ−p),T 33 −1
2
g 33 T= (T 22 −1
2
g 33 T)sin^2 θ, T 10 −1
2
g 10 T=g 00 g 11 Prc.