404 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
In (7.1.43) we haveg 11 =α(r), g 22 =r^2 , g 33 =r^2 sin^2 θ, gij=0 fori 6 =j.By (7.1.5) we can get the Levi-Civita connection as
(7.1.45)
Γ^221 =Γ^212 =
1
r
, Γ^233 =−sinθcosθ, Γ^331 =Γ^313 =1
r,
Γ^332 =Γ^332 =
cosθ
sinθ, Γ^122 =−
r
α, Γ^133 =−
r
αsin^2 θ,Γ^111 =
1
2 αdα
dr
, Γkij=0 for others.We deduce from (7.1.45) the explicit form of the Ricci curvature tensor (7.1.4):(7.1.46)
R 11 =−
1
αrdα
dr, R 22 =
1
α−
r
2 α^2dα
dr− 1 ,
R 33 =R 22 sin^2 θ, Rij= 0 ∀i 6 =j.Based on (7.1.45) and (7.1.46) we can obtain the expressions of the differential operators
(7.1.2)-(7.1.8) as follows:
1) The Laplace-Beltrami operator∆uk= (∆ur,∆uθ,∆uφ):∆uθ=1
r^2[
1
sinθ∂
∂ θ(
sinθ
∂uθ
∂ θ)
+
1
sin^2 θ∂^2 uθ
∂ φ^2(7.1.47)
+
1
α∂
∂r(
r^2
∂uθ
∂r)
+
2
r∂ur
∂ θ−
2cosθ
sinθ∂uφ
∂ φ−
1
sin^2 θuθ]
+
1
αr^2[
2
∂
∂r(ruθ)−α′
2 α∂
∂r(r^2 uθ)]
,
∆uφ=1
r^2[
1
sinθ∂
∂ θ(
sinθ∂uφ
∂ θ)
+
1
sin^2 θ∂^2 uφ
∂ φ^2+
1
α∂
∂r(
r^2∂uφ
∂r)
(7.1.48)
+
2cosθ
sinθ∂uφ
∂ θ− 2 uφ+2cosθ
sin^3 θ∂uθ
∂ φ+
2
rsin^2 θ∂ur
∂ φ]
+
1
αr^2[
2
∂
∂r(ruφ)−α′
2 α∂
∂r(r^2 uφ)]
,
∆ur=1
r^2[
1
sinθ∂
∂ θ(
sinθ∂ur
∂ θ)
+
1
sin^2 θ∂^2 ur
∂ φ^2+
1
α∂
∂r(
r^2∂ur
∂r)]
(7.1.49)
−
2
αr^2[
ur+rcosθ
sinθ
uθ+r∂uθ
∂ θ
+r∂uφ
∂ φ−
r^2
2∂
∂r(
α′
2 α
ur