Gravity and Strings

The metric
Appendix F
Connections and curvature components
F.1For some d = 4 metrics
F.1.1 General static, spherically symmetric metrics (I)
ds 2 = gtt(r)dt 2 + grr(r)dr 2 − r 2 d 2 (2)
leads to the Levi-Cività connection components
and the Ricci tensor
Ɣtt r =− 1
2 ∂r gtt/grr, Ɣtr t = 1
2 ∂r gtt/gtt, Ɣrr r = 1
2 ∂r grr/grr,
Ɣrθ θ = 1/r, Ɣrϕ ϕ = 1/r, Ɣθθ r = r/grr,
Ɣθϕ ϕ = cos θ/sin θ, Ɣϕϕ r = sin 2 θƔθθ r , Ɣϕϕ θ =−sin θ cos θ,
√
gttκ
Rtt =−
′
√
−grr
Rθθ =− rg′ tt
2grrgtt
+ g′ √
tt
−grrκ
, Rrr =
rgrr
′
√ −
gtt
g′ tt
,
rgrr
+ rg′ rr
2g 2 rr
− 1 + 1
, Rϕϕ = sin
grr
2 θ Rθθ,
where the prime indicates partial derivatization with respect to r and κ is
The Ricci scalar is
R = 2
κ ′
√
−grrgtt
If we choose the Vierbein basis
κ = 1
g
2
′ tt
√
−grrgtt
(F.1)
(F.2)
(F.3)
. (F.4)
− 2
ln −
rgrr
gtt
′
+
grr
2
r 2
1 + 1
. (F.5)
grr
et 0 = √ gtt, er 1 = √ −grr, eθ 2 = r, eϕ 3 = r sin θ, (F.6)
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