General Relativity Formulary

General Relativity Formulary

1) Differential geometry

Metric connection (Christoffel symbols) Γ^α_βγ ≡ 1

2g^αδ(∂_βg_γδ+∂γg_βδ−∂_δg_βγ). Covariant derivative

V^α;β =V^α,β+ Γ^α_βγV^γ V_α;β =V_α,β−Γ^γ_αβV_γ. Riemann curvature tensor

R^α_βγδ≡∂_γΓ^α_βδ−∂_δΓ^α_βγ+ Γ^λ_βδΓ^α_λγ−Γ^λ_βγΓ^α_δλ , or, with the first index lowered

R_αβγδ =g_αλR^λ_βγδ

= 1

2 ∂_βγ² g_αδ+∂_αδ² g_βγ−∂_αγ² g_βδ−∂_βδ² g_αγ +gµν Γ^µ_αδΓ^ν_βγ −Γ^µ_αγΓ^ν_βδ

. Ricci tensor and Ricci scalar

R_αβ ≡R^γ_αγβ , R≡g^αβR_αβ . Bianchi identities

R_αβγδ;λ+R_αβλγ;δ+R_αβδλ;γ = 0. 2) General Relativity

Geodesic equations d²x^α

dp² + Γ^α_βγdx^β dp

dx^γ dp = 0. Einstein equations

G_µν = 8πG T_µν ,

where G_µν is the Einstein tensor defined by Gµν ≡Rµν−1

2R gµν ,

and Tµν is the energy-momentum tensor given by T_µν≡ 2

√−g

δS_matter δg^µν . Variation of the determinant of the metric

δg=g g^µνδg_µν =−g g_µνδg^µν. 3) Energy-momentum tensor of point-particle

T^µν = m

√−g Z

dτ dz^µ dτ

dz^ν

dτ δ⁴(x−z(τ)).

4) Schwarzschild solution ds²=

1−2GM r

dt²− 1

1−^2GM_r dr²−r² dθ²+ sin²θ dφ² . Geodesic equations for the Schwarzschild geometry

¨t+B⁰ Bt˙r˙= 0

¨ r+1

2 B⁰

At˙²+1 2

A⁰ Ar˙²− r

A

θ˙²−rsin²θ A

φ˙²= 0 θ¨+2

rr˙θ˙−sinθcosθφ˙²= 0 φ¨+2

rr˙φ˙+ 2 cotθθ˙φ˙= 0, where

B(r) = 1−2GM

r , A(r) =B⁻¹(r).

Motion in the Schwarzschild metric A(r)

dr dφ

2

J² r⁴ +J²

r² − 1

B(r) =−E dτ²=EB²(r)dt², r²dφ

dt =J B(r) Massive particle

1 2

dr dτ

2

+ 1 2

1−2GM

r 1 +J²/E r²

= 1 2E Massless particle (dλ=B(r)dt)

1 2

dr dλ

2

+1 2

1−2GM r

J² r² = 1

2 5) Linearized theory

g_µν =η_µν+h_µν R⁽¹⁾_µν = 1

2

−hµν+∂_λ∂µh^λν+∂_λ∂νh^λµ−∂µ∂νh

R⁽¹⁾=−h+∂µ∂νh^µν

Einstein equations for physical amplitudes h₁₂= 8πG T₁₂

h= 4πG(T₂₂−T₁₁) 6) Numerical values

G= 6.67·10⁻¹¹m³kg⁻¹s⁻² c'3·10⁸m/s

M'2·10³⁰kg, r'7·10⁵km M⊕'6·10²⁴kg, r⊕'6000 km MN S '2M, rN S '12 km