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

R_αβγδ =g_αλR^λ_βγδ

= 1

2 ∂_αγ² gβδ+∂_βδ² gαγ−∂_βγ² gαδ−∂_αδ² gβγ

+gµν Γ^µ_αγΓ^ν_βδ−Γ^µ_αδΓ^ν_βγ . 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,

where p = τ for a massive particle and p = x⁰ for a massless particle.

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

√−g

δSmatter

δg^µν . Variation of the determinant of the metric

δg=g g^µνδgµν =−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) Static isotropic spacetime

ds² =B(r)dt²−A(r)dr²−r² dθ²+ sin²θ dφ² .

The corresponding Geodesic equations

¨t+B⁰ Bt˙r˙= 0

¨ r+1

2 B⁰

At˙²+1 2

A⁰ Ar˙²− r

Aθ˙²−rsin²θ A φ˙²= 0 θ¨+2

rr˙θ˙−sinθcosθφ˙²= 0 φ¨+2

rr˙φ˙+ 2 cotθθ˙φ˙= 0, 4.1) Schwarzschild solution

B(r) = 1−2GM

r , A(r) =B⁻¹(r).

Motion in Schwarzschild solution A(r)

dr dφ

2

J²

E²r⁴ + J²

E²r² − 1

B(r) =−m² E² dτ² = m²B²(r)

E² dt², r²dφ dt = J

EB(r) Radial equation for a massive particle

1 2

dr dτ

2

+1 2

1−2GM

r 1 +J²/m² r²

= E² 2m² Radial equation for a massless particle (dλ=B(r)dt)

1 2

dr dλ

2

+1 2

1−2GM r

J² E²r² = 1

2

5) Numerical values

G= 6.67·10⁻¹¹m³kg⁻¹s⁻² M'2·10³⁰kg, r'7·10⁵km M⊕'6·10²⁴kg, r⊕'6000 km M_{N S} '2M, r_{N S} '12 km