The_Cambridge_Handbook_of_Physics_Formulas

3.3 Gravitation
67
General relativity a
Line element ds 2 = g µν dx µ dx ν = −dτ 2 (3.45)
Christoffel
symbols and
covariant
differentiation
Γ α βγ = 1 2 gαδ (g δβ,γ +g δγ,β −g βγ,δ ) (3.46)
φ ;γ = φ ,γ ≡ ∂φ/∂x γ (3.47)
A α ;γ = A α ,γ +Γ α βγA β (3.48)
B α;γ = B α,γ −Γ β αγB β (3.49)
ds invariant interval
dτ proper time interval
g µν metric tensor
dx µ differential of x µ
Γ α βγ Christoffel symbols
,α partial diff. w.r.t. x α
;α covariant diff. w.r.t. x α
φ scalar
A α contravariant vector
B α covariant vector
3
Riemann tensor
R α βγδ =Γ α µγΓ µ βδ −Γα µδΓ µ βγ
+Γ α βδ,γ −Γ α βγ,δ (3.50)
B µ;α;β −B µ;β;α = R γ µαβ B γ (3.51)
R α βγδ
Riemann tensor
R αβγδ = −R αβδγ ; R βαγδ = −R αβγδ (3.52)
R αβγδ +R αδβγ +R αγδβ = 0 (3.53)
Geodesic
equation
Geodesic
deviation
Dv µ
Dλ = 0 (3.54)
where
DA µ
Dλ ≡ dAµ
dλ +Γµ αβ Aα v β (3.55)
v µ
λ
tangent vector
(= dx µ /dλ)
affine parameter (e.g., τ
for material particles)
D 2 ξ µ
Dλ 2 = −Rµ αβγ vα ξ β v γ (3.56) ξ µ geodesic deviation
Ricci tensor R αβ ≡ R σ ασβ = g σδ R δασβ = R βα (3.57) R αβ Ricci tensor
Einstein tensor G µν = R µν − 1 2 gµν R (3.58)
Einstein’s field
equations
G µν =8πT µν (3.59)
Perfect fluid T µν =(p+ρ)u µ u ν +pg µν (3.60)
Schwarzschild
solution
(exterior)
(
ds 2 =− 1− 2M ) (
dt 2 + 1− 2M ) −1
dr 2
r
r
+r 2 (dθ 2 +sin 2 θ dφ 2 ) (3.61)
G µν Einstein tensor
R Ricci scalar (= g µν R µν )
T µν
p
ρ
u ν
stress-energy tensor
pressure (in rest frame)
density (in rest frame)
fluid four-velocity
M spherically symmetric
mass (see page 183)
(r,θ,φ) spherical polar coords.
t time
Kerr solution (outside a spinning black hole)
ds 2 = − ∆−a2 sin 2 θ
ϱ 2
dt 2 −2a 2Mrsin2 θ
ϱ 2 dt dφ
+ (r2 +a 2 ) 2 −a 2 ∆sin 2 θ
ϱ 2 sin 2 θdφ 2 + ϱ2
∆ dr2 +ϱ 2 dθ 2 (3.62)
J angular momentum
(along z)
a ≡ J/M
∆ ≡ r 2 −2Mr+a 2
ϱ 2 ≡ r 2 +a 2 cos 2 θ
a General relativity conventionally uses the (−1,1,1,1) metric signature and “geometrized units” in which G = 1 and
c = 1. Thus, 1kg = 7.425 × 10 −28 m etc. Contravariant indices are written as superscripts and covariant indices as
subscripts. Note also that ds 2 means (ds) 2 etc.