Lecture Notes, October 1, 2014
Elisa Todarello
1
Parametrization of geodesics
The equation for a geodesic
dxµ
∇µ
dλ
∂xν
∂λ
=0
can be derived from an action principle. Two possible actions are:
r
r
Z
Z
∂xµ ∂xν
∂xµ ∂xν
I1 = dλ −gµν
= dτ −gµν
,
∂λ ∂λ
∂τ ∂τ
Z
1
∂xµ ∂xν
I2 =
dλ gµν
.
2
∂λ ∂λ
I2 is not invariant under a change of parameter.
Z
1
1
∂xµ ∂xν
δI1 =
dλ q
δ −gµν
=
2
∂λ
∂λ
∂xµ ∂xν
−gµν ∂λ ∂λ


Z
µ
ν
dx
1
dx

= dλ δxσ gσν
∇µ  q
dλ
∂xµ ∂xν dλ
−g
µν ∂λ ∂λ
Z
=
dxµ
dτ δx gσν
∇µ
dτ
σ
dxν
dτ
,
q
µ ∂xν
since −gµν ∂x
dλ = dτ .
∂λ ∂λ
So, if we use the action I1 , the geodesic turns out to be parametrised by
τ , while in I2 , λ can be any affine parameter. The two parametrisations are
the same is we set λ = τ in I2 .
If we want to change the parameter in I2
dλ =
dλ
dα ,
dα
where α is not affinely parametrised, we get
dxµ dxν
dxν
∇
= f (α)
dα dα
dα
where
f (α) =
dα
dλ
−2
d2 xα
.
dλ2
So, in general, “force terms” like f (α) can be removed by reparametrisation.
2
Let
Parallel transport
dxµ
dλ
= kµ ,
λ=0
where λ = 0 corresponds to the point p. Then
xµ
λ=1
≡ expp (k)
defines the exponential map, where xµ solves the geodesic equation.
Then
expp (αk) = xν λ=α .
All curves, especially geodesics, either continue without stopping or hit a
singularity, e.g. black holes, white holes (black holes in the past), big bang
singularity.
3
Riemann tensor
Consider parallel transport of a vector along a loop. Even if the loop is
infinitesimally small, we cannot drop the second derivatives of g, i.e. consider
the loop flat, because they determine the curvature.
Consider a “square” loop of sides Aµ , B ν , then the change in a vector v ρ
when transported along the loop must have the following form
δv ρ = Rρσµν Aµ B ν v σ ,
where Rρσµν is a tensor.
A way to derive an expression for Rρσµν is to calculate
[∇ν , ∇ν ]v ρ ,
which means to see what happens when we transport a vector along direction
µ and then ν and viceversa.
[∇ν , ∇ν ]v ρ = ∇µ ∇ν v ρ − ∇ν ∇µ v ρ = ∂µ (∇ν v ρ ) + Γρµσ ∇ν v σ − Γσµν ∇σ v σ ,
[∇ν , ∇ν ]v ρ
= ∇µ ∇ν v ρ − ∇ν ∇µ v ρ
= ∂µ (∇ν v ρ ) + Γρµσ ∇ν v σ − Γσµν ∇σ v σ − (µ ↔ ν)
= ∂µ ∂ν v ρ + ∂µ (Γρνσ )v σ + Γρνσ ∂µ v σ + Γρµσ (∂v σ + Γσνλ v λ )
−Γσµν ∂σ v ρ − Γσµν Γρσλ v λ − (µ ↔ ν)
λ
= (∂µ Γρνσ − ∂ν Γρνσ + Γρµλ Γλνσ − Γσνλ Γλµσ )v σ − Tµν
∇λ v ρ .
The Riemann tensor is
Rρσµν = ∂µ Γρνσ − ∂ν Γρνσ + Γρµλ Γλνσ − Γσνλ Γλµσ
Properties of Rρσµν
• Anti-symmetry in µν
Rρσ(µν) = 0
• It is a tensor
• depends on Γ, not directly on g (we can define it for any connection).
The torsion tensor would not appear if we were using the metric connection.
Torsion free definition
R(x, y)z ≡ ∇x ∇y z − ∇y ∇x z − ∇[x,y] z
We can adopt this as a definition for any Γ. Torsion
T (x, y) ≡ ∇x y − ∇y x − [x, y] .
T is zero for any metric compatible connection.