Lecture Notes, September 30, 2014
Elisa Todarello
1
Covariant Derivatives
We define the covariant derivative as
∇µ v ν =
∂v ν
+ Γνµσ v σ .
∂xµ
(1)
For this equation to hold, the connection Γ has to transform in a given
way under coordinate transformations. The connection does not transform
as a tensor, but according to the rule:
0
0
Γνµ0 σ0
0
∂xν ∂xµ ∂xσ ν
∂xµ ∂xσ ∂ 2 xν
.
=
Γ
+
∂xν ∂xµ0 ∂xσ0 µσ ∂xµ0 ∂xσ0 ∂xµ ∂xσ
(2)
Notice that no reference to a metric tensor is needed to derive this transformation rule. An object Γ that satisfies (2) is a connection and the covariant
derivative ∇µ has the properties a derivative should have:
1. Linearity
∇(S + T ) = ∇S + ∇T
2. Leibniz rule
∇(S ⊗ T ) = ∇S ⊗ T + S ⊗ ∇T .
Connections exist, and there may be more than one. For instance, if Γνµσ
is a connection, then also Γνσµ is a connection, as it can be seen from the
symmetry of the transformation equation (1). The difference Γνµσ − Γνσµ is a
ν
tensor Tµσ
, called the torsion tensor.
We need to add more requirements in order to give a complete definition
of connection Γ.
3. Compatibility with ∇µ φ = ∂µ φ
To check this, we need to create a scalar as φ = v µ ωµ
4. ∇µ commutes with index contraction
Then we can write
?
∇µ φ = (∇µ v ν )ων + v ν (∇µ ων ) = (∂µ v ν )ων + v ν (∂µ ων ) .
If this equation is true for any v ν , then we can say
∇µ ων = ∂µ ω ν − Γσµν ωσ .
Assuming requirement (3) is satisfied, we can extend the result to a
generic tensor
∇σ T µ1 µ2 ...µkν1 ν2 ...νl
= ∂σ T µ1 µ2 ...µkν1 ν2 ...νl
+ Γµσλ1 T λµ2 ...µkν1 ν2 ...νl + Γµσλ2 T µ1 λ...µkν1 ν2 ...νl + . . .
− Γλσν1 T µ1 µ2 ...µkλν2 ...νl − Γλσν2 T µ1 µ2 ...µkν1 λ...νl − . . . .
If there are two connections, Γνµσ and Γ̃νµσ , then
ν
Sµσ
= Γνµσ − Γ̃νµσ
is a tensor.
Adding two more conditions, we obtain the full definition of the metric compatible connection (which is unique). This connection is also called
Christoffel symbol, or Riemann connection.
5. No torsion
ν
Tµσ
=0
6. Relation with metric on a manifold (M, g)
∇µ g σν = 0
Hence,
∇λ µνρσ = 0
2
Parallel transport
In flat space, if we want to compare two vectors (take the difference of the
two), we can simply use the head-to-tail rule. This cannot be done in a
curved space. We need to introduce the concept of parallel transport along
a curve.
We want to find some operator such that:
D µ1 µ2 ...µk
T
ν1 ν2 ...νl = 0 .
Dλ
Define
D
dxµ
≡
∇µ .
Dλ
dλ
In fact, the equation for a geodesics is
ν
dxµ
∂x
∇µ
=0 .
dλ
∂λ
For a vector v µ
D µ
v
Dλ
The factor
dxν
dλ
=
=
=
dxν
∇ν v µ
dλν
dx ∂v µ
+ Γµνσ vσ
dλ ∂xν
ν
dv ν
+ Γµνσ dx
vσ
dλ
dλ
.
depends on the curve along which we are transporting.
We can write the above relation as a matrix equation
(v µ )0 + M µσ v σ = 0 .
This equation stresses the fact that (v µ )0 depends an all components of v µ .