# Lecture Notes, September 30, 2014 1 Covariant Derivatives Elisa Todarello

```Lecture Notes, September 30, 2014
Elisa Todarello
1
Covariant Derivatives
We define the covariant derivative as
∇&micro; v ν =
∂v ν
+ Γν&micro;σ v σ .
∂x&micro;
(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
Γν&micro;0 σ0
0
∂xν ∂x&micro; ∂xσ ν
∂x&micro; ∂xσ ∂ 2 xν
.
=
Γ
+
∂xν ∂x&micro;0 ∂xσ0 &micro;σ ∂x&micro;0 ∂xσ0 ∂x&micro; ∂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 ∇&micro; 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 Γν&micro;σ
is a connection, then also Γνσ&micro; is a connection, as it can be seen from the
symmetry of the transformation equation (1). The difference Γν&micro;σ − Γνσ&micro; is a
ν
tensor T&micro;σ
, called the torsion tensor.
We need to add more requirements in order to give a complete definition
of connection Γ.
3. Compatibility with ∇&micro; φ = ∂&micro; φ
To check this, we need to create a scalar as φ = v &micro; ω&micro;
4. ∇&micro; commutes with index contraction
Then we can write
?
∇&micro; φ = (∇&micro; v ν )ων + v ν (∇&micro; ων ) = (∂&micro; v ν )ων + v ν (∂&micro; ων ) .
If this equation is true for any v ν , then we can say
∇&micro; ων = ∂&micro; ω ν − Γσ&micro;ν ωσ .
Assuming requirement (3) is satisfied, we can extend the result to a
generic tensor
∇σ T &micro;1 &micro;2 ...&micro;kν1 ν2 ...νl
= ∂σ T &micro;1 &micro;2 ...&micro;kν1 ν2 ...νl
+ Γ&micro;σλ1 T λ&micro;2 ...&micro;kν1 ν2 ...νl + Γ&micro;σλ2 T &micro;1 λ...&micro;kν1 ν2 ...νl + . . .
− Γλσν1 T &micro;1 &micro;2 ...&micro;kλν2 ...νl − Γλσν2 T &micro;1 &micro;2 ...&micro;kν1 λ...νl − . . . .
If there are two connections, Γν&micro;σ and Γ̃ν&micro;σ , then
ν
S&micro;σ
= Γν&micro;σ − Γ̃ν&micro;σ
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
T&micro;σ
=0
6. Relation with metric on a manifold (M, g)
∇&micro; g σν = 0
Hence,
∇λ &micro;νρσ = 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 &micro;1 &micro;2 ...&micro;k
T
ν1 ν2 ...νl = 0 .
Dλ
Define
D
dx&micro;
≡
∇&micro; .
Dλ
dλ
In fact, the equation for a geodesics is
ν
dx&micro;
∂x
∇&micro;
=0 .
dλ
∂λ
For a vector v &micro;
D &micro;
v
Dλ
The factor
dxν
dλ
=
=
=
dxν
∇ν v &micro;
dλν
dx ∂v &micro;
+ Γ&micro;νσ vσ
dλ ∂xν
ν
dv ν
+ Γ&micro;νσ dx
vσ
dλ
dλ
.
depends on the curve along which we are transporting.
We can write the above relation as a matrix equation
(v &micro; )0 + M &micro;σ v σ = 0 .
This equation stresses the fact that (v &micro; )0 depends an all components of v &micro; .
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