T1: 4. Covariant Derivative

Question 1

In what case is the coordinate derivative of a vector, a tensor?

Answer 1

Under linear coordinate transformations; the non-linear term in the product rule disappears.

Question 2

Define the covariant derivative ∇ (or connection)

Answer 2

A linear operator which takes a vector to a rank [1,1] tensors and acts like a derivative.

Question 3

What does it mean for a covariant derivative to ‘act like a derivative’

Answer 3

Obeys the Leibniz (product) rule for derivatives, with some scalar function f:

∇(fV) = (df)V+f∇V

Question 4

What define a connection to be ‘torision-free’?

Answer 4

The connection coefficients are symmetric in the lower indices.

Question 5

What property does a ‘torsion-free’ connection have?

Answer 5

Its action on a scalar (function) is commutative.
[∇_µ, ∇_ν]f = 0

Question 6

What two properties does the Levi-Civita connection have?

Answer 6

1. It is torsion-free (symmetry in lower indices of connection coeffs/Christoffels)
2. It is ‘metric compatible’.

Question 7

What does it mean for a connection to be ‘metric compatible’?

Answer 7

Its action on the metric vanishes:

∇_λ g_µν = 0

Question 8

For a torsion-free metric, how many equations does the cov deriv of the metric give?

Answer 8

The number of components of the connection coeffs/Christoffels.

Question 9

Does the covariant derivative commute with raising/lowering indices

Answer 9

Yes!

Question 10

State the covariant derivative ∇_µ V^ν (on a vector)

Answer 10

∇_µ V^ν = ∂µV^ν + Γ^ν(µλ) V^λ

Question 11

State the covariant derivative ∇_µ ω_ν (on a covector)

Answer 11

∇_ν ω_µ = ∂v ω_µ - Γ^λ(vµ) ω_λ

Question 12

Generally, how many connection coefficients does an n dimensional space have?

Answer 12

n^3

Question 13

Give an expression for the Christoffel

Answer 13

Γ_μν^σ = 1/2 g^σλ(∂μ g_λν + ∂ν g_μλ - ∂λ g_μν)

Question 14

How do we find an expression for the Christoffels?

Answer 14

Give cov derivative of metric and find difference.