T1: 6. Curvature

Question 1

How do we measure curvature?

Answer 1

The extent to which the commutator of the covariant derivative on a tensor is non-zero.

Question 2

Define the Riemann tensor (vector)

Answer 2

R_μνρ^λ in the equation

[∇_μ,∇_ν ] V^λ = R_μνρ^λ V^ρ

Question 3

Define the Riemann tensor (covector)

Answer 3

R_μνλ^ρ in the equation

[∇_μ,∇_ν ] ω_λ = -R_μνλ^ρ ω_ρ

Question 4

Define the Riemann tensor (words)

Answer 4

The extent to which a metric generalises the Minkowski metric.

Question 5

What is the Riemann tensor in flat space?

Answer 5

Zero; all Christoffels are zero in flat space.

Question 6

What does it mean for a vector field W^μ to be ‘covariantly constant’? (For all points p)

Answer 6

Its covariant derivative is zero: ∇_ν W^μ=0

Question 7

What does covariant constance mean for the Riemann tensor?

Answer 7

It must be zero, such that the constance holds for all p in the field.

Question 8

What does it mean for a vector field W^μ to be ‘covariantly constant along’ the curve γ?

Answer 8

V^λ ∇_λ W^μ=0

Question 9

How does covariant constance along a curve differ from at a point?

Answer 9

Along a curve we only require W to be constant in the direction of the curve, rather than in all directions.

Question 10

Define the holonomy

Answer 10

The way vectors change under parallel transport around a closed loop that characterises the curvature of space.

Question 11

Define geodesic deviation

Answer 11

The failure of initially parallel geodesics to remain parallel.

Question 12

State the geodesic deviation equation

Answer 12

A^ν = U^λ V^ρ V^μ R_ρμλ^ν

Question 13

What does the geodesic deviation equation tell us?

Answer 13

The only way for A^v = 0 is if the Riemann tensor is zero. This means parallel lines only remain parallel in flat space.

Question 14

What are the conditions for Riemann normal coordinates?

Answer 14

The metric equals the Minkowski metric at a point and the coordinate derivative of the metric is zero.

Question 15

What are the three main symmetries of the Riemann tensor (w/lower indices)

Answer 15

1. exchange of indices 1 and 2 give a minus sign
2. exchange of the first and second pairs of indices
3. Sum of rotation of last three indices equals zero

Question 16

State the Bianchi identity

Answer 16

∇([λ) R(ρσ]μν) = 0

Consider cov derivative on Riemann tensor (lower indices). Sum of rotation of first three indices =0.

Question 17

Define the Ricci tensor

Answer 17

R_μν = R^λ_μλν = R_vμ

Question 18

Define the Ricci scalar

Answer 18

R = g_μνR^μν

Question 19

Write down the Riemann tensor

Answer 19

Check notes! :)

Question 20

How do we get our mini-Riemann in RNC?

Answer 20

The christoffels at a point vanish, while their derivatives do not, since these give a second derivative of the metric.

Question 21

How many independent components does the Riemann tensor have?

Answer 21

1/12 n^2(n^2 -1)

Question 22

How can we check the Bianchi identity in RNC?

Answer 22

All cov derivs are partials since christoffels vanish. These all commute on the derivative version of Riemann tensor

Question 23

State the contracted Bianchi identity

Answer 23

∇_µ R^µ_ρ = 1/2∇_ρ R

Question 24

Define the Einstein tensor

Answer 24

G_