T1: 6. Curvature Flashcards
How do we measure curvature?
The extent to which the commutator of the covariant derivative on a tensor is non-zero.
Define the Riemann tensor (vector)
R_μνρ^λ in the equation
[∇_μ,∇_ν ] V^λ = R_μνρ^λ V^ρ
Define the Riemann tensor (covector)
R_μνλ^ρ in the equation
[∇_μ,∇_ν ] ω_λ = -R_μνλ^ρ ω_ρ
Define the Riemann tensor (words)
The extent to which a metric generalises the Minkowski metric.
What is the Riemann tensor in flat space?
Zero; all Christoffels are zero in flat space.
What does it mean for a vector field W^μ to be ‘covariantly constant’? (For all points p)
Its covariant derivative is zero: ∇_ν W^μ=0
What does covariant constance mean for the Riemann tensor?
It must be zero, such that the constance holds for all p in the field.
What does it mean for a vector field W^μ to be ‘covariantly constant along’ the curve γ?
V^λ ∇_λ W^μ=0
How does covariant constance along a curve differ from at a point?
Along a curve we only require W to be constant in the direction of the curve, rather than in all directions.
Define the holonomy
The way vectors change under parallel transport around a closed loop that characterises the curvature of space.
Define geodesic deviation
The failure of initially parallel geodesics to remain parallel.
State the geodesic deviation equation
A^ν = U^λ V^ρ V^μ R_ρμλ^ν
What does the geodesic deviation equation tell us?
The only way for A^v = 0 is if the Riemann tensor is zero. This means parallel lines only remain parallel in flat space.
What are the conditions for Riemann normal coordinates?
The metric equals the Minkowski metric at a point and the coordinate derivative of the metric is zero.
What are the three main symmetries of the Riemann tensor (w/lower indices)
- exchange of indices 1 and 2 give a minus sign
- exchange of the first and second pairs of indices
- Sum of rotation of last three indices equals zero
State the Bianchi identity
∇([λ) R(ρσ]μν) = 0
Consider cov derivative on Riemann tensor (lower indices). Sum of rotation of first three indices =0.
Define the Ricci tensor
R_μν = R^λ_μλν = R_vμ
Define the Ricci scalar
R = g_μνR^μν
Write down the Riemann tensor
Check notes! :)
How do we get our mini-Riemann in RNC?
The christoffels at a point vanish, while their derivatives do not, since these give a second derivative of the metric.
How many independent components does the Riemann tensor have?
1/12 n^2(n^2 -1)
How can we check the Bianchi identity in RNC?
All cov derivs are partials since christoffels vanish. These all commute on the derivative version of Riemann tensor
State the contracted Bianchi identity
∇_µ R^µ_ρ = 1/2∇_ρ R
Define the Einstein tensor
G_
