T1: 7 Symmetry

Question 1

Define symmetry (or isometry) under coordinate transformation x ̃^μ(x^ρ) for a scalar field ϕ(x^ρ)

Answer 1

ϕ(x^μ = x^μ _0) = ϕ(x ̃^μ = x^μ _0)

The coordinate transformation leaves the value of the field at the value of the coordinate unchanged.

Question 2

What defines a continuous symmetry?

Answer 2

A parameter in the coordinate transform

Question 3

State the infinitesimal coordinate transform

Answer 3

x ̃^μ = x^μ _0 + ϵX^μ

Question 4

State Killing’s equation

Answer 4

∇_μ X_ν + ∇_ν X_μ = 0

Question 5

What special statement can we make given a Killing vector field W_μ and tangent vector field V^μ?

Answer 5

The quantity W_μ V^μ is constant along the geodesic (d/dλ of it =0)

Question 6

To first order in ϵ, what is the condition for a scalar field ϕ to be invariant under symmetry?

Answer 6

X^λ ∂_λ ϕ = 0