PHYSICS 131 hw-8-soln

# PHYSICS 131 hw-8-soln - HOMEWORK SET 8 PHYS 131 Instructor...

This preview shows pages 1–2. Sign up to view the full content.

HOMEWORK SET 8, PHYS. 131 Instructor: Anthony Zee Email: [email protected] TA: Kevin Moore Email: [email protected] Note: I will be using the convention where primes will be put on indicies rather than tensors. (eg. x μ is x μ ± in a primed coordinate system). This is more conveneient in cases where you want to write diferent parts oF a tensor in diferent coordinate systems - although that won’t be necessary here. 1. IF V μ is a tensor, show V μ is a tensor . We start with the defnition V μ = g μν V ν , and use the transFormation laws For the metric and V μ : V μ = V σ ± ∂x ν ∂x σ ± g μν = g μ ± ν ± ∂x μ ± ∂x μ ∂x ν ± ∂x ν Thus we get V μ = ± g μ ± ν ± ∂x μ ± ∂x μ ∂x ν ± ∂x ν ² ³ V σ ± ∂x ν ∂x σ ± ´ Then using ∂x ν ± ∂x ν ∂x ν ∂x σ ± = δ ν ± σ ± We obtain V μ = ∂x μ ± ∂x μ V μ ± Which is the required tensor transFormation law. 2. Show T ρμ ρσλ (a tensor contracted over ρ ) transForms as a tensor S μ σλ . ±irst we write down how S μ σλ transForms: S μ ± σ ± λ ± = ∂x μ ± ∂x μ ∂x σ ∂x σ ± ∂x λ ∂x λ ± S μ σλ Now how T ρμ ρσλ transForms: T ρ ± μ ± ρ ± σ ± λ ± = ∂x ρ ± ∂x ρ ∂x μ ± ∂x μ ∂x ρ ∂x ρ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

### Page1 / 3

PHYSICS 131 hw-8-soln - HOMEWORK SET 8 PHYS 131 Instructor...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online