PHYSICS 131 hw-8-soln

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

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

View Full Document Right Arrow Icon
HOMEWORK SET 8, PHYS. 131 Instructor: Anthony Zee Email: zee@kitp.ucsb.edu TA: Kevin Moore Email: kmoore@physics.ucsb.edu 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
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
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...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online