PHYSICS 131 hw-4-soln

PHYSICS 131 hw-4-soln - HOMEWORK SET 4, PHYS. 131...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
HOMEWORK SET 4, PHYS. 131 Instructor: Anthony Zee Email: [email protected] TA: Kevin Moore Email: [email protected] 7.2. The following line element corresponds to Fat spacetime: dS 2 = - dt 2 +2 dxdt + dy 2 + dz 2 ±ind a coordinate transformation that puts the line element in the usual Fat space form. Here we notice that only the x and t coordinates are mixed, so we can take our coordinate transformation to only involve these coordinates. Aside: You may recognize the form of this problem from classical mechanics. What we’re doing is similar to doing a coordinate transformation to uncouple a system of coupled harmonic oscillators So our general coordinate transformations look like: t ± = t ± ( t, x ) x ± = x ± ( t, x ) y ± = yz ± = z Now, we want the metric written in the new coordinates to look like: dS 2 = - dt ± 2 + dx ± 2 + dy ± 2 + dz ± 2 So we use the general tensor transformation formula to get g αβ = ∂x α ± α β ± β g α ± β ± We can write down 3 relevant independent equations from this: g xx = 0 = - ± ∂t ± ² 2 + ± ± ² 2 g xt = 1 = - ± ± ± ² + ± ± ± ² g tt = - 1= - ± ± ² 2 + ± ± ² 2 These equations can basically be solved by inspection, giving the solution: t ± = t - x, x ± = x 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
8.2. In usual spherical coordinates the metric on a two-dimensional sphere is dS 2 = a 2 ( 2 + sin 2 θdφ 2 ) where a is constant. (a) Calculate the Christofel symbols “by hand”. The metric is given by g AB = a 2 ± 10 0 sin 2 θ ² The only element oF g AB that has non-vanishing derivatives is g φφ , and it de- pends only on θ . The only nonvanishing Christofel symbols will be given by g φA Γ A φθ = g φφ Γ φ θφ = 1 2 ± ∂g φφ ∂θ + φθ ∂φ - θφ ² = a 2 sin θ cos θ g θB Γ B φφ = g θθ Γ θ φφ = 1 2 ± θφ + θφ - φφ ² = - a 2 sin θ cos θ
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/15/2008 for the course PHYS 131 taught by Professor Kevin during the Spring '08 term at UCSB.

Page1 / 6

PHYSICS 131 hw-4-soln - HOMEWORK SET 4, PHYS. 131...

This preview shows document pages 1 - 3. Sign up to view the full document.

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