PHYSICS 131 hw-4-soln

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

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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 flat spacetime: dS 2 = - dt 2 + 2 dxdt + dy 2 + dz 2 Find a coordinate transformation that puts the line element in the usual flat 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 = y z = 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 x α x β x β g α β We can write down 3 relevant independent equations from this: g xx = 0 = - t x 2 + x x 2 g xt = 1 = - t x t t + x x x t g tt = - 1 = - t t 2 + x t 2 These equations can basically be solved by inspection, giving the solution: t = t - x, x = x 1
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8.2. In usual spherical coordinates the metric on a two-dimensional sphere is dS 2 = a 2 ( d θ 2 + sin 2 θ d φ 2 ) where a is constant. (a) Calculate the Christo ff el symbols “by hand”. The metric is given by g AB = a 2 1 0 0 sin 2 θ The only element of g AB that has non-vanishing derivatives is g φφ , and it de- pends only on θ . The only nonvanishing Christo ff el symbols will be given by g φ A Γ A φθ = g φφ Γ φ θφ = 1 2 g φφ ∂θ + g φθ ∂φ - g θφ ∂θ = a 2 sin θ cos θ g θ B
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