Spacetime and Geometry · Problem 8

Gravitation — Sean Carroll

Physics — General Relativity · three-sphere-curvature

Problem

The metric for the three-sphere in coordinates xμ=(ψ,θ,ϕ)x^\mu = (\psi, \theta, \phi) can be written ds2=dψ2+sin⁡2ψ(dθ2+sin⁡2θ dϕ2).(3.220)ds^2 = d\psi^2 + \sin^2\psi(d\theta^2 + \sin^2\theta\, d\phi^2). \tag{3.220} (a) Calculate the Christoffel connection coefficients. Use whatever method you like, but it is good practice to get the connection coefficients by varying the integral (3.49). (b) Calculate the Riemann tensor, Ricci tensor, and Ricci scalar. (c) Show that (3.191) is obeyed by this metric, confirming that the three-sphere is a maximally symmetric space (as you would expect).

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