Spacetime and Geometry · Problem 16

Gravitation — Sean Carroll

Physics — General Relativity · three-sphere-orthonormal-frame

Problem

Consider again the metric on a three-sphere,

ds^2 = d\psi^2 + \sin^2\psi(d\theta^2 + \sin^2\theta\, d\phi^2). \tag{3.226}

In this problem we make use of noncoordinate bases, discussed in Appendix J. In an orthonormal frame of one-forms

\hat{\theta}^{(a)}

the metric would become

ds^2 = \hat{\theta}^{(1)} \otimes \hat{\theta}^{(1)} + \hat{\theta}^{(2)} \otimes \hat{\theta}^{(2)} + \hat{\theta}^{(3)} \otimes \hat{\theta}^{(3)}. \tag{3.227}

(a) Find such an orthonormal frame of one-forms, such that the matrix

e^\mu_a

is diagonal. Don't worry about covering the entire manifold. (b) Compute the components of the spin connection by solving

de^a + \omega^a{}_b \wedge e^b = 0

. (c) Compute the components of the Riemann tensor

R^\rho{}_{\sigma\mu\nu}

in the coordinate basis adapted to

x^\mu

by computing the components of the curvature two-form

R^a{}_{b\mu\nu}

and then converting.