A two-sphere of radius m carries the metric

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Curvature: the Riemann tensor, geodesic deviation and the Einstein tensor

Problem

A two-sphere of radius

a = 3.0\times10^6

m carries the metric

ds^2 = a^2(d\theta^2 + \sin^2\theta\, d\phi^2)

, with non-zero Christoffel symbols

\Gamma^\theta{}_{\phi\phi} = -\sin\theta\cos\theta

and

\Gamma^\phi{}_{\theta\phi} = \Gamma^\phi{}_{\phi\theta} = \cot\theta

(all others zero). ((a)) Use the commutator definition

R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}

to compute

R^\theta{}_{\phi\theta\phi}

, then lower the index to give

R_{\theta\phi\theta\phi}

and evaluate it at

\theta = 30^\circ

. ((b)) Read off the Gaussian curvature

K

. ((c)) A separate accelerated wedge is described (in geometric units,

c=1

) by the Rindler metric with

\Gamma^\xi{}_{\eta\eta} = a_{\rm p}(1+a_{\rm p}\xi)

and

\Gamma^\eta{}_{\eta\xi} = \Gamma^\eta{}_{\xi\eta} = a_{\rm p}/(1+a_{\rm p}\xi)

for a proper acceleration of

9.8\ {\rm m\,s^{-2}}

, which in these units is

a_{\rm p} = 9.8/c^2 = 1.09\times10^{-16}\ {\rm m^{-1}}

(all other components zero). Compute

R^\xi{}_{\eta\xi\eta}

and state what this proves about the wedge.

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