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=1.5×107a = 1.5\times10^7 m carries the metric ds2=a2(dθ2+sin⁡2θ dϕ2)ds^2 = a^2(d\theta^2 + \sin^2\theta\, d\phi^2), with non-zero Christoffel symbols Γθϕϕ=−sin⁡θcos⁡θ\Gamma^\theta{}_{\phi\phi} = -\sin\theta\cos\theta and Γϕθϕ=Γϕϕθ=cot⁡θ\Gamma^\phi{}_{\theta\phi} = \Gamma^\phi{}_{\phi\theta} = \cot\theta (all others zero). ((a)) Compute RθϕθϕR^\theta{}_{\phi\theta\phi} from the commutator definition, lower the index and evaluate RθϕθϕR_{\theta\phi\theta\phi} at θ=60∘\theta = 60^\circ. ((b)) Read off the Gaussian curvature KK. ((c)) A Rindler wedge (geometric units, c=1c=1) has Γξηη=ap(1+apξ)\Gamma^\xi{}_{\eta\eta} = a_{\rm p}(1+a_{\rm p}\xi) and Γηηξ=Γηξη=ap/(1+apξ)\Gamma^\eta{}_{\eta\xi} = \Gamma^\eta{}_{\xi\eta} = a_{\rm p}/(1+a_{\rm p}\xi) for a proper acceleration of 50 m s−250\ {\rm m\,s^{-2}}, that is ap=5.56×10−16 m−1a_{\rm p} = 5.56\times10^{-16}\ {\rm m^{-1}} in these units. Compute RξηξηR^\xi{}_{\eta\xi\eta} and state what it proves.

Browse the problem library