A spherical glass marble has radius

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Curvature

Problem

A spherical glass marble has radius a=2.00 cma = 2.00\ \text{cm}, with surface metric ds2=a2(dθ2+sin⁡2θ dϕ2)\mathrm{d}s^2 = a^2(\mathrm{d}\theta^2 + \sin^2\theta\, \mathrm{d}\phi^2). Its Riemann tensor has Rθϕθϕ=sin⁡2θR^\theta{}_{\phi\theta\phi} = \sin^2\theta. ((a)) Use the symmetries of the Riemann tensor to find RϕθϕθR^\phi{}_{\theta\phi\theta}, then contract to find RθθR_{\theta\theta} and RϕϕR_{\phi\phi} at θ=45∘\theta = 45^\circ. ((b)) Find the scalar curvature RR of the marble's surface, in m−2\text{m}^{-2}.

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