A rocket accelerates so that an observer riding at fixed spatial coordinate measures…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · Differential geometry of curved spacetime

The Geometry of Gravity, Ch The covariant derivative, parallel transport and geodesics

Problem

A rocket accelerates so that an observer riding at fixed spatial coordinate ξ=0\xi = 0 measures proper acceleration ap=2g⊕=19.6 m s−2a_{\rm p} = 2g_\oplus = 19.6\ \mathrm{m\,s^{-2}}, and the rocket's coordinates (x0,ξ)(x^0,\xi), with x0=ctx^0 = ct, are related to an inertial frame exactly as in the Rindler construction, giving the metric ds2=−(1+apξc2)2(dx0)2+dξ2.ds^2 = -\left(1+\frac{a_{\rm p}\xi}{c^2}\right)^2 (dx^0)^2 + d\xi^2. ((a)) Using the Christoffel formula, compute Γξ00\Gamma^\xi{}_{00} and Γ00ξ\Gamma^0{}_{0\xi} at ξ=10 m\xi = 10\ \mathrm{m}. ((b)) Using the slow-motion reduction of the geodesic equation (as used for Newtonian gravity), find the coordinate acceleration d2ξ/dt2d^2\xi/dt^2 of a particle released momentarily at rest at ξ=10 m\xi = 10\ \mathrm{m}, and compare it with apa_{\rm p}. ((c)) Using the gravitational redshift result for a static field (established from time dilation between two observers held at fixed height in a uniform field), find the fractional frequency shift Δν/ν\Delta\nu/\nu for a signal sent from ξ=10 m\xi = 10\ \mathrm{m} down to ξ=0\xi = 0.

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