A rocket of proper length burns from rest with constant proper acceleration along the…

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

Relativity and Gravitation, Ch Accelerated observers, hyperbolic motion and the equivalence principle

Problem

A rocket of proper length

h = 240\ \mathrm{m}

burns from rest with constant proper acceleration

a = 12.0\ \mathrm{m\,s^{-2}}

along the

+x

axis of the inertial launch frame

S

, starting at

t = 0

. A laser fixed at the tail emits light of frequency

\nu_{\rm em} = 4.740\times10^{14}\ \mathrm{Hz}

, measured by a clock at the tail, aimed forwards at a detector at the nose. Take

c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}

G = 6.674\times10^{-11}\ \mathrm{N\,m^{2}\,kg^{-2}}

, and for the Earth

M = 5.972\times10^{24}\ \mathrm{kg}

, radius

6.37\times10^{6}\ \mathrm{m}

. Work everything to first order in

ah/c^{2}

and say so; ignore all gravitational fields for the rocket parts. ((a)) Find the redshift

z

of the light received at the nose, and the resulting decrease

\nu_{\rm em} - \nu_{\rm rec}

in frequency measured by a clock at the nose, in Hz. ((b)) When the tail clock has recorded a proper time

\tau = 2.00\times10^{7}\ \mathrm{s}

of burn, find the rocket's speed in

S

as a multiple of

c

, and the time elapsed in

S

since launch. ((c)) By the Einstein equivalence principle the crew cannot tell their cabin from a cabin at rest on the Earth's surface by any local experiment. One experiment that does distinguish them is this: release two small marbles at rest, separated by

240\ \mathrm{m}

along the vertical. In the rocket their relative acceleration is zero. Just outside the Earth's surface, what is the magnitude of their relative acceleration, and is it a stretch or a squeeze?

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