A survey rocket of proper length burns in deep space

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 survey rocket of proper length

250\ \mathrm{m}

burns in deep space, where no gravitational field acts, with constant proper acceleration

a = 15.0\ \mathrm{m\,s^{-2}}

along the

+x

axis of the launch frame

S

. It starts from rest at

x = 0

t = \tau = 0

, with

\tau

the proper time of the rocket's tail clock; the nose is the forward end. Take

c = 2.998\times 10^{8}\ \mathrm{m\,s^{-1}}

and

1\ \mathrm{yr} = 3.156\times 10^{7}\ \mathrm{s}

. ((a)) Working to first order in

ah/c^2

, find the fraction by which the nose clock runs fast compared with the tail clock, and the time by which it gets ahead during one year of tail-clock time. ((b)) When the tail clock reads

\tau = 0.50\ \mathrm{yr}

, find the rocket's speed in

S

(as a multiple of

c

) and the elapsed time

t

S

, in years. ((c)) How far behind the launch point, in metres, is the rocket's Rindler horizon?

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