A rocket of proper length is rigid: every part of it keeps the same proper distance…

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 = 400\ \mathrm{m}

is rigid: every part of it keeps the same proper distance from every other part, so all parts share one and the same horizon. The nose starts from rest at

x = 0

in the inertial frame

S

t = 0

and has constant proper acceleration

a_{\rm N} = 9.80\ \mathrm{m\,s^{-2}}

along

+x

; the tail starts at

x = -h

. Thrust points from tail to nose, and there is no gravitating matter anywhere. Take

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

. ((a)) Starting from

ct = (c^2/a)\sinh(a\tau/c)

and

x = (c^2/a)[\cosh(a\tau/c) - 1]

, show that the worldline obeys

(x + c^2/a)^2 - c^2t^2 = c^4/a^2

and that its asymptote is

x = ct - c^2/a

, so the horizon of an observer with proper acceleration

a

lies a proper distance

c^2/a

behind his starting point. ((b)) Hence show that the tail's proper acceleration is

a_{\rm T} = a_{\rm N}/(1 - a_{\rm N}h/c^2)

, and expand this to first order in

a_{\rm N}h/c^2

. Compare the ratio

a_{\rm T}/a_{\rm N}

with the nose-to-tail clock rate ratio of this chapter. ((c)) Evaluate the fractional rate difference between the two clocks, and the difference

a_{\rm T} - a_{\rm N}

\mathrm{m\,s^{-2}}

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