A rocket undergoes hyperbolic motion with constant proper acceleration

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

The Geometry of Gravity, Ch Accelerated observers, Rindler coordinates and the equivalence principle

Problem

A rocket undergoes hyperbolic motion with constant proper acceleration aa, so its worldline is x(τ)=(c2/a)cosh⁡(aτ/c)x(\tau) = (c^2/a)\cosh(a\tau/c), ct(τ)=(c2/a)sinh⁡(aτ/c)ct(\tau) = (c^2/a)\sinh(a\tau/c), with τ\tau the rocket's proper time measured from the vertex of the hyperbola. At proper time τ=5.0 s\tau = 5.0\ \mathrm{s}, the four-velocity component u1=dx/dτu^1 = dx/d\tau is measured to be 1.60×109 m s−11.60\times10^{9}\ \mathrm{m\,s^{-1}} (this exceeds cc because u1u^1 is a rate with respect to proper time, not an ordinary speed). ((a)) Using u0=ccosh⁡(aτ/c)u^0 = c\cosh(a\tau/c) and u1=csinh⁡(aτ/c)u^1 = c\sinh(a\tau/c), find the proper acceleration aa of the rocket. ((b)) Adapt Rindler coordinates (η,ξ)(\eta,\xi) to this rocket, and evaluate g00=−(1+aξ/c2)2g_{00} = -(1+a\xi/c^2)^2 at ξ=100 m\xi = 100\ \mathrm{m} (the nose, if the rocket's tail sits at ξ=0\xi=0). ((c)) Using the Rindler metric, find the fractional rate difference 1+aℓ/c2−11+a\ell/c^2 - 1 between a clock at the nose (ξ=100 m\xi=100\ \mathrm{m}) and one at the tail of this rocket.

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