A rocket of proper length (tail at

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 of proper length LL (tail at ξ=0\xi=0, nose at ξ=L\xi=L) holds constant proper acceleration aa. Its onboard computer needs the nose and tail clocks to stay synchronised to a fractional tolerance ϵ\epsilon: the exact excess Δτnose/Δτtail−1\Delta\tau_{\rm nose}/\Delta\tau_{\rm tail}-1 must not exceed ϵ\epsilon. ((a)) Using the chapter's exact clock-rate result, derive the maximum steady proper acceleration amax⁡(L,ϵ)a_{\max}(L,\epsilon) that keeps a rocket of length LL within tolerance ϵ\epsilon. ((b)) Evaluate amax⁡a_{\max} for L=50 mL=50\ \mathrm{m} and ϵ=1.00×10−9\epsilon=1.00\times10^{-9}, and again for L=100 mL=100\ \mathrm{m} at the same tolerance. State in one sentence why doubling LL has that effect on amax⁡a_{\max}. ((c)) For a rocket held at a=9.81 m s−2a=9.81\ \mathrm{m\,s^{-2}} and the same tolerance ϵ=1.00×10−9\epsilon=1.00\times10^{-9}, find the maximum length Lmax⁡L_{\max} that keeps it in tolerance, and compare Lmax⁡L_{\max} with the horizon distance c2/ac^2/a found earlier in the chapter for this rocket, expressing the comparison as a single ratio.

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