The chapter's opening states that a rocket held at 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

The chapter's opening states that a rocket held at constant proper acceleration a′a' traces x2−c2t2=c4/a′2x^2-c^2t^2=c^4/a'^2 (after shifting the origin). A different member of the family, at fixed Rindler coordinate ξ≠0\xi\ne0, sits permanently at that label as η\eta ranges over every value. ((a)) Show that the family member at fixed ξ\xi is itself a rocket of constant proper acceleration, and derive its proper acceleration aξa_\xi in terms of aa and ξ\xi. ((b)) For a rocket with a=9.81 m s−2a=9.81\ \mathrm{m\,s^{-2}} (tail at ξ=0\xi=0) and a nose at ξ=100 m\xi=100\ \mathrm{m}, evaluate the fractional difference (a−aξ)/a(a-a_\xi)/a between the nose's own proper acceleration and the tail's, and the absolute difference in m s−2\mathrm{m\,s^{-2}}. ((c)) Compare the fractional difference from (b) with the clock-rate excess aℓ/c2a\ell/c^2 already found for the same rocket and the same 100 m100\ \mathrm{m}, and state the relation between the two in one sentence.

Browse the problem library