The Rindler map built in the chapter sends to inertial coordinates by , .

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 Rindler map built in the chapter sends (η,ξ)(\eta,\xi) to inertial coordinates (ct,x)(ct,x) by ct=(c2/a+ξ)sinh⁡(aη/c)ct=(c^2/a+\xi)\sinh(a\eta/c), x=(c2/a+ξ)cosh⁡(aη/c)x=(c^2/a+\xi)\cosh(a\eta/c). ((a)) Show that this map is one-to-one onto the wedge x>∣ct∣x>|ct|: derive the inverse formulas ξ(x,t)\xi(x,t) and η(x,t)\eta(x,t), and explain why every event of the wedge receives exactly one pair (η,ξ)(\eta,\xi) with ξ>−c2/a\xi>-c^2/a, and no event outside the wedge receives one at all. ((b)) A rocket has proper acceleration a=10 m s−2a=10\ \mathrm{m\,s^{-2}}. Find the Rindler coordinates (η,ξ)(\eta,\xi) of the event with inertial coordinates x=1.50×1016 mx=1.50\times10^{16}\ \mathrm{m}, ct=9.00×1015 mct=9.00\times10^{15}\ \mathrm{m}.

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