The Rindler time coordinate was built so that the tail clock

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 time coordinate η\eta was built so that the tail clock, at ξ=0\xi=0, reads η\eta itself. Build a second set of coordinates, (η′,ξ)(\eta',\xi), in which instead the clock at the nose, ξ=ξ0\xi=\xi_0, reads η′\eta' directly. ((a)) Using the clock-rate result dτ0=(1+aξ0/c2) dηd\tau_0=(1+a\xi_0/c^2)\,d\eta for the nose clock, define η′\eta' in terms of η\eta so that η′=τ0\eta'=\tau_0 exactly, and substitute into the Rindler line element to derive the new metric component g00′(ξ)g'_{00}(\xi). ((b)) Check that g00′(ξ0)=−1g'_{00}(\xi_0)=-1, and find g00′(0)g'_{00}(0), the value at the tail, for a=9.81 m s−2a=9.81\ \mathrm{m\,s^{-2}} and ξ0=100 m\xi_0=100\ \mathrm{m}. Give the deviation g00′(0)+1g'_{00}(0)+1 to three significant figures.

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