A rocket undergoes constant proper acceleration along the -axis
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 constant proper acceleration a=10ms−2 along the x-axis, adapted to Rindler coordinates (η,ξ) with ds2=−(1+aξ/c2)2c2dη2+dξ2+dy2+dz2. Someone examines g00=−(1+aξ/c2)2 and argues: "g00 vanishes at ξ=−c2/a. A clock held at fixed ξ there would need an infinite coordinate interval dη to tick through any proper time, so nothing can ever cross that surface: it is a physical wall in spacetime. And since gravity itself is 'just a coordinate effect' by the equivalence principle, the same argument shows gravity's effects can be removed by a change of coordinates over the whole of spacetime at once, not just locally."
((a)) State whether the claim that ξ=−c2/a is a physical wall is correct, and identify the error, using the fact that Rindler coordinates were built from an inertial coordinate system that covers the whole of Minkowski spacetime.
((b)) Give the numerical value of ξ at which g00 vanishes, in metres.
((c)) A satellite is released from the rocket at η=0 at ξ0=50m, momentarily at rest relative to the rocket at that instant, and afterwards moves at constant velocity (no more thrust). Using the inertial coordinates from which the Rindler chart was built, find the inertial time t at which the satellite's worldline crosses the line x=ct, and state in one sentence what the satellite experiences there.
((d)) Say, referring to the weak and Einstein forms of the equivalence principle, why the closing claim (that gravity's effects vanish over the whole of spacetime at once, not just locally) misapplies the principle.