PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · Einstein's field equations
The Geometry of Gravity, Ch Linearised gravity: weak fields, gravitomagnetism and the wave equation
Problem
A rocket has proper acceleration ap=50ms−2. In the Rindler coordinates (η,ξ) built on its worldline, the metric is ds2=−(1+apξ/c2)2c2dη2+dξ2+dy2+dz2, and h00=−(1+apξ/c2)2+1 exceeds 1 in magnitude once ξ exceeds about 7.5×1014 m. Someone argues: "For ξ beyond that value ∣h00∣>1, so the perturbation is not small any more, and that must mean the spacetime there is strongly curved."
((a)) Say what the Riemann tensor of the spacetime described by these Rindler coordinates actually is, and use it to say whether the claim is correct.
((b)) State the criterion that does correctly decide whether the linear approximation ∣hμν∣≪1 can be trusted in some region, and explain in one or two sentences why it does not reduce to a statement about hμν alone.