Two objects of different composition are released from rest and fall freely through a…
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 The equivalence principle under test
Problem
Two objects of different composition are released from rest and fall freely through a height ℓ=10.0 m near the Earth's surface, where g⊕=9.81ms−2.
((a)) If universality of free fall holds exactly, both objects fall with acceleration g⊕. Compute the time t0 each takes to reach the bottom, treating the fall as non-relativistic and uniform.
((b)) Suppose universality of free fall is violated at the level of a torsion-balance bound, ε=2(a1−a2)/(a1+a2)=1.0×10−13, so the two objects fall with a1=g⊕(1+ε/2) and a2=g⊕(1−ε/2). Using t=2ℓ/a and a first-order expansion in ε, find the difference Δt=t2−t1 in their arrival times.
((c)) Rindler coordinates give the differential ageing between clocks fixed at the two ends of a single rigid accelerating rocket whose tail has proper acceleration a. The free-fall argument that treats free fall as the worldline of maximal proper time implicitly assumes the same worldline for every falling body. Explain briefly which of these two results is disturbed by a non-zero ε and which is not.