A Mössbauer source is mounted at the top of a cathedral spire of height m and the…
PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Atomic, Molecular & Optical Physics · The equivalence principle and gravity as geometry
Relativity and Gravitation, Ch Experimental foundations: free fall, Lorentz invariance and clock tests
Problem
A Mössbauer source is mounted at the top of a cathedral spire of height h=62.0 m and the absorber at the bottom. Take g=9.81ms−2 as uniform over the spire and over the flight described below, and c=2.998×108ms−1. All shifts are to be computed to first order in Φ/c2 and in v2/c2. Separately, a caesium clock is flown on a research aircraft along a north-south meridian route at a constant ground speed u=250ms−1 and a constant altitude 11.0 km above the airfield, for an elapsed time of 8.00 h as read by the clock left at the airfield. Work in a frame in which the airfield is at rest and ignore the Earth's rotation entirely; ignore the climb and descent.
((a)) Calculate the magnitude of the predicted fractional frequency shift ∣Δν/ν∣ of the gamma rays received at the bottom of the spire.
((b)) Calculate the upward source speed that would exactly compensate this shift by a first-order Doppler shift.
((c)) Calculate the predicted reading difference (flown clock minus airfield clock) in ns at the end of the flight, and state whether it agrees with a measured value of +29±7 ns.
((d)) Name the pillar of the Einstein equivalence principle that the spire measurement tests, and the pillar that the velocity term of the flight comparison tests.