Take and . All accelerations, heights and time intervals are quoted in the Earth…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · The equivalence principle and gravity as geometry

Relativity and Gravitation, Ch Experimental foundations: free fall, Lorentz invariance and clock tests

Problem

Take

g = 9.81\ \mathrm{m\,s^{-2}}

and

c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}

. All accelerations, heights and time intervals are quoted in the Earth (laboratory) frame; air resistance is ignored; the gravitational frequency shift is used to first order in

gh/c^{2}

. ((a)) Two test masses of different composition are found to fall in the laboratory with accelerations that are each

9.81\ \mathrm{m\,s^{-2}}

to three significant figures but differ from one another by

3.0\times10^{-11}\ \mathrm{m\,s^{-2}}

. Compute the Eötvös parameter

\eta

. ((b)) In the Pound and Rebka geometry the source sits at the foot of a tower of height

22.5\ \mathrm{m}

and the absorber at the top. (i) Compute the predicted fractional frequency shift

|\Delta\nu|/\nu

. (ii) The measured value is

(2.57\pm0.26)\times10^{-15}

. By how much does the measured value exceed the prediction? Give the excess as a multiple of

10^{-16}

. ((c)) A navigation satellite's clock is found to gain a net

39.2\ \mu\mathrm{s}

per day on ground clocks. If a receiver ignored this offset for one day, the ranging error would be

c

times the timing error. Compute that error in kilometres.

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