Take and . Heights and speeds are quoted in the Earth (laboratory) frame; the drop…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · 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}}

. Heights and speeds are quoted in the Earth (laboratory) frame; the drop tube is evacuated so air resistance is absent; all rate differences are used to first order in

gh/c^{2}

and in

v^{2}/c^{2}

. ((a)) Two masses of different composition with Eötvös parameter

\eta = 3.0\times10^{-13}

are released side by side from rest and fall

18.0\ \mathrm{m}

in an evacuated tube. Using the chapter's result that the vertical separation on arrival is

\eta h

, compute that separation in metres. ((b)) In the tower experiment the emitted line is brought into resonance with the absorber by moving the source with the speed

v = gh/c

. Compute

v

for a tower of height

h = 22.5\ \mathrm{m}

. ((c)) A satellite clock moves at

3.87\ \mathrm{km\,s^{-1}}

relative to the (non-rotating) Earth-centred frame. Using only the first-order velocity term

v^{2}/2c^{2}

, compute the amount by which this clock falls behind an Earth-centred clock in one day, in microseconds.

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