Part of a laboratory redshift test uses a vertical evacuated tube of height m with…

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

Part of a laboratory redshift test uses a vertical evacuated tube of height

h = 27.0

m with the gamma-ray source at the bottom and the absorber at the top; take

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

as uniform over the tube. A clock is also flown on a space station in a circular orbit of radius

r = 6.771\times10^{6}

m about the Earth's centre and compared with an identical clock at the Earth's surface, at radius

R = 6.371\times10^{6}

m. Take the Earth's mass

M = 5.972\times10^{24}

kg,

G = 6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}

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

and

\Phi = -GM/r

with

\Phi \to 0

at infinity. Work in the non-rotating frame in which the Earth's centre is at rest, ignore the Earth's rotation (so the surface clock is at rest at radius

R

), and work to first order in

\Phi/c^2

and

v^2/c^2

. ((a)) Calculate the magnitude of the predicted fractional frequency shift

|\Delta\nu/\nu|

for the

27.0

m tube. ((b)) Calculate the predicted offset of the station clock relative to the surface clock, in

\mathrm{\mu s}

per day, giving the gravitational and velocity contributions separately, and state whether it agrees with a measured loss of

24.6 \pm 0.5\ \mathrm{\mu s}

per day. ((c)) Which pillars of the Einstein equivalence principle does the station-clock comparison test, and which one does it not test?

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