A group reports three separate measurements

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 group reports three separate measurements. Throughout, take the local free-fall acceleration at the site as

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

and

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

. All accelerations, heights and elapsed times below are quoted in the Earth (laboratory) frame, and air resistance is ignored throughout. Use the first-order gravitational rate results of this chapter, that is, work to first order in

gh/c^{2}

. ((a)) An aluminium mass and a copper mass are released in the laboratory and fall with accelerations

g_1 = 9.8120040\ \mathrm{m\,s^{-2}}

and

g_2 = 9.8120036\ \mathrm{m\,s^{-2}}

. Compute the Eötvös parameter

\eta

. ((b)) Gamma rays are emitted by a source at the foot of a tower of height

30.0\ \mathrm{m}

and received by a detector at the top. Compute the predicted magnitude of the fractional frequency shift

|\Delta\nu|/\nu

between emission and reception. ((c)) An atomic clock is carried in an aircraft that circles at a constant altitude of

9.00\ \mathrm{km}

above a ground station for

36.0\ \mathrm{h}

of ground-clock time, and is then compared with the clock that stayed at the station. Ignore all velocity effects and Earth's rotation, and use only the first-order gravitational rate difference

gh/c^{2}

. (i) Predict, to the nearest nanosecond, the reading of the flown clock minus the reading of the ground clock. (ii) The measured offset is

(138 \pm 15)\ \mathrm{ns}

. By how many nanoseconds (nearest nanosecond) does the measured value differ from your prediction?

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