A modern repeat of the Hafele and Keating experiment carries a caesium clock eastward…

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

A modern repeat of the Hafele and Keating experiment carries a caesium clock eastward around the world on scheduled flights, while an identical clock stays on the ground. Model the flight as follows: the aircraft flies at constant latitude

45.0^\circ\ \mathrm{N}

, at a constant altitude

h = 9.40\ \mathrm{km}

above the ground clock, with a constant ground speed

u = 240\ \mathrm{m\,s^{-1}}

eastward, for a total elapsed time

T = 41.2\ \mathrm{h}

read on the ground clock. The ground clock sits at sea level at the same latitude and is carried round by the Earth's rotation. Data: Earth's radius

R = 6.371\times10^{6}\ \mathrm{m}

; Earth's rotation period (sidereal)

86\,164\ \mathrm{s}

;

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

, taken uniform between ground and aircraft;

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

. Work in the non-rotating frame centred on the Earth, in which each clock accumulates proper time at the rate

\mathrm{d}\tau/\mathrm{d}t = 1 + \Phi/c^{2} - v^{2}/2c^{2}

to first order in

\Phi/c^2

and

v^2/c^2

. ((a)) Predict the offset

\Delta\tau = \tau_{\rm flown} - \tau_{\rm ground}

at the end of the flight, in nanoseconds. ((b)) The flown clock is found to read

-31\ \mathrm{ns}

relative to the ground clock, with an experimental uncertainty of

\pm 9\ \mathrm{ns}

. Express the difference between measurement and prediction as a multiple of that uncertainty, and say whether the two agree.

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