In a drop tower an evacuated capsule is released at the Earth's surface and falls…

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

Relativity and Gravitation, Ch Gravity as curved spacetime: redshift and local inertial frames

Problem

In a drop tower an evacuated capsule is released at the Earth's surface and falls freely for

4.7\ \mathrm{s}

before being caught. Inside it, two small test masses are released at rest relative to the capsule, one directly above the other, separated by

L = 2.0\ \mathrm{m}

along the vertical. Along the vertical the tidal field of the Earth gives the two masses a relative acceleration of magnitude

|\mathcal{E}|L

with

|\mathcal{E}| = 2GM_\oplus/r^3

, a stretch, and the drift of their separation from its initial value grows as

\tfrac12|\mathcal{E}|LT^2

while it stays small. The tracking system resolves changes of separation of

\delta = 0.10\ \mathrm{mm}

. Take

M_\oplus = 5.972\times10^{24}\ \mathrm{kg}

r = R_\oplus = 6.371\times10^{6}\ \mathrm{m}

and

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

. ((a)) Estimate the time

T

for which the capsule counts as an inertial frame of this size at this resolution. ((b)) Find the drift of the separation accumulated over the whole

4.7\ \mathrm{s}

drop, and say whether the capsule passes as inertial for the entire drop.

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