A spacecraft is in free fall towards the Earth

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Newtonian gravitation as a field theory

The Geometry of Gravity, Ch The tidal field: what free fall cannot remove

Problem

A spacecraft is in free fall towards the Earth, at distance

r = 1.0\times10^7\,\mathrm{m}

from the Earth's centre (

GM_\oplus = 3.986\times10^{14}\,\mathrm{m^3\,s^{-2}}

). Two identical balls are released at rest

S = 2.00\,\mathrm{m}

apart, aligned along the radial direction, symmetrically about the capsule's centre. An engineer notices the balls drift apart and argues: "The capsule is not a perfect free-fall frame because there is a residual relative acceleration of

2GM_\oplus S/r^3

between the balls. If I fire the capsule's thrusters to give the whole capsule a uniform extra acceleration

\vec a_{\rm extra}

of the right size, directed radially, I can cancel this drift and make the capsule a perfect free-fall frame again." Find the error in this reasoning, and calculate the relative acceleration between the balls (i) before the thrusters fire and (ii) after the thrusters fire with

|\vec a_{\rm extra}| = 2GM_\oplus S/r^3

applied identically to the whole capsule.

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