Two small test masses fall freely side by side near the Earth

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Curvature: the Riemann tensor, geodesic deviation and the Einstein tensor

Problem

Two small test masses fall freely side by side near the Earth, momentarily at rest relative to each other and

S = 2.0

m apart along the vertical

x

direction of their common rest frame. In that frame, with

x^0 = ct

and

u^\mu \approx (c,0,0,0)

, the curvature components needed are

R^x{}_{0x0} = -3.40\times10^{-23}\ \mathrm{m^{-2}}

and

R^y{}_{0y0} = +1.70\times10^{-23}\ \mathrm{m^{-2}}

, with

y

horizontal; every other

R^i{}_{0j0}

is zero. Take

c = 3.00\times10^8\ \mathrm{m\,s^{-1}}

. ((a)) Write the

x

component of the geodesic deviation equation for this pair, and say which factor in it is fixed by the geometry, which by the masses' motion and which by their separation. ((b)) Find their relative acceleration

\ddot S

, and say whether they drift apart or together. ((c)) Repeat (b) for a second pair, the same distance apart along

y

instead.

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