Treat the Moon as a point mass kg at distance m from the centre of the Earth

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Einstein's field equations

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

Problem

Treat the Moon as a point mass

M_{\rm Moon} = 7.342\times10^{22}

kg at distance

D = 3.844\times10^8

m from the centre of the Earth. Use

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

and

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

. ((a)) Using the Newtonian tidal field

\mathcal E_{ij} = \partial_i\partial_j\Phi

of the Moon's potential, together with the weak-field relation

\Gamma^i{}_{00} = \partial_i\Phi/c^2

obtained from a slow particle's geodesic equation, find

R^r{}_{0r0}

for the lunar field at the Earth's centre. ((b)) Two small test masses are released at diametrically opposite points of the Earth's surface, on the Earth-Moon line, separated by

S = 2R_\oplus

with

R_\oplus = 6.371\times10^6

m, and fall freely. Using the geodesic deviation equation in its Newtonian limit, find the part of their initial relative radial acceleration caused by the Moon's field.

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