A spherical asteroid has mass kg and radius km

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · The equivalence principle and gravity as geometry

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

Problem

A spherical asteroid has mass M=3.00×1020M = 3.00 \times 10^{20} kg and radius R=300R = 300 km. Clock A sits on its surface; clock B is held at rest at r=900r = 900 km from the centre by station-keeping thrusters, on the same radial line. Both clocks are permanently at rest relative to the asteroid, which does not rotate. The potential outside the asteroid is Φ=−GM/r\Phi = -GM/r, zero at infinity, and all results are to be given to first order in Φ/c2\Phi/c^2. Take G=6.674×10−11 N m2 kg−2G = 6.674 \times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}, c=2.998×108 m s−1c = 2.998 \times 10^{8}\ \mathrm{m\,s^{-1}} and 11 year =3.156×107= 3.156 \times 10^{7} s. ((a)) Compute the fractional difference in rate between the two clocks, and say which runs faster. ((b)) Over one year, by how much do the two clocks drift apart? ((c)) Clock A drives a transmitter of frequency 1.001.00 GHz (measured at A). In one year of B's own proper time, how many fewer crests does B receive than A emits in one year of A's proper time? What would this number be if spacetime were flat and the configuration static?

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