Two small probes fall freely towards a black hole of mass

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Newtonian gravitation as a field theory

Relativity and Gravitation, Ch Tidal forces and the Newtonian tidal field

Problem

Two small probes fall freely towards a black hole of mass M=5.0 M⊙M = 5.0\,M_\odot, treated as a Newtonian point mass. Take 1 M⊙=1.989×1030 kg1\,M_\odot = 1.989\times10^{30}\ \mathrm{kg} and G=6.674×10−11 N m2 kg−2G = 6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}. Both probes are at the same distance rr from the centre, so their separation ξ\xi is perpendicular to the radius (transverse). The separation is tiny compared with rr, so first-order (tidal) expressions may be used, and the probes' own gravity is ignored. ((a)) With ξ=4.0\xi = 4.0 m and r=2.00×106r = 2.00\times10^{6} m, find the magnitude of the relative acceleration of the two probes, and say whether they approach or separate. ((b)) A rigid instrument of length 1.01.0 m is now carried radially (along the line to the hole). At what distance rr does the tidal stretch across its length reach 100 m s−2100\ \mathrm{m\,s^{-2}}? ((c)) Repeat (b) for a hole of mass 50 M⊙50\,M_\odot, using the way your answer to (b) depends on MM.

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