An astronaut of height falls freely

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Relativistic stars and compact objects

Relativity and Gravitation, Ch Escape velocity, the Schwarzschild radius and the idea of a black hole

Problem

An astronaut of height ξ=1.8 m\xi = 1.8\ \mathrm{m} falls freely, feet first, towards a spherically symmetric non-rotating black hole of mass M=30 M⊙M = 30\,M_\odot, with M⊙=1.989×1030 kgM_\odot = 1.989 \times 10^{30}\ \mathrm{kg}. Her body is aligned radially. Using the Newtonian tidal result of Chapter 7, the radial stretch between head and feet at radius rr is Δa=2GMξ/r3\Delta a = 2GM\xi/r^3. Evaluate it at r=rs=2GM/c2r = r_s = 2GM/c^2. Use G=6.674×10−11 N m2 kg−2G = 6.674 \times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}} and c=2.998×108 m s−1c = 2.998 \times 10^{8}\ \mathrm{m\,s^{-1}}.

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