A student writes two claims about black holes

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Relativistic stars and compact objects · Black holes

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

Problem

A student writes two claims about black holes. Claim 1: "A black hole is a body whose escape velocity exceeds cc. A light pulse aimed radially outward from just inside the horizon therefore climbs, slows down, stops and falls back, exactly like a stone thrown up from the Earth." Claim 2: "The horizon of a 2×107 M⊙2\times10^{7}\,M_\odot hole is far more dangerous to an astronaut than the horizon of a 25 M⊙25\,M_\odot hole, because the massive hole's gravity is so much stronger." Take the astronaut's body to have radial extent ξ=1.8 m\xi = 1.8\ \mathrm{m} and use the Newtonian tidal field of Chapter 7, Δa=2GMξ/r3\Delta a = 2GM\xi/r^3, evaluated at r=rs=2GM/c2r = r_s = 2GM/c^2. Take G=6.674×10−11 N m2 kg−2G = 6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}, c=2.998×108 m s−1c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}, M⊙=1.989×1030 kgM_\odot = 1.989\times10^{30}\ \mathrm{kg}. Which one of the following statements is correct? (a) Both claims are right: light is slowed by the hole's gravity, and the stretch across 1.8 m1.8\ \mathrm{m} at the horizon grows with mass. (b) Claim 1 is right and Claim 2 is wrong: light does slow and fall back, but the stretch goes as 1/M21/M^2, giving about 3×107 m s−23\times10^{7}\ \mathrm{m\,s^{-2}} for the 25 M⊙25\,M_\odot hole and about 5×10−5 m s−25\times10^{-5}\ \mathrm{m\,s^{-2}} for the 2×107 M⊙2\times10^{7}\,M_\odot hole. (c) Both claims are wrong: the locally measured speed of light is always cc, gravity changes its frequency and not its speed, and the horizon stretch goes as 1/M21/M^2, being about 3×107 m s−23\times10^{7}\ \mathrm{m\,s^{-2}} for the 25 M⊙25\,M_\odot hole against about 5×10−5 m s−25\times10^{-5}\ \mathrm{m\,s^{-2}} for the 2×107 M⊙2\times10^{7}\,M_\odot hole, so the stellar-mass hole is the lethal one. (d) Claim 1 is wrong but Claim 2 is right: the stretch is proportional to MM, so the 2×107 M⊙2\times10^{7}\,M_\odot hole gives about 8×1058\times10^{5} times the stretch of the 25 M⊙25\,M_\odot hole, some 2×1013 m s−22\times10^{13}\ \mathrm{m\,s^{-2}}.

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