In 1783 Michell set the Newtonian escape speed of a body equal to ,

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Relativistic stars and compact objects · Relativistic stars and compact objects

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

Problem

In 1783 Michell set the Newtonian escape speed of a body equal to

c

\tfrac12 c^2 = GM/R

, and so found a radius from which "light could not escape". A student prefers a different Newtonian argument: since light of energy

E

carries

E/c^2

where a mass would sit, the student writes the condition for light to be bound as

mc^2 = GMm/R

and obtains a radius smaller than Michell's. Consider a black hole of mass

M = 12\,M_\odot

, with

M_\odot = 1.989\times10^{30}\ \mathrm{kg}

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

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

. ((a)) Compute the radius the student's condition gives. ((b)) Compute the Schwarzschild radius

r_s = 2GM/c^2

, the radius that the full theory of gravity also gives for the horizon. ((c)) An astronaut of radial extent

\xi = 1.7\ \mathrm{m}

falls freely through the horizon. Using the Newtonian tidal field of Chapter 7,

\Delta a = 2GM\xi/r^3

, compute the tidal stretch along the astronaut's body at

r = r_s

. In your solution, state in one or two sentences what is wrong with both Newtonian arguments, even though one of them delivers the accepted radius.

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