Define the mean density inside the horizon of a non-rotating black hole of mass as

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

Define the mean density inside the horizon of a non-rotating black hole of mass

M

\bar\rho = M/\left(\tfrac{4}{3}\pi r_s^{3}\right)

, with

r_s = 2GM/c^{2}

. Estimate the properties of the hole whose mean density equals that of air at room temperature and pressure,

\bar\rho = 1.2\ \mathrm{kg\,m^{-3}}

. ((a)) Estimate its mass, in kg and in solar masses. ((b)) Estimate its Schwarzschild radius, in m and in astronomical units (

1\ \mathrm{au} = 1.496 \times 10^{11}\ \mathrm{m}

). ((c)) Evaluate the compactness parameter

GM/rc^{2}

r = r_s

, and state whether the first-order redshift formula

z \approx \Delta\Phi/c^{2}

of Chapter 9, with

\Delta\Phi = \Phi_{\rm rec} - \Phi_{\rm em}

, may be applied to light emitted from just outside this horizon. Use

G = 6.674 \times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}

c = 2.998 \times 10^{8}\ \mathrm{m\,s^{-1}}

and

M_\odot = 1.989 \times 10^{30}\ \mathrm{kg}

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