Appendix/Ramblings/InsideOut

Looking Outward, From Inside a Black Hole

[Written by J. E. Tohline, early morning of 13 October 2017]  The relationship between the mass, $M$, and radius, $R$, of a black hole is,

$\frac{2GM}{c^{2}R}$

$=$

$1.$

The mean density of matter inside a black hole of mass $M$ is, therefore,

$\overline{\rho }$	$=$	$\frac{3M}{4\pi R^3}=\frac{3M}{4\pi }[\frac{2GM}{c^2}{]}^{-3}=\frac{3c^6}{2^5\pi G^3{M}^2}$
	$\approx$	$\left[\frac{(3\times 10^{10}{)}^6}{2^5(\frac{2}{3}\times 10^{-7}{)}^3(2\times 10^{33}{)}^2{M}_{\odot }^2}\right]\mathrm{g}{\mathrm{c}\mathrm{m}}^{-3}$
	$\approx$	$\left[\frac{3^{6+3}\times 10^{60}}{2^{10}(10^{66-21})M_{\odot }^2}\right]\mathrm{g}{\mathrm{c}\mathrm{m}}^{-3}$
	$\approx$	$\left[\frac{3^9\times 10^{60}}{2^{10}(10^{45})M_{\odot }^2}\right]\mathrm{g}{\mathrm{c}\mathrm{m}}^{-3}$
	$\approx$	$\left[\frac{2\times 10^{16}}{M_{\odot }^2}\right]\mathrm{g}{\mathrm{c}\mathrm{m}}^{-3}.$

We are accustomed to imagining that the interior of a black hole (BH) must be an exotic environment because a one solar-mass BH has a mean density that is on the order of, but larger than, the density of nuclear matter. From the above expression, however, we see that a $10^9{M}_{\odot }$ BH has a mean density that is less than that of water (1 gm/cm³). And the mean density of a BH having the mass of the entire universe must be very small indeed. This leads us to the following list of questions.

Enumerated Questions

1. Can we construct a Newtonian structure out of normal matter that has a mass of, say, $10^9{M}_{\odot }$ whose equilibrium radius is much less than the radius of the BH horizon associated with that object? Does it necessarily have a mean temperature whose associated sound speed is super-relativistic?
2. Who else in the published literature has explored questions along these lines?

See Also

• Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154) — On the Dynamics of Revolving Fluids