User:Tohline/Appendix/Ramblings/InsideOut

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Looking Outward, From Inside a Black Hole

Whitworth's (1981) Isothermal Free-Energy Surface
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[Written by J. E. Tohline, early morning of 13 October 2017]  The relationship between the mass, <math>~M</math>, and radius, <math>~R</math>, of a black hole is,

<math>~\frac{2GM}{c^2 R}</math>

<math>~=</math>

<math>~1 \, .</math>

The mean density of matter inside a black hole of mass <math>~M</math> is, therefore,

<math>~\bar\rho</math>

<math>~=</math>

<math>~ \frac{3M}{4\pi R^3} = \frac{3M}{4\pi} \biggl[ \frac{2GM}{c^2}\biggr]^{-3} = \frac{3c^6}{2^5\pi G^3 M^2} </math>

 

<math>~\approx</math>

<math>~ \biggl[\frac{(3 \times 10^{10})^6}{2^5 (\tfrac{2}{3}\times 10^{-7})^3 (2\times 10^{33})^2 M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3} </math>

 

<math>~\approx</math>

<math>~ \biggl[\frac{3^{6+3} \times 10^{60}}{2^{10} (10^{66-21}) M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3} </math>

 

<math>~\approx</math>

<math>~ \biggl[\frac{3^{9} \times 10^{60}}{2^{10} (10^{45}) M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3} </math>

 

<math>~\approx</math>

<math>~ \biggl[\frac{2 \times 10^{16}}{M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3} \, . </math>

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 <math>~10^9 M_\odot</math> 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, <math>~10^9 M_\odot</math> whose equilibrium radius is much less than the radius of the BH horizon associated with that object?
  2. Who else in the published literature has explored questions along these lines?

See Also


 

Whitworth's (1981) Isothermal Free-Energy Surface

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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation