Appendix/Ramblings/InsideOut - Revision history

Joel2: Created page with "FORCETOC =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,

$~\frac{2GM}{c^2 R}$
2024-07-05T23:48:40Z

Created page with "FORCETOC   =Looking Outward, From Inside a Black Hole= [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, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{2GM}{c^2 R}</math> </td> <td align="cent..."

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FORCETOC 

=Looking Outward, From Inside a Black Hole=

[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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{2GM}{c^2 R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 \, .</math>
</td>
</tr>
</table>
</div>
The mean density of matter inside a black hole of mass <math>~M</math> is, therefore,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\bar\rho</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<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>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<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>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
\biggl[\frac{3^{6+3} \times 10^{60}}{2^{10} (10^{66-21}) M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
\biggl[\frac{3^{9} \times 10^{60}}{2^{10} (10^{45}) M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2 \times 10^{16}}{M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3} \, .
</math>
</td>
</tr>
</table>
</div>
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/cm<sup>3</sup>). 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==

<ol>
<li>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? Does it necessarily have a mean temperature whose associated sound speed is super-relativistic?</li>
<li>Who else in the published literature has explored questions along these lines?
</ol>

=See Also=
* [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] — ''On the Dynamics of Revolving Fluids''

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