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The relationship between the mass, <math>~M</math>, and radius, <math>~R</math>, of a black hole is, | [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, | ||
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<math>~\frac{ | <math>~\frac{2GM}{c^2 R}</math> | ||
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\frac{3M}{4\pi R^3} | \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> | ||
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<math>~=</math> | <math>~\approx</math> | ||
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<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} | |||
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<math>~\approx</math> | |||
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<math>~ | |||
\biggl[\frac{3^{6+3} \times 10^{60}}{2^{10} (10^{66-21}) M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3} | |||
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<math>~\approx</math> | |||
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<math>~ | <math>~ | ||
\frac{ | \biggl[\frac{3^{9} \times 10^{60}}{2^{10} (10^{45}) M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3} | ||
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<math>~ | <math>~\approx</math> | ||
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<math>~ | <math>~ | ||
\frac{ | \biggl[\frac{2 \times 10^{16}}{M_\odot^2}\biggr] ~\mathrm{g}~\mathrm{cm}^{-3} \, . | ||
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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= | =See Also= | ||
Latest revision as of 11:47, 13 October 2017
Looking Outward, From Inside a Black Hole
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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
- 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?
- 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
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