Editing Apps/DysonWongTori (section)

=====Exterior Solution=====
The ''exterior'' solution is:

<div align="center" id="Wong1973Interior">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')\biggr|_{\mathrm{for}~\eta^' \le \eta_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2^{3 / 2}}{3\pi^2} \biggl(\frac{q}{a}\biggr) \frac{\sinh^3 \eta_0}{\cosh\eta_0} \biggl\{ 
(\cosh \eta^' - \cos \theta^')^{1 / 2} 
~\sum\limits_{n=0}^\infty \epsilon_n \cos(n\theta^') 
P_{n-1 / 2}(\cosh\eta^') C_n(\cosh\eta_0) 
\biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eqs. (2.59) & (2.60) combined
  </td>
</tr>
</table>
</div>
where,

<div align="center" id="Wong1973Interior">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
(n+\tfrac{1}{2})Q_{n+1/2} (\cosh\eta_0)Q^2_{n-1/2} (\cosh\eta_0)
- (n-\tfrac{3}{2})Q_{n-1/2} (\cosh\eta_0)Q^2_{n+1/2} (\cosh\eta_0) \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.63)
  </td>
</tr>
</table>
</div>

[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] goes on to say that <font color="#009999">&hellip; for the case of a very thin ring (i.e., <math>~\eta_0 \rightarrow \infty</math>), the exterior solution has contributions mostly from the first term in the expansion of the series &hellip;  By considering the asymptotic values as <math>~\eta_0 \rightarrow \infty</math>, one obtains the potential at a point exterior to a thin ring given by</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')\biggr|_{\mathrm{for}~\eta^' \le \eta_0}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
2^{-3 / 2} \biggl(\frac{q}{a}\biggr) (\cosh\eta^' - \cos\theta^')^{1 / 2} K\biggl( \frac{\tanh(\eta^'/2)}{\cosh(\eta^'/2)} \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>
He goes on to state that this expression <font color="#009999">can be shown to be identical to the result for a thin ring obtained in a simple integration without using the toroidal coordinates &hellip;</font> namely,  
<!-- Referencing [https://www.its.caltech.edu/~kip/index.html/PubScans/II-4.pdf Thorne (1965)], Wong states that the exterior potential this"thin ring" approximation gives an exterior potential of the form,-->

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~U(\rho^', z^')</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2q}{\pi} \biggr) \frac{K(k)}{[(z^')^2 + (d + \rho^')^2]^{1 / 2}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.67)
  </td>
</tr>
</table>
where,
<div align="center">
<math>~k^2 \equiv \frac{4d \rho^'}{[(z^')^2 + (d + \rho^')^2]^{1 / 2}} \, .</math>
</div>

After making the substitution, <math>~q \rightarrow (-GM)</math>, we see that this is, indeed, the expression that has been derived, above, for the [[#TRApproximation|gravitational potential in the thin ring approximation]], <math>~\Phi_\mathrm{TR}</math>.


<table border="1" align="center" cellpadding="10" width="90%">
<tr><td align="center">Thin Ring Approximation as Presented by [https://www.its.caltech.edu/~kip/index.html/PubScans/II-4.pdf Thorne (1965)]</td></tr>
<tr><td align="left">
Interestingly, for an example of a derivation of the [[Apps/DysonWongTori#Thin_Ring_Approximation|thin ring approximation, which we have reviewed above]], rather than referencing [https://archive.org/details/foundationsofpot033485mbp Kellogg (1929)] or [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential MacMillan (1958)], Wong (1973) points to an article by [https://www.its.caltech.edu/~kip/index.html/PubScans/II-4.pdf Thorne (1965)], which was published in the [http://adsabs.harvard.edu/abs/1965qssg.conf.....R ''Proceedings of the 1<sup>st</sup> Texas Symposium on Relativistic Astrophysics'' (1965), eds., I. Robinson, A. Schild, &amp; E. L. Schucking (Chicago: Chicago University Press)]. Presumably this reflects Wong's personal interactions with [https://en.wikipedia.org/wiki/John_Archibald_Wheeler J. A. Wheeler's] research group at Princeton University, as Thorne was a student of Wheeler.

<table border="0" align="right" width="410px" cellpadding="10">
<tr><td align="center">
'''Fig. 1 extracted unmodified from [https://www.its.caltech.edu/~kip/index.html/PubScans/II-4.pdf Thorne (1965)]'''<br />
"''The Instability of a Toroidal Magnetic Geon Against Gravitational Collapse'''"<p></p>
[http://adsabs.harvard.edu/abs/1965qssg.conf.....R ''Proceedings of the 1<sup>st</sup> Texas Symposium on Relativistic Astrophysics'' (1965), eds., Robinson, et al. (Chicago: Chicago University Press)]
</td></tr>
<tr><td align="center">
[[File:Thorne65Fig1.png|400px|To be inserted: Fig. 1 from Thorne (1965)]]
</td></tr></table>
But Thorne actually presents a solution to a general-relativistic-based initial-value equation that reads,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Delta^2\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2\pi G\psi^5 T_0^0 \, .</math>
  </td>
</tr>
</table>
He sets up his problem of interest in the context of a <font color="#009999">toroidal magnetic geon at a moment of time symmetry as seen in the base metric</font> <math>~(x,y,z)</math>; see his Figure 1, reproduced there, on the right.  More specifically, he imagines a situation where existing magnetic field lines are <font color="#009999">entirely contained within a torus of major radius <math>~b</math> and minor radius <math>~a</math>, as measured in the base metric &hellip; </font> and with <math>~b \gg a</math>.  As Thorne points out, this <font color="#009999">&hellip; initial value equation is just Poisson's equation with <math>~-2\pi G\psi^5T_0^0</math> as the source of the "conformal correction factor" <math>~\psi</math>.</font>
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\psi_{r>a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \frac{1}{\pi} \frac{M_m}{M} \frac{M_m}{[(z^')^2 + \rho^' + b^')^2]^{1 / 2}} K\biggl( \biggl[ \frac{4\rho^' b^'}{(z^')^2 + (\rho^' + b^')^2} \biggr]^{1 / 2} \biggr)
</math>
  </td>
</tr>
</table>

</td></tr></table>

<!--
====Other Tidbits====

Reference [17] in Wong (1973) points to an article by Kip S. Thorne in the Proceedings titled, "Quasi-Stellar Sources and Gravitational Collapse," edited by I. Robinson ''et al.'', Chapter I1.7, University of Chicago Press, Chicago, 1965.  More completely, a pointer to the complete proceedings is,
* [http://adsabs.harvard.edu/abs/1965qssg.conf.....R I. Robinson, A. Schild, &amp; E. L. Schucking (1965)], ''Proceedings of the 1<sup>st</sup> Texas Symposium on Relativistic Astrophysics'', (Chicago:  University of Chicago Press).
While a pointer to the specific article by Thorne inside the proceedings is,
* [https://www.its.caltech.edu/~kip/index.html/PubScans/II-4.pdf Kip S. Thorne (1965)], ''The Instability of a Toroidal Magnetic Geon Against Gravitational Collapse," Chapter 7 of Robinson ''et al.''
-->