Editing Apps/DysonWongTori (section)

====The Coulomb Potential====
[[File:MovieWongN4b.gif|300px|right|frame|Contribution to potential by mode n = 3 (magnified by 100)]]As [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] reminds us, the Coulomb potential, <math>~U({\vec{r}}~')</math>, at a point <math>~{\vec{r}}~'</math> due to an arbitrary charge distribution, <math>~\rho({\vec{r}})</math>, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~U({\vec{r}}~')</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\iiint \frac{\rho(\vec{r}) d^3r}{|~\vec{r} - {\vec{r}}^{~'} ~|} \, .</math>
  </td>
</tr>
</table>
</div>
Referencing, for example, equation (3) of [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl and Tohline (1999)], we see that if we let <math>~\rho({\vec{r}})</math> represent a ''mass'' distribution instead of a ''charge'' distribution, this identical expression will give the Newtonian gravitational potential if we simply multiply through by (conventionally, the negative of) the gravitational constant, <math>~G</math>.

From the [[#volume|above expression for the differential volume element in toroidal coordinates]], the right-hand side of this expression for the potential becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~U(\eta^',\theta^',\psi^')</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0 a^3
\iiint \frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|} \biggl[ \frac{\Theta(\upsilon) \sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi~ 
\, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.52)
  </td>
</tr>
</table>
</div>
Next, Wong (1973) points out that in toroidal coordinates the Green's function is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 }
\sum\limits_{m,n} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \cos[m(\psi - \psi^')] \cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta
\end{cases}\, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.53)
  </td>
</tr>
</table>
</div>
where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are "<font color="darkgreen">Legendre functions of the first and second kind with order <math>~n - \tfrac{1}{2}</math> and degree <math>~m</math> (toroidal harmonics)</font>," and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>.  


<table border="1" align="center" cellpadding="8" width="70%">
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  <th align="center" bgcolor="yellow">
LaTeX mathematical expressions cut-and-pasted directly from
<br />
NIST's ''Digital Library of Mathematical Functions''
  </th>
</tr>
<tr>
  <td align="left">
Note that, according to [http://dlmf.nist.gov/14.19 &sect;14.19(iii) of NIST's ''Digital Library of Mathematical Functions''],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P^{m}_{n-\frac{1}{2}} \bigl(\cosh\xi\bigr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\Gamma \bigl(n+m+\frac{1}{2}
\bigr)(\sinh\xi)^{m}}{2^{m}\pi^{1/2}\Gamma\bigl(n-m+\frac{1}{2}\bigr)\Gamma 
\bigl(m+\frac{1}{2}\bigr)}~ \int_{0}^{\pi}\frac{(\sin\phi)^{2m}}{(\cosh\xi+ 
\cos\phi\sinh\xi)^{n+m+(1/2)}}\mathrm{d}\phi,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{Q}^{m}_{n-\frac{1}{2}} \bigl(\cosh\xi \bigr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\Gamma\bigl(n+\frac{1}{2}\bigr)}{\Gamma \bigl(n+m+\tfrac{1}{2}\bigr)
\Gamma \bigl(n-m+\frac{1}{2}\bigr)}~\int_{0}^{\infty}\frac{\cosh \bigl(mt \bigr)}{(\cosh\xi+\cosh t \sinh\xi)^{n+(1/2)}}\mathrm{d}t \, .
</math>
  </td>
</tr>
</table>

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

After plugging this Green's function into the expression for the potential, then integrating over the azimuthal angle &#8212; which is permitted, here, because the density distribution, <math>~\rho(\vec{r})</math>, is assumed to be axisymmetric &#8212; [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] obtains,

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

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\rho_0 a^2 (\cosh \eta^' - \cos \theta^')^{1 / 2} \sum\limits_n \epsilon_n \int_{\eta_0}^\infty d\eta 
\int_{-\pi}^{\pi} \biggl[\frac{\cos[n(\theta - \theta^')]}{(\cosh\eta - \cos\theta)^{5 / 2}} \biggr]d\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sinh\eta ~\begin{cases}P_{n-1 / 2}(\cosh\eta) ~Q_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P_{n-1 / 2}(\cosh\eta^') ~Q_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta
\end{cases}\, 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.55)
  </td>
</tr>
</table>
</div>
which is valid for any azimuthal angle, <math>~\psi^'</math>.  Notice that the step function, <math>~\Theta(\upsilon)</math>, no longer explicitly appears in this expression for the Coulomb (or gravitational) potential; it has been used to establish the specific limits on the "radial" coordinate integration.  Next, he completes the integration over the angle, <math>~\theta</math>, to obtain,

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

<tr>
  <td align="right">
<math>~U(\eta^',\theta^')</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2^{9 / 2}\rho_0 a^2}{3} (\cosh \eta^' - \cos \theta^')^{1 / 2} \sum\limits_n \epsilon_n \cos(n\theta^') \int_{\eta_0}^\infty d\eta 
\biggl[ \frac{Q^2_{n-1 / 2}(\cosh\eta)}{\sinh\eta} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \begin{cases}P_{n-1 / 2}(\cosh\eta) ~Q_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P_{n-1 / 2}(\cosh\eta^') ~Q_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta
\end{cases}\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.57)
  </td>
</tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
TO BE DONE:


As we have suggested in an [[2DStructure/ToroidalCoordinates#Special_Case|accompanying discussion]], this expression should match our separate "special case" evaluation of the potential of a uniform-density torus if this expression is evaluated at <math>~\eta^' = +\infty</math>, and all values of the polar angle, <math>~\theta^'</math>.
</td></tr></table>

Finally, [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] was able to complete the integration over the radial coordinate, <math>~\eta</math> to obtain an expression for the potential &#8212; most generally, his equation (2.59) &#8212; at all ''interior'' as well as all ''exterior'' coordinate positions, <math>~(\eta^', \theta^')</math>.  

=====Interior Solution=====
The ''interior'' 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^' \ge \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\{ 
- \frac{3\pi^2}{2^{5/ 2}} \biggl[ \frac{\sinh^2\eta^'}{(\cosh \eta^' - \cos \theta^')^2} \biggr]
+~ (\cosh \eta^' - \cos \theta^')^{1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits_{n=0}^\infty \epsilon_n \cos(n\theta^') 
Q_{n-1 / 2}(\cosh\eta^') B_n(\cosh\eta_0) 
\biggr\} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.65)
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~B_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
(n+\tfrac{1}{2})P_{n+1/2} (\cosh\eta_0)Q^2_{n-1/2} (\cosh\eta_0)
- (n-\tfrac{3}{2})P_{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. (2.62)
  </td>
</tr>
</table>
</div>

=====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.''
-->