Editing PGE/RotatingFrame (section)

=====Foundation=====
Suppose the 3-component vector, <math>\vec\Omega</math>, represents a general time-dependent rotation of the <math>(x_1, x_2, x_3)</math>-frame with respect to the inertial frame.  In this context, Chandrasekhar introduces a (3 &times; 3) matrix, <math>\mathbf{\Omega^*}</math>, whose nine components can be expressed in terms of the three components of <math>\vec\Omega</math> via the relations,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>(\Omega^*)_{ij}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\epsilon_{ijk}\Omega_k \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (6a)</td>
</tr>
</table>
Alternatively, we may write,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>(\Omega^*)_{ik}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\epsilon_{ikj}\Omega_j = -~\epsilon_{ijk}\Omega_j\, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="60%" cellpadding="8"><tr><td align="left">

Both of these expressions make use of the three-element [https://en.wikipedia.org/wiki/Levi-Civita_symbol#Definition Levi-Civita tensor], <math>\epsilon_{ijk}</math>. Its six nonzero component values are &hellip;
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center"><math>ij k</math></td>
  <td align="center"><math>\epsilon_{ijk}</math></td>
  <td rowspan="4" bgcolor="lightgrey">&nbsp;</td>
  <td align="center"><math>ij k</math></td>
  <td align="center"><math>\epsilon_{ijk}</math></td>
</tr>
<tr>
  <td align="center">123</td>
  <td align="center" rowspan="3">+1</td>
  <td align="center">132</td>
  <td align="center" rowspan="3">-1</td>
</tr>
<tr>
  <td align="center">312</td>
  <td align="center">321</td>
</tr>
<tr>
  <td align="center">231</td>
  <td align="center">213</td>
</tr>
</table> 
Hence, the six nonzero components of the matrix, <math>\mathbf{\Omega^*}</math>, are,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>(\Omega^*)_{12}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_3\, ;</math>
  </td>

<td align="center">&nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>(\Omega^*)_{13}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\Omega_2\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(\Omega^*)_{21}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\Omega_3\, ;</math>
  </td>

<td align="center">&nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>(\Omega^*)_{23}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_1\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(\Omega^*)_{31}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_2\, ;</math>
  </td>

<td align="center">&nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>(\Omega^*)_{32}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\Omega_1\, .</math>
  </td>
</tr>
</table>

----

<table border="0" align="center" cellpadding="10" width="30px">

<tr>
  <td align="center" colspan="3" bgcolor="white"><math>\mathbf{\Omega^*}</math><br /><font size="-1">(3 &times; 3 matrix)</font></td>
</tr>

<tr>
  <td align="center" width="10px" bgcolor="lightblue"><math>0</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>\Omega_3</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>-~\Omega_2</math></td>
</tr>

<tr>
  <td align="center" width="10px" bgcolor="lightblue"><math>-~\Omega_3</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>0</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>\Omega_1</math></td>
</tr>

<tr>
  <td align="center" width="10px" bgcolor="lightblue"><math>\Omega_2</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>-~\Omega_1</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>0</math></td>
</tr>
</table>

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

<span id="Product">For later use,</span> we note as well that for an arbitrary vector &#8212; call it, <math>\vec{Q}</math> &#8212; the individual components of the product, <math>\mathbf{\Omega^*} \vec{Q}</math>, are given by the expression,

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

<tr>
  <td align="right">
<math>(\mathbf{\Omega^*}\vec{Q} )_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
(\Omega^*)_{ij}Q_j
=
(\epsilon_{ijk}\Omega_k)Q_j
\, .
</math>
  </td>
</tr>
</table>
Compare, for example, Eqs. (17) and (19) in &sect;25 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].

Now, if the motion of the moving frame relative to the inertial frame is specified entirely by the vector <math>\vec\Omega</math>, Chandrasekhar proves that any time-dependent vector defined in the inertial frame &#8212; call it <math>\vec{F}</math> &#8212; will obey the following operator relation:
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[
\mathbf{T}\frac{d}{dt} 
-
\biggl(\frac{d}{dt} - \mathbf{\Omega^*}\biggr)\mathbf{T}
\biggr]\vec{F}
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (11)</td>
</tr>
</table>