Exercise 2.07

Prolate spheroidal coordinates can be used to simplify the Kepler problem in celestial mechanics. They are related to the usual cartesian coordinates $(x, y, z)$ of Euclidean three-space by $$ \begin{align} x&=\sinh \chi\,\sin \theta\,\cos\phi \\ y&=\sinh \chi\,\sin \theta\,\sin\phi \\ z&=\cosh \chi\,\cos \theta\, \end{align} $$ Restrict your attention to the plane $y = 0$ and answer the following questions.
(a) What is the coordinate transformation matrix $\partial x^{\mu}/\partial x^{\nu'}$ relating $(x, z)$ to $(\chi,\theta)$?
(b) What does the line element $ds^2$ look like in prolate spheroidal coordinates?

(a) The $y=0$ plane is defined by $\phi=0$ in the prolate spheroidal coordinates. We obtain the following matrix. $$ \frac{ \partial x^{\mu} }{ \partial x^{\nu'} } =\begin{pmatrix} \cosh \chi \sin \theta & \sinh \chi \cos \theta \\ \sinh \chi \cos \theta & -\cosh \chi \sin \theta \end{pmatrix}=M $$ (b) $$ ds^{2}=dx_{\mu}dx^{\mu}=dx_{\nu'} \frac{ \partial x^{\nu'} }{ \partial x^{\mu} } \frac{ \partial x^{\mu} }{ \partial x^{\lambda'} }dx^{\lambda'} $$ Computing the central matrix, $$ \frac{ \partial x^{\nu'} }{ \partial x^{\mu} }\frac{ \partial x^{\mu} }{ \partial x^{\lambda'} } = M^{T}M = \frac{\cosh(2 \chi)-\cos(2 \theta)}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ we obtain $$ ds^{2}= \frac{\cosh(2 \chi)-\cos(2 \theta)}{2}(d\chi^{2}+d\theta^{2}) $$