Hyperboloid Embedding

$\begin{displaymath} R^2 = x^2+y^2+z^2-w^2 \end{displaymath}$

(1)

$\begin{displaymath} 2x{dx\over dw} + 2y{dy\over dw} + 2z{dz\over dw} - 2w = 0. \end{displaymath}$

(2)

Since

$\begin{displaymath} {\bf r} \equiv x\hat{\bf x} + y\hat{\bf y} + z\hat{\bf z}, \end{displaymath}$

(3)

$\begin{displaymath} dw = {x\,dx + y\,dy + z\,dz\over w} = {{\bf r}\cdot d{\bf r}\over\sqrt{r^2-R^2}}. \end{displaymath}$

(4)

To stay on the surface of the Hyperboloid,

$\displaystyle ds^2$	$\textstyle =$	$\displaystyle dx^2+dy^2+dz^2-dw^2$
	$\textstyle =$	$\displaystyle dx^2+dy^2+dz^2 - {r^2\,dr^2\over r^2-R^2}$
	$\textstyle =$	$\displaystyle dr^2+r^2d\Omega^2 + {dr^2\over 1 - {R^2\over r^2}}.$	(5)