sandbar is the bar in crossing the bar.
sandbar is the bar in crossing the bar.
what is the bar in crossing the bar?
sandbar
crossing the bar









































"crossing the bar" is an 1889 poem by alfred, lord tennyson. it is considered that tennyson wrote it in elegy; the narrator uses an extended metaphor to compare death with crossing the "sandbar" between river of life, with its outgoing "flood", and the ocean that lies beyond [death], the "boundless deep", to which we return.









































section::::overview.
crossing the bar
in space, the möbius/lorentz transformations can be written as:









































\eta' & =\gamma\xi+\delta\eta









































\right.\\









































\hline \left.\begin{matrix}\mathbf{u}=\left(\begin{matrix}x_{1} & x_{2}\\









































\end{matrix}\right)=\left(\begin{matrix}\bar{\xi}\xi & \xi\bar{\eta}\\









































\bar{\xi}\eta & \bar{\eta}\eta









































\end{matrix}\right)=\left(\begin{matrix}x_{0}+x_{3} & x_{1}-ix_{2}\\









































\end{matrix}\right)\\









































\end{matrix}\right|\begin{matrix}\mathbf{d}=\left(\begin{matrix}\alpha & \beta\\









































\gamma & \delta









































\end{matrix}\right)\\









































\end{matrix}\\









































\hline \mathbf{u}'=\mathbf{d}\cdot\mathbf{u}\cdot\bar{\mathbf{d}}^{\mathrm{t}}=\begin{align}x_{1}^{\prime} & =x_{1}\alpha\bar{\alpha}+x_{2}\alpha\bar{\beta}+x_{3}\bar{\alpha}\beta+x_{4}\beta\bar{\beta}\\









































x_{2}^{\prime} & =x_{1}\bar{\alpha}\gamma+x_{2}\bar{\alpha}\delta+x_{3}\bar{\beta}\gamma+x_{4}\bar{\beta}\delta\\









































x_{3}^{\prime} & =x_{1}\alpha\bar{\gamma}+x_{2}\alpha\bar{\delta}+x_{3}\beta\bar{\gamma}+x_{4}\beta\bar{\delta}\\









































\hline \begin{align}x_{3}^{\prime}x_{2}^{\prime}-x_{1}^{\prime}x_{4}^{\prime} & =x_{3}x_{2}-x_{1}x_{4}=0\\









































thus:









































or in line with equation () one can substitute formula_45 so that the möbius/lorentz transformations become related to the unit sphere formula_46:









































\eta' & =\gamma\xi+\delta\eta









































\right.\\









































\hline \begin{matrix}\mathbf{u}=\left(\begin{matrix}x_{1} & x_{2}\\









































\end{matrix}\right)=\left(\begin{matrix}\bar{\xi}\xi & \xi\bar{\eta}\\









































\bar{\xi}\eta & \bar{\eta}\eta









































\end{matrix}\right)=\left(\begin{matrix}1+u_{3} & u_{1}-iu_{2}\\









































\end{matrix}\right)\\
history of lorentz transformations