Hyperbolic angle

[touqra]

Is there a general way of writing the Lorentz transformation for (2+1) dimension or higher, in terms of its hyperbolic angle, sinh and cosh ?

[Mentz114]

This is a boost in the x direction with velocity \beta
\left[ \begin{array}{ccc}
\cosh(\beta) & \sinh(\beta) & 0 \\\
\sinh(\beta) & \cosh(\beta) & 0 \\\
0 & 0 & 1 \end{array} \right]

[Peeter]

Lut, isn't that a boost in the -x direction?

[touqra]

This is a boost in the x direction with velocity \beta
\left[ \begin{array}{ccc}
\cosh(\beta) & \sinh(\beta) & 0 \\\
\sinh(\beta) & \cosh(\beta) & 0 \\\
0 & 0 & 1 \end{array} \right]


What about in a 2D plane ? I don't think it's just a matrix multiplication between x-axis boost and y-axis boost, or is it ? At least, addition of velocity is not the case.

[jtbell]

For three dimensions (but without hyperbolic angles) see Wikipedia (http://en.wikipedia.org/wiki/Lorentz_transformation). Scroll down to where it says "More generally for a boost in an arbitrary direction."

[Rasalhague]

Lut, isn't that a boost in the -x direction?

Yes, unless I'm mistaken, a boost in the positive x direction (output frame moving in the positive x direction wrt the input frame) has minus signs in front of the sinhs. Also, I think the input for cosh and sinh should be the velocity parameter or "rapidity", sometimes written phi, which is artanh(beta) = artanh(v/c), rather than the velocity itself - artanh being the inverse hyperbolic tangent.

http://en.wikipedia.org/wiki/Rapidity

[DrGreg]

Under the convention that a 4-vector is written as

\left[ \begin{array}{c}
ct \\\
\textbf{x}
\end{array} \right]


(where x is the spatial 3-vector) it is

\left[ \begin{array}{ccc}
\cosh \psi & & -\textbf{e}^T \sinh \psi \\\
-\textbf{e} \sinh \psi & & \textbf{I} + (\cosh \psi - 1) \textbf{ee}^T}
\end{array} \right]


where \textbf{e} c \tanh \psi is the 3-velocity vector of the boost (e being a unit 3-vector in the spatial direction of the velocity.)

If you want an answer entirely in trig-angles and hyperbolic-angles, write e in spherical polar coordinates:

\textbf{e} = \left[ \begin{array}{c}
\cos \phi \sin \theta \\\
\sin \phi \sin \theta \\\
\cos\theta
\end{array} \right]


and expand the matrix!

[Mentz114]

isn't that a boost in the -x direction?

Only for retardyons moving at sub-light speeds:wink:.

You're right, sinh(beta) is negative for beta < 1.