Hyperbolic angle

**touqra** · Jun16-09, 04:15 PM

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** · Jun16-09, 05:59 PM

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

**Peeter** · Jun16-09, 06:08 PM

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

**touqra** · Jun16-09, 06:57 PM

Originally Posted by Mentz114

This is a boost in the x direction with velocity $LaTeX Code: \\beta$
$LaTeX Code: \\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** · Jun16-09, 08:11 PM

For three dimensions (but without hyperbolic angles) see Wikipedia. Scroll down to where it says "More generally for a boost in an arbitrary direction."

**Rasalhague** · Jun16-09, 08:49 PM

Originally Posted by Peeter

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** · Jun16-09, 09:51 PM

Under the convention that a 4-vector is written as

$LaTeX Code: \\left[ \\begin{array}{c} ct \\\\\\ \\textbf{x} \\end{array} \\right] $

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

$LaTeX Code: \\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 $LaTeX Code: \\textbf{e} c \\tanh \\psi$ 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:

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

and expand the matrix!

**Mentz114** · Jun17-09, 09:03 AM

Originally Posted by Peeter

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

Only for retardyons moving at sub-light speeds

.

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