Lorentz Transformation & Hyperbolic Angle in (2+1) Dimensions

[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 $\beta$
$$\left[ \begin{array}{ccc}<br /> \cosh(\beta) & \sinh(\beta) & 0 \\\<br /> \sinh(\beta) & \cosh(\beta) & 0 \\\<br /> 0 & 0 & 1 \end{array} \right]$$

 

[Peeter]

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

 

[touqra]

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

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. 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

$$\left[ \begin{array}{c}<br /> ct \\\<br /> \textbf{x} <br /> \end{array} \right]$$​

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

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

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:

$$\textbf{e} = \left[ \begin{array}{c}<br /> \cos \phi \sin \theta \\\<br /> \sin \phi \sin \theta \\\<br /> \cos\theta <br /> \end{array} \right]$$​

and expand the matrix!

 

[Mentz114]

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

Only for retardyons moving at sub-light speeds.

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