# Lorentz boost and equivalence with 3d hyperbolic rotations

1. May 4, 2010

### chwie

I was thinking that if i have for example a boost in the direction of x, then the boost should be equivalent to an hyperbolic rotation of the y and z axes in the other direction. I don't know if it's true or not. Then I want to know if somebody knows this result or why is false?

I was thinking also that is possible to make an homomorphism between some subgroup of the space rotations by complex angle (six parameter group) and some subgroup of the Lorentz transform (six parameter group). Also i don't know if that homomorphism is possible and I'm too lazy like to try to do the proof it myself. If somebody knows why is impossible to make such homomorphism or if there is some theorem that proof this, please let me know. Thanks in advance.

2. May 4, 2010

Are you talking about something like the http://en.wikipedia.org/wiki/Algebra_of_physical_space" [Broken]

Last edited by a moderator: May 4, 2017
3. May 4, 2010

### chwie

Nope, that is the homomorphism between 2x2 unimodular matrix and the restricted Lorentz group, but thanks for the help.

Well i was myself trying to proof about the equivalence of the boost and the 3d hyperbolic rotations, but the problem is to find the hyperbolic rotation that leaves the magnitude of the z and y vector invariant, and can be done using imaginaries terms, but the physics is not equivalent, then i was wrong. I don't know why I thought was possible, in an 4D euclidean space is possible to do similar relations, but because of the pseudo-euclidean metric is not possible in this case.

Also about the Lorentz group isomorphism and the rotation with complex angle is another way to said the representation of the pseudo-orthogonal matrix in four dimension. Then that is the subgroup of the rotations with complex angle which is isomorphic with the Lorentz group. Which is the most common representation. I was hoping in finding a more general one, but was not possible.