## Proceedings of the International Conference on Geometry, Integrability and Quantization

### On the Geometry Induced by Lorentz Transformations in Pseudo-Euclidean Spaces

Abraham Ungar

#### Abstract

The Lorentz transformations of order $(m,n)$ in pseudo-Euclidean spaces with indefinite inner product of signature $(m,n)$ are extended in this work from $m=1$ and $n\ge1$ to all $m,n\ge1$. A parametric realization of the Lorentz transformation group of any order $(m,n)$ is presented, giving rise to generalized gyrogroups and gyrovector spaces called bi-gyrogroups and bi-gyrovector spaces. The latter, in turn, form the setting for generalized analytic hyperbolic geometry that underlies generalized balls called eigenballs.

#### Article information

Dates
First available in Project Euclid: 15 December 2015

https://projecteuclid.org/ euclid.pgiq/1450194168

Digital Object Identifier
doi:10.7546/giq-17-2016-360-368

Mathematical Reviews number (MathSciNet)
MR3445441

Zentralblatt MATH identifier
1346.83006

#### Citation

Ungar, Abraham. On the Geometry Induced by Lorentz Transformations in Pseudo-Euclidean Spaces. Proceedings of the Seventeenth International Conference on Geometry, Integrability and Quantization, 360--368, Avangard Prima, Sofia, Bulgaria, 2016. doi:10.7546/giq-17-2016-360-368. https://projecteuclid.org/euclid.pgiq/1450194168