##
**Beyond the Einstein addition law and its gyroscopic Thomas precession. The theory of gyrogroups and gyrovector spaces.**
*(English)*
Zbl 0972.83002

Fundamental Theories of Physics. 117. Dordrecht: Kluwer Academic Publishers. xlii, 413 p. (2001).

Generalizing the addition rule for special relativistic 3-velocities, the author introduces an operator \(\text{gyr}\), and builds up a whole theory of gyrogroups and gyrovector spaces. He makes these things plausible by comparing with the Thomas precession, and then develops several theorems about gyrogroups.

From the space-time point of view, however, the \(3+1\)-decomposition is only secondary after fixing a system of reference; the primary notion is the 4-velocity. Nowadays, almost all papers on relativity write the velocity in the 4-velocity form, and there the addition rule of velocities is much simpler than in the 3-velocity picture. So I conclude that the gyrogroups may have some interest for themselves, but they are probably not very helpful in dealing with physical problems connected with the addition of velocities.

The reference list contains 230 carefully listed items; however, it contains several misprints.

From the space-time point of view, however, the \(3+1\)-decomposition is only secondary after fixing a system of reference; the primary notion is the 4-velocity. Nowadays, almost all papers on relativity write the velocity in the 4-velocity form, and there the addition rule of velocities is much simpler than in the 3-velocity picture. So I conclude that the gyrogroups may have some interest for themselves, but they are probably not very helpful in dealing with physical problems connected with the addition of velocities.

The reference list contains 230 carefully listed items; however, it contains several misprints.

Reviewer: Hans-Jürgen Schmidt (Potsdam)

### MSC:

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

83A05 | Special relativity |

51P05 | Classical or axiomatic geometry and physics |

83C10 | Equations of motion in general relativity and gravitational theory |

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |