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Relativistic Chasles' theorem and the conjugacy classes of the inhomogeneous Lorentz group

### Abstract

This work is devoted to the relativistic generalization of Chasles' theorem, namely, to the proof that every proper orthochronous isometry of Minkowski spacetime, which sends some point to its chronological future, is generated through the frame displacement of an observer which moves with constant acceleration and constant angular velocity. The acceleration and angular velocity can be chosen either aligned or perpendicular, and in the latter case the angular velocity can be chosen equal or smaller than the acceleration. We start reviewing the classical Euler's and Chasles' theorems both in the Lie algebra and group versions. We recall the relativistic generalization of Euler's theorem and observe that every (infinitesimal) transformation can be recovered from information of algebraic and geometric type, the former being identified with the conjugacy class and the latter with some additional geometric ingredients (the screw axis in the usual non-relativistic version). Then the proper orthochronous inhomogeneous Lorentz Lie group is studied in detail. We prove its exponentiality and identify a causal semigroup and the corresponding Lie cone. Through the identification of new Ad-invariants we classify the conjugacy classes, and show that those which admit a causal representative have special physical significance. These results imply a classification of the inequivalent Killing vector fields of Minkowski spacetime which we express through simple representatives. Finally, we arrive at the mentioned generalization of Chasles' theorem.

© 2013 American Institute of Physics

Received 29 July 2012
Accepted 11 January 2013
Published online 15 February 2013

Article outline:

I. INTRODUCTION
II. EULER'S AND CHASLES' THEOREMS
A. Infinitesimal (Lie algebra) formulation and screw product
III. THE LORENTZ GROUP
A. The Lie algebra and its orbits
B. Lorentzian extension of Euler's theorem
IV. THE INHOMOGENEOUS LORENTZ GROUP
A. The causal semigroup of *ISO*(1, 3)^{↑}
B. The Lie algebra and its interpretation
C. Exponentiality of *ISO* (1, 3)^{↑}
D. Ad-invariants and Lie algebra orbits
E. The Lie wedge
1. The strict inclusion exp *L*(*I*) *I* and the causal cone of *F*
F. The causal orbits
G. Lorentzian extension of Chasles' theorem
V. CONCLUSIONS

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2013-02-15

2016-02-11

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