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Feb 2013

Volume 54, Issue 2, Articles (02xxxx)

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J. Math. Phys. 54, 021506 (2013); http://dx.doi.org/10.1063/1.4790887 (24 pages)

Gui-Qiang Chen, Vaibhav Kukreja, and Hairong Yuan
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back to top General Relativity and Gravitation

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

E. Minguzzi

J. Math. Phys. 54, 022501 (2013); http://dx.doi.org/10.1063/1.4789950 (38 pages)

Online Publication Date: 15 February 2013

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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.
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03.30.+p Special relativity
02.20.Sv Lie algebras of Lie groups
02.40.Dr Euclidean and projective geometries

Invariant classification of vacuum pp-waves

R. Milson, D. McNutt, and A. Coley

J. Math. Phys. 54, 022502 (2013); http://dx.doi.org/10.1063/1.4791691 (29 pages)

Online Publication Date: 20 February 2013

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We solve the equivalence problem for vacuum pp-wave spacetimes by employing the Karlhede algorithm. Our main result is a suite of Cartan invariants that allows for the complete invariant classification of the vacuum pp-waves. In particular, we derive the invariant characterization of the G2 and G3 sub-classes in terms of these invariants. It is known [J. M. Collins, “The Karlhede classification of type N vacuum spacetimes,” Class. Quantum Grav. 8, 1859–1869 (1991)10.1088/0264-9381/8/10/011] that the invariant classification of vacuum pp-waves requires at most the fourth order covariant derivative of the curvature tensor, but no specific examples requiring the fourth order were known. Using our comprehensive classification, we prove that the q ⩽ 4 bound is sharp and explicitly describe all such maximal order solutions.
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04.20.Jb Exact solutions
02.10.Ud Linear algebra
04.20.Gz Spacetime topology, causal structure, spinor structure
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