275 research outputs found
Relativistic Chasles' theorem and the conjugacy classes of the inhomogeneous Lorentz group
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 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.Comment: Latex2e, 49 pages. v2: few typos correcte
The dual horospherical Radon transform for polynomials, Mosc
Abstract. Let X = G/K be a semisimple symmetric space of noncompact type. A horosphere in X is an orbit of a maximal unipotent subgroup of G. The set Hor X of all horospheres is a homogeneous space of G. The horospherical Radom transform suggested by I. M. Gelfand and M. I. Graev in 1959 takes any function Ď• on X to a function on Hor X obtained by integrating Ď• over horospheres. We explicitly describe the dual transform in terms of its action on polynomial functions on Hor X
Future asymptotic expansions of Bianchi VIII vacuum metrics
Bianchi VIII vacuum solutions to Einstein's equations are causally
geodesically complete to the future, given an appropriate time orientation, and
the objective of this article is to analyze the asymptotic behaviour of
solutions in this time direction. For the Bianchi class A spacetimes, there is
a formulation of the field equations that was presented in an article by
Wainwright and Hsu, and in a previous article we analyzed the asymptotic
behaviour of solutions in these variables. One objective of this paper is to
give an asymptotic expansion for the metric. Furthermore, we relate this
expansion to the topology of the compactified spatial hypersurfaces of
homogeneity. The compactified spatial hypersurfaces have the topology of
Seifert fibred spaces and we prove that in the case of NUT Bianchi VIII
spacetimes, the length of a circle fibre converges to a positive constant but
that in the case of general Bianchi VIII solutions, the length tends to
infinity at a rate we determine.Comment: 50 pages, no figures. Erronous definition of Seifert fibred spaces
correcte
Simple Space-Time Symmetries: Generalizing Conformal Field Theory
We study simple space-time symmetry groups G which act on a space-time
manifold M=G/H which admits a G-invariant global causal structure. We classify
pairs (G,M) which share the following additional properties of conformal field
theory: 1) The stability subgroup H of a point in M is the identity component
of a parabolic subgroup of G, implying factorization H=MAN, where M generalizes
Lorentz transformations, A dilatations, and N special conformal
transformations. 2) special conformal transformations in N act trivially on
tangent vectors to the space-time manifold M. The allowed simple Lie groups G
are the universal coverings of SU(m,m), SO(2,D), Sp(l,R), SO*(4n) and E_7(-25)
and H are particular maximal parabolic subgroups. They coincide with the groups
of fractional linear transformations of Euklidean Jordan algebras whose use as
generalizations of Minkowski space time was advocated by Gunaydin. All these
groups G admit positive energy representations. It will also be shown that the
classical conformal groups SO(2,D) are the only allowed groups which possess a
time reflection automorphism; in all other cases space-time has an intrinsic
chiral structure.Comment: 37 pages, 4 Table
A holomorphic representation of the Jacobi algebra
A representation of the Jacobi algebra by first order differential operators with polynomial
coefficients on the manifold is presented. The
Hilbert space of holomorphic functions on which the holomorphic first order
differential operators with polynomials coefficients act is constructed.Comment: 34 pages, corrected typos in accord with the printed version and the
Errata in Rev. Math. Phys. Vol. 24, No. 10 (2012) 1292001 (2 pages) DOI:
10.1142/S0129055X12920018, references update
On Non-Abelian Symplectic Cutting
We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact
groups. By using a degeneration based on the Vinberg monoid we give, in good
cases, a global quotient description of a surgery construction introduced by
Woodward and Meinrenken, and show it can be interpreted in algebro-geometric
terms. A key ingredient is the `universal cut' of the cotangent bundle of the
group itself, which is identified with a moduli space of framed bundles on
chains of projective lines recently introduced by the authors.Comment: Various edits made, to appear in Transformation Groups. 28 pages, 8
figure
Causal structures and causal boundaries
We give an up-to-date perspective with a general overview of the theory of
causal properties, the derived causal structures, their classification and
applications, and the definition and construction of causal boundaries and of
causal symmetries, mostly for Lorentzian manifolds but also in more abstract
settings.Comment: Final version. To appear in Classical and Quantum Gravit
Signatures in Shape Analysis: an Efficient Approach to Motion Identification
Signatures provide a succinct description of certain features of paths in a
reparametrization invariant way. We propose a method for classifying shapes
based on signatures, and compare it to current approaches based on the SRV
transform and dynamic programming.Comment: 7 pages, 3 figures. Conference paper for Geometric Science of
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