8 research outputs found
Deformation and Contraction of Symmetries in Special Relativity
This dissertation gives an account of the fundamental principles underlying two conceptionally different ways of embedding Special Relativity into a wider context. Both of them root in the attempt to explore the full scope of the Relativity Postulate. The first approach uses Lie algebraic analysis alone, but already yields a whole range of alternative kinematics that are all in a quantifiable sense near to those in Special Relativity, while being rather far away in a qualitative way. The corresponding models for spacetime are seen to be four-dimensional versions of the prototypical planar geometries associated with the work of Cayley and Klein. The close relationship between algebraic and geometric methods displayed by these considerations is being substantialized in terms of light-like spacetime extensions. The second direction of departures from Special Relativity stresses and develops the algebraic view on spacetime by considering Hopf instead of Lie algebras as candidates for the description of kinematical transformations and hence spacetime symmetry. This approach is motivated by the belief in the existence of a quantum theory of gravity, and the assumption that such manifests itself in nonlinear modifications of the laws of Special Relativity at length scales comparable to the Planck length. The twofold character of this work, and the presentation of an example for the fully geometric character of a specific Hopf algebraic deformation of the PoincareI algebra, enable a conclusion that speculates on a possible relationship between the two developed viewpoints via the technique of nonlinear realizations. A non-perturbative approach to the latter is given which generalizes to all the considered geometries
Hamilton geometry: Phase space geometry from modified dispersion relations
We describe the Hamilton geometry of the phase space of particles whose
motion is characterised by general dispersion relations. In this framework
spacetime and momentum space are naturally curved and intertwined, allowing for
a simultaneous description of both spacetime curvature and non-trivial momentum
space geometry. We consider as explicit examples two models for Planck-scale
modified dispersion relations, inspired from the -de Sitter and
-Poincar\'e quantum groups. In the first case we find the expressions
for the momentum and position dependent curvature of spacetime and momentum
space, while for the second case the manifold is flat and only the momentum
space possesses a nonzero, momentum dependent curvature. In contrast, for a
dispersion relation that is induced by a spacetime metric, as in General
Relativity, the Hamilton geometry yields a flat momentum space and the usual
curved spacetime geometry with only position dependent geometric objects.Comment: 32 pages, section on quantisation of the theory added, comments on
additin of momenta on curved momentum spaces extende
Hamilton Geometry - Phase Space Geometry from Modified Dispersion Relations
Quantum gravity phenomenology suggests an effective modification of the
general relativistic dispersion relation of freely falling point particles
caused by an underlying theory of quantum gravity. Here we analyse the
consequences of modifications of the general relativistic dispersion on the
geometry of spacetime in the language of Hamilton geometry. The dispersion
relation is interpreted as the Hamiltonian which determines the motion of point
particles. It is a function on the cotangent bundle of spacetime, i.e. on phase
space, and determines the geometry of phase space completely, in a similar way
as the metric determines the geometry of spacetime in general relativity. After
a review of the general Hamilton geometry of phase space we discuss two
examples. The phase space geometry of the metric Hamiltonian
and the phase space geometry of the first order q-de
Sitter dispersion relation of the form which is suggested from quantum gravity phenomenology. We
will see that for the metric Hamiltonian the geometry of phase space is
equivalent to the standard metric spacetime geometry from general relativity.
For the q-de Sitter Hamiltonian the Hamilton equations of motion for
point particles do not become autoparallels but contain a force term, the
momentum space part of phase space is curved and the curvature of spacetime
becomes momentum dependent.Comment: 6 page
Planck-scale-modified dispersion relations in homogeneous and isotropic spacetimes
The covariant understanding of dispersion relations as level sets of Hamilton functions on phase space enables us to derive the most general dispersion relation compatible with homogeneous and isotropic spacetimes. We use this concept to present a Planck-scale deformation of the Hamiltonian of a particle in Friedman-Lemaître-Robertson-Walker (FLRW) geometry that is locally identical to the κ-Poincaré dispersion relation, in the same way as the dispersion relation of point particles in general relativity is locally identical to the one valid in special relativity. Studying the motion of particles subject to such a Hamiltonian, we derive the redshift and lateshift as observable consequences of the Planck-scale deformed FLRW universe. © 2017 American Physical Society
Deformation und Kontraktion von Symmetrien in der Speziellen Relativitätstheorie
This dissertation gives an account of the fundamental principles underlying two conceptionally different ways of embedding Special Relativity into a wider context. Both of them root in the attempt to explore the full scope of the Relativity Postulate. The first approach uses Lie algebraic analysis alone, but already yields a whole range of alternative kinematics that are all in a quantifiable sense near to those in Special Relativity, while being rather far away in a qualitative way. The corresponding models for spacetime are seen to be four-dimensional versions of the prototypical planar geometries associated with the work of Cayley and Klein. The close relationship between algebraic and geometric methods displayed by these considerations is being substantialized in terms of light-like spacetime extensions. The second direction of departures from Special Relativity stresses and develops the algebraic view on spacetime by considering Hopf instead of Lie algebras as candidates for the description of kinematical transformations and hence spacetime symmetry. This approach is motivated by the belief in the existence of a quantum theory of gravity, and the assumption that such manifests itself in nonlinear modifications of the laws of Special Relativity at length scales comparable to the Planck length. The twofold character of this work, and the presentation of an example for the fully geometric character of a specific Hopf algebraic deformation of the PoincareI algebra, enable a conclusion that speculates on a possible relationship between the two developed viewpoints via the technique of nonlinear realizations. A non-perturbative approach to the latter is given which generalizes to all the considered geometries