arXiv subjects

Christian Pfeifer

Publications and source records attributed to Christian Pfeifer.

86 records · Page 5Linked to original sources

arXiv2018

Finlser geometric extension of Einstein gravity

We present our Finsler spacetime formalism which extends the standard formulation of Finsler geometry to be applicable in physics. Finsler spacetimes are viable non-metric geometric backgrounds for physics; they guarantee well defined causality, the propagation of light on a non-trivial null structure, a clear notion of physical observers and the existence of physical field theories determining the geometry of space-time dynamically in terms of an extended gravitational field equation. Here we give the precise definition of Finsler spacetimes and the notion of well-defined observers, their measurements, the transformations between them and how to formulate action based field theories.

gr-qc↗

arXiv2017

Curved spacetimes with local $κ$-Poincaré dispersion relation

We use our previously developed identification of dispersion relations with Hamilton functions on phase space to locally implement the $κ$-Poincaré dispersion relation in the momentum spaces at each point of a generic curved spacetime. We use this general construction to build the most general Hamiltonian compatible with spherical symmetry and the Plank-scale-deformed one such that in the local frame it reproduces the $κ$-Poincaré dispersion relation. Specializing to Planck-scale-deformed Schwarzschild geometry, we find that the photon sphere around a black hole becomes a thick shell since photons of different energy will orbit the black hole on circular orbits at different altitudes. We also compute the redshift of a photon between different observers at rest, finding that there is a Planck-scale correction to the usual redshift only if the observers detecting the photon have different masses.

gr-qc↗

arXiv2017

Quantum Energy Inequalities in Pre-Metric Electrodynamics

Pre-metric electrodynamics is a covariant framework for electromagnetism with a general constitutive law. Its lightcone structure can be more complicated than that of Maxwell theory as is shown by the phenomenon of birefringence. We study the energy density of quantized pre-metric electrodynamics theories with linear constitutive laws admitting a single hyperbolicity double-cone and show that averages of the energy density along the worldlines of suitable observers obey a Quantum Energy Inequality (QEI) in states that satisfy a microlocal spectrum condition. The worldlines must meet two conditions: (a) the classical weak energy condition must hold along them, and (b) their velocity vectors have positive contractions with all positive frequency null covectors (we call such trajectories `subluminal'). After stating our general results, we explicitly quantize the electromagnetic potential in a translationally invariant uniaxial birefringent crystal. Since the propagation of light in such a crystal is governed by two nested lightcones, the theory shows features absent in ordinary (quantized) Maxwell electrodynamics. We then compute a QEI bound for worldlines of inertial `subluminal' observers, which generalizes known results from the Maxwell theory. Finally, it is shown that the QEIs fail along trajectories that have velocity vectors which are timelike with respect to only one of the lightcones.

math-ph↗

arXiv2017

Geodesics and the magnitude-redshift relation on cosmologically symmetric Finsler spacetimes

We discuss the geodesic motion of both massive test particles, following timelike geodesics, and light, following null geodesics, on Finsler spacetimes with cosmological symmetry. Using adapted coordinates on the tangent bundle of the spacetime manifold, we derive the general form of the geodesic equation. Further, we derive a complete set of constants of motion. As an application of these findings, we derive the magnitude-redshift relation for light propagating on a cosmologically symmetric Finsler background, both for a general Finsler spacetime and for particular examples, such as spacetimes equipped with Bogoslovsky and Randers length measures. Our results allow a confrontation of these geometries with observations of the magnitude and redshift of supernovae.

gr-qc↗

arXiv2016

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 Hamiltonian we derive the redshift and lateshift as observable consequences of the Planck-scale deformed FLRW universe.

gr-qc↗

arXiv2016

Electromagnetic Potential in Pre-Metric Electrodynamics: Causal Structure, Propagators and Quantization

An axiomatic approach to electrodynamics reveals that Maxwell electrodynamics is just one instance of a variety of theories for which the name electrodynamics is justified. They all have in common that their fundamental input are Maxwell's equations $\textrm{d} F = 0$ (or $F = \textrm{d} A$) and $\textrm{d} H = J$ and a constitutive law $H = # F$ which relates the field strength two-form $F$ and the excitation two-form $H$. A local and linear constitutive law defines what is called local and linear pre-metric electrodynamics whose best known application are the effective description of electrodynamics inside media including, e.g., birefringence. We analyze the classical theory of the electromagnetic potential $A$ before we use methods familiar from mathematical quantum field theory in curved spacetimes to quantize it in a locally covariant way. Our analysis of the classical theory contains the derivation of retarded and advanced propagators, the analysis of the causal structure on the basis of the constitutive law (instead of a metric) and a discussion of the classical phase space. This classical analysis sets the stage for the construction of the quantum field algebra and quantum states. Here one sees, among other things, that a microlocal spectrum condition can be formulated in this more general setting.

math-ph↗

arXiv2015

The tangent bundle exponential map and locally autoparallel coordinates for general connections with application to Finslerian geometries

We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinates known from metric geometry to Finsler geometric manifolds and it turns out that they are identical to the Douglas-Thomas normal coordinates introduced earlier. We expand the fundamental geometry function of a Finsler spacetime in these new coordinates and find that it is constant to quadratic order. The quadratic order term comes with the non-linear curvature of the manifold. From physics these coordinates may be interpretation as the realisation of an Einstein elevator in Finslerian spacetime geometries.

math-ph↗

arXiv2015

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 $q$-de Sitter and $κ$-Poincaré 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.

gr-qc↗

arXiv2014

Radar orthogonality and radar length in Finsler and metric spacetime geometry

The radar experiment connects the geometry of spacetime with an observers measurement of spatial length. We investigate the radar experiment on Finsler spacetimes which leads to a general definition of radar orthogonality and radar length. The directions radar orthogonal to an observer form the spatial equal time surface an observer experiences and the radar length is the physical length the observer associates to spatial objects. We demonstrate these concepts on a forth order polynomial Finsler spacetime geometry which may emerge from area metric or pre-metric linear electrodynamics or in quantum gravity phenomenology. In an explicit generalisation of Minkowski spacetime geometry we derive the deviation from the euclidean spatial length measure in an observers rest frame explicitly.

gr-qc↗

arXiv2013

Local spacetime effects on gyroscope systems

We give a precise theoretical description of initially aligned sets of orthogonal gyroscopes which are transported along different paths from some initial point to the same final point in spacetime. These gyroscope systems can be used to synchronize separated observers' spatial frames by free fall along timelike geodesics. We find that initially aligned gyroscope systems, or spatial frames, lose their synchronization due to the curvature of spacetime and their relative motion. On the basis of our results we propose a simple experiment which enables observers to determine locally whether their spacetime is described by a rotating Kerr or a non-rotating Schwarzschild metric.

gr-qc↗

arXiv2012

Finsler spacetimes and gravity

We consider the geometry of spacetime based on a non-metric, Finslerian, length measure, which, in terms of physics, represents a generalized clock. Our defnition of Finsler spacetimes ensure a well defined notion of causality, a precise description of observers and a geometric background for field theories. Moreover we present our Finsler geometric extension of the Einstein equations, which determine the geometry of Finsler spacetimes dynamically.

gr-qc↗

arXiv2012

Beyond the speed of light on Finsler spacetimes

As a prototypical massive field theory we study the scalar field on the recently introduced Finsler spacetimes. We show that particle excitations exist that propagate faster than the speed of light recognized as the boundary velocity of observers. This effect appears already in Finsler spacetime geometries with very small departures from Lorentzian metric geometry. It switches on for a sufficiently large ratio of the particle four-momentum and mass, and is the consequence of a modified version of the Coleman-Glashow velocity dispersion relation. The momentum dispersion relation on Finsler spacetimes is shown to be the same as on metric spacetimes, which differs from many quantum gravity models. If similar relations resulted for fermions on Finsler spacetimes, these generalized geometries could explain the potential observation of superluminal neutrinos claimed by the Opera Collaboration.

gr-qc↗

arXiv2012

Finsler geometric extension of Einstein gravity

We construct gravitational dynamics for Finsler spacetimes in terms of an action integral on the unit tangent bundle. These spacetimes are generalizations of Lorentzian metric manifolds which satisfy necessary causality properties. A coupling procedure for matter fields to Finsler gravity completes our new theory that consistently becomes equivalent to Einstein gravity in the limit of metric geometry. We provide a precise geometric definition of observers and their measurements, and show that the transformations by means of which different observers communicate form a groupoid that generalizes the usual Lorentz group. Moreover, we discuss the implementation of Finsler spacetime symmetries. We use our results to analyze a particular spacetime model that leads to Finsler geometric refinements of the linearized Schwarzschild solution.

gr-qc↗

arXiv2011

Causal structure and electrodynamics on Finsler spacetimes

We present a concise new definition of Finsler spacetimes that generalize Lorentzian metric manifolds and provide consistent backgrounds for physics. Extending standard mathematical constructions known from Finsler spaces we show that geometric objects like the Cartan non-linear connection and its curvature are well-defined almost everywhere on Finsler spacetimes, also on their null structure. This allows us to describe the complete causal structure in terms of timelike and null curves; these are essential to model physical observers and the propagation of light. We prove that the timelike directions form an open convex cone with null boundary as is the case in Lorentzian geometry. Moreover, we develop action integrals for physical field theories on Finsler spacetimes, and tools to deduce the corresponding equations of motion. These are applied to construct a theory of electrodynamics that confirms the claimed propagation of light along Finsler null geodesics.

gr-qc↗