176 research outputs found
The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric
No full textIn a series of recent papers by the author and collaborators, the relations existing between
the metric properties of Randers spaces and the conformal geometry of
stationary Lorentzian manifolds were discovered and investigated. These
relations were called Stationary-to-Randers Correspondence (SRC).
In this paper we focus on one aspect of SRC, the equality between the
index of a geodesic in a Randers space and that of its lightlike lift in
the associated conformal stationary spacetime. Moreover we make some
remarks about regularity of the energy functional of a Finsler metric on
the infinite dimensional manifold of curves connecting two points, in
connection with infinite dimensional techniques in Morse Theory
Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness
No full textBy using stationary-to-Randers correspondence (SRC, see Caponio et al.
in Rev Mat Iberoamericana 27:919–952, 2011), a characterization of light and time-
convexity of the boundary of a region of a standard stationary (n + 1)-spacetime is
obtained, in terms of the convexity of the boundary of a domain in a Finsler n or
(n + 1)-space of Randers type. The latter convexity is analyzed in depth and, as a
consequence, the causal simplicity and the existence of causal geodesics confined in
the region and connecting a point to a stationary line are characterized. Applications to
asymptotically flat spacetimes include the light-convexity of hypersurfaces S n−1 (r ) ×
R, where S n−1 (r ) is a sphere of large radius in a spacelike section of an end, as well
as the characterization of their time-convexity with natural physical interpretations.
The lens effect of both light rays and freely falling massive particles with a finite
lifetime, (i.e., the multiplicity of such connecting curves) is characterized in terms of
the focalization of the geodesics in the underlying Randers manifolds
An intrinsic Fermat principle on stationary Lorentzian manifolds and applications
No full textAbstractIn this paper a Fermat principle for Lorentzian manifold endowed with a timelike Killing vector field is formulated. This principle is applied to obtain existence and multiplicity results on the number of light rays joining an event with an integral curve of the Killing vector field
Time-like solutions to the Lorentz force equation in time-dependent electromagnetic and gravitational fields
No full textAbstractWe find existence and multiplicity results for time-like spatially periodic trajectories of massive particles carrying an electric charge q and subjected to time-dependent gravitational and electromagnetic fields. Such trajectories are obtained by projecting, on the base space–time, time-like geodesics with respect to a suitable Kaluza–Klein metric
On the Analyticity of Static Solutions of a Field Equation in Finsler Gravity
No full textIt is well-known that static vacuum solutions of Einstein equations are analytic in suitable coordinates. We ask here for an extension of this result in the context of Finsler gravity. We consider Finsler spacetimes that retain several properties of static Lorentzian spacetimes, are Berwald and have vanishing Ricci scalar
A note on the Sagnac effect in general relativity as a Finslerian effect
No full textThe geometry of the Sagnac effect in a stationary region of a spacetime is
reviewed with the aim of emphasizing the role of asymmetry of a Finsler metric
defined on a spacelike hypersurface associated to a stationary splitting and
related to future-pointing null geodesics of the spacetime. We show also that
an analogous asymmetry comes into play in the Sagnac effect for timelike
geodesics.Comment: AMSLaTeX, 6 pages; v3: a misprint in Eq. (3) and in the equation at
the end of the paper corrected; these two corrections are not included in the
published versio
Standard static Finsler spacetimes
No full textWe introduce the notion of a standard static Finsler spacetime RxM where the base M is a Finsler manifold. We prove some results which connect causality with the Finslerian geometry of the base extending analogous ones for static and stationary Lorentzian spacetimes
On Finsler spacetimes with a timelike Killing vector field
No full textWe study Finsler spacetimes and Killing vector fields taking care of the fact that the generalised metric tensor associated to the Lorentz–Finsler function L is in general well defined only on a subset of the slit tangent bundle. We then introduce a new class of Finsler spacetimes endowed with a timelike Killing vector field that we call stationary splitting Finsler spacetimes. We characterize when a Finsler spacetime with a timelike Killing vector field is locally a stationary splitting. Finally, we show that the causal structure of a stationary splitting is the same of one of two Finslerian static spacetimes naturally associated to the stationary splitting
Trajectories of Affine Control Systems and Geodesics of a Spacetime with a Causal Killing Vector Field (vol 29, pg 1409, 2023)
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