10 research outputs found
Finsler geodesics in the presence of a convex function and their applications
We obtain a result about the existence of only a finite number of geodesics
between two fixed non-conjugate points in a Finsler manifold endowed with a
convex function. We apply it to Randers and Zermelo metrics. As a by-product,
we also get a result about the finiteness of the number of lightlike and
timelike geodesics connecting an event to a line in a standard stationary
spacetime.Comment: 16 pages, AMSLaTex. v2 is a minor revision: title changed, references
updated, typos fixed; it matches the published version. This preprint and
arXiv:math/0702323v3 [math.DG] substitute arXiv:math/0702323v2 [math.DG
A note on the existence of standard splittings for conformally stationary spacetimes
Let be a spacetime which admits a complete timelike conformal Killing
vector field . We prove that splits globally as a standard
conformastationary spacetime with respect to if and only if is
distinguishing (and, thus causally continuous). Causal but non-distinguishing
spacetimes with complete stationary vector fields are also exhibited. For the
proof, the recently solved "folk problems" on smoothability of time functions
(moreover, the existence of a {\em temporal} function) are used.Comment: Metadata updated, 6 page
Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold
Given a Riemannian manifold M and a hypersurface H in M, it is well known
that infinitesimal convexity on a neighborhood of a point in H implies local
convexity. We show in this note that the same result holds in a semi-Riemannian
manifold. We make some remarks for the case when only timelike, null or
spacelike geodesics are involved. The notion of geometric convexity is also
reviewed and some applications to geodesic connectedness of an open subset of a
Lorentzian manifold are given.Comment: 14 pages, AMSLaTex, 2 figures. v2: typos fixed, added one reference
and several comments, statement of last proposition correcte
On the definition and examples of cones and finsler spacetimes
The authors warmly acknowledge Professor Daniel Azagra (Universidad Complutense, Madrid) his advise on approximation of convex functions as well as Profs. Kostelecky (Indiana University), Fuster (University of Technology, Eindhoven), Stavrinos (University of Athens), Pfeifer (University of Tartu), Perlick (University of Bremen) and Makhmali (Institute of Mathematics, Warsaw) their comments on a preliminary version of the article. The careful revision by the referee is also acknowledged. This work is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Region de Murcia, Spain, by Fundacion Seneca, Science and Technology Agency of the Region de Murcia. MAJ was partially supported by MINECO/FEDER project reference MTM2015-65430-P and Fundacion Seneca project reference 19901/GERM/15, Spain and MS by Spanish MINECO/ERDF project reference MTM2016-78807-C2-1-P.A systematic study of (smooth, strong) cone structures C and Lorentz–Finsler metrics L is carried out. As a link between both notions, cone triples (Ω,T,F), where Ω (resp. T) is a 1-form (resp. vector field) with Ω(T)≡1 and F, a Finsler metric on ker(Ω), are introduced. Explicit descriptions of all the Finsler spacetimes are given, paying special attention to stationary and static ones, as well as to issues related to differentiability. In particular, cone structures C are bijectively associated with classes of anisotropically conformal metrics L, and the notion of cone geodesic is introduced consistently with both structures. As a non-relativistic application, the time-dependent Zermelo navigation problem is posed rigorously, and its general solution is provided.MINECO/FEDER project, Spain
MTM2015-65430-PFundacion Seneca
19901/GERM/15Spanish MINECO/ERDF project
MTM2016-78807-C2-1-
The causal structure of spacetime is a parameterized Randers geometry
There is a by now well-established isomorphism between stationary
4-dimensional spacetimes and 3-dimensional purely spatial Randers geometries -
these Randers geometries being a particular case of the more general class of
3-dimensional Finsler geometries. We point out that in stably causal
spacetimes, by using the (time-dependent) ADM decomposition, this result can be
extended to general non-stationary spacetimes - the causal structure (conformal
structure) of the full spacetime is completely encoded in a parameterized
(time-dependent) class of Randers spaces, which can then be used to define a
Fermat principle, and also to reconstruct the null cones and causal structure.Comment: 8 page
On the energy functional on Finsler manifolds and applications to stationary spacetimes
In this paper we first study some global properties of the energy functional
on a non-reversible Finsler manifold. In particular we present a fully detailed
proof of the Palais--Smale condition under the completeness of the Finsler
metric. Moreover we define a Finsler metric of Randers type, which we call
Fermat metric, associated to a conformally standard stationary spacetime. We
shall study the influence of the Fermat metric on the causal properties of the
spacetime, mainly the global hyperbolicity. Moreover we study the relations
between the energy functional of the Fermat metric and the Fermat principle for
the light rays in the spacetime. This allows us to obtain existence and
multiplicity results for light rays, using the Finsler theory. Finally the case
of timelike geodesics with fixed energy is considered.Comment: 23 pages, AMSLaTeX. v4 matches the published versio
Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness
By using Stationary-to-Randers correspondence (SRC), 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 stationary hypersurfaces which
project in a spacelike section of an end onto a sphere of large radius, 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.Comment: AMSLaTex, 41 pages. v2 is a major revision: new discussions on
physical applicability of the results, especially to asymptotically flat
spacetimes; references adde