A Morse theory for massive particles and photons in general relativity

@article{Giannoni1999AMT,
  title={A Morse theory for massive particles and photons in general relativity},
  author={Fabio Giannoni and Antonio Masiello and Paolo Piccione},
  journal={Journal of Geometry and Physics},
  year={1999},
  volume={35},
  pages={1-34}
}
View PDF on arXiv
14 Citations
Background Citations
3
Methods Citations
3

14 Citations

Gravitational lensing in general relativity via bifurcation theory

In a Lorentzian manifold, conjugate points along a lightlike are endpoints of homotopies of lightlike geodesics, up to first order infinitesimals. When studying phenomena on a very large scale in…

Applications of Calculus of Variations to General Relativity

We present some global results on Lorentzian geometry obtained by using global variational methods. In particular some results on the geodesic connectedness of Lorentzian manifolds and on the…

On the Finiteness of Light Rays Between a Source and an Observer on Conformally Stationary Space-Times

We use a convexity condition to prove the finiteness of the number of light rays joining a pointlike source with a pointlike observer in a stationary relativistic spacetime. The result is extended to…

Existence and multiplicity results for massive particles trajectories in a universe with boundary

In an open time-convex region Λ of a strongly causal Lorentzian manifold (M,g), we consider an event p and a timelike, injective curve γ. We look for geodesics connecting p and γ in Λ and satisfying…

Global variational methods in general relativity with applications to gravitational lensing

In this note we give an idea of two of the more important theories concerning global variational methods, namely Ljusternik‐Schnirelmann Theory and Morse Theory. Thanks to them it is possible, in…

A note on the boundary of a static Lorentzian manifold

Functional regularity properties for light rays in general relativity

In this note we consider some functionals restricted to the nonholonomic constraint satisfied by lightlike curves in Lorentzian manifolds. These functionals are related to the arrival time map, whose…

Genericity of Nondegeneracy for Light Rays in Stationary Spacetimes

Given a Lorentzian manifold (M, g), an event p and an observer U in M, then p and U are light conjugate if there exists a lightlike geodesic γ : [0, 1] → M joining p and U whose endpoints are…

Time Extremizing Trajectories of Massive and Massless Objects in General Relativity

This is a review article about recent results concerning one-dirnensional variational problems in Lorentzian geometry see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. We will discuss from a mathematical…

Some Variational Problems in Semi-Riemannian Geometry

In this contribution we are concerned with global properties of geodesics on semi-Riemannian manifolds obtained by studying the variational properties of the action functional. Applications to…

References

SHOWING 1-10 OF 32 REFERENCES

On a Fermat principle in general relativity. A Ljusternik-Schnirelmann theory for light rays

SummaryIn this paper existence and multiplicity results for lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds are proved under intrinsic assumptions. Such…

On Fermat's principle in general relativity. I. The general case

The following version of Fermat's principle is proven to hold on an arbitrary Lorentzian manifold (i.e., without any kind of symmetry or causality condition being required): Among all lightlike…

AN APPLICATION OF THE TOPOLOGICAL DEGREE TO GRAVITATIONAL LENSES

In this letter we provide a new proof of a general theorem on gravitational lenses, first proven by Burke (1981) for the special case of thin lenses. The theorem states that a transparent…

Fermat Principle in Arbitrary Gravitational Fields

The Fermat principle is reviewed and used to derive the zigzag path constructed for massive and massless particles in order to determine if these paths are a suitable approximation to the first order…

Variational methods in Lorentzian geometry

Appliies variational methods and critical point theory on infinite dimenstional manifolds to some problems in Lorentzian geometry which have a variational nature, such as existence and multiplicity…

A Variational Theory for Light Rays in Stably Causal Lorentzian Manifolds: Regularity and Multiplicity Results

Abstract:This paper is dedicated to the study of light rays joining an event p with a timelike curve γ in a light–convex subset &\Lambda; of a stably causal Lorentzian manifold . We set up a…

Global Lorentzian Geometry

Introduction - Riemannian themes in Lorentzian geometry connections and curvature Lorentzian manifolds and causality Lorentzian distance examples of space-times completness and extendibility…

Multiplane gravitational lensing. I. Morse theory and image counting

The image counting problem for gravitational lensing by general matter deflectors distributed over finitely many lens planes is considered. Counting formulas and lower bounds are found via Morse…

A Morse theory for light rays on stably causal lorentzian manifolds

Dans cet article, nous presentons une theorie de Morse pour les rayons lumineux joignant un evenement p a une courbe γ de genre temps dans un espace-temps stablement causal avec bords. Quelques…

Conjugate points on spacelike geodesics or pseudo-self-adjoint Morse-Sturm-Liouville systems

This paper develops the basic theory of conjugate points along geodesies in manifolds with indefinite metric; equivalently, that of conjugate points for Morse-Sturm-Liouville systems which are…