Fermat Principle in Finsler Spacetimes

It is shown that, on a manifold with a Finsler metric of Lorentzian signature, the lightlike geodesics satisfy the following variational principle. Among all lightlike curves from a point (emission event) to a timelike curve (worldline of receiver), the lightlike geodesics make the arrival time stationary. Here ``arrival time'' refers to a parametrization of the timelike curve. This variational principle can be applied (i) to the vacuum light rays in an alternative spacetime theory, based on Finsler geometry, and (ii) to light rays in an anisotropic non-dispersive medium with a general-relativistic spacetime as background.Comment: 18 pages, submitted to Gen. Rel. Gra

A Characterisation of the Weylian Structure of Space-Time by Means of Low Velocity Tests

The compatibility axiom in Ehlers, Pirani and Schild's (EPS) constructive axiomatics of the space-time geometry that uses light rays and freely falling particles with high velocity, is replaced by several constructions with low velocity particles only. For that purpose we describe in a space-time with a conformal structure and an arbitrary path structure the radial acceleration, a Coriolis acceleration and the zig-zag construction. Each of these quantities give effects whose requirement to vanish can be taken as alternative version of the compatibility axiom of EPS. The procedural advantage lies in the fact, that one can make null-experiments and that one only needs low velocity particles to test the compatibility axiom. We show in addition that Perlick's standard clock can exist in a Weyl space only.Comment: to appear in Gen.Rel.Gra

Kerr black hole lensing for generic observers in the strong deflection limit

We generalize our previous work on gravitational lensing by a Kerr black hole in the strong deflection limit, removing the restriction to observers on the equatorial plane. Starting from the Schwarzschild solution and adding corrections up to the second order in the black hole spin, we perform a complete analytical study of the lens equation for relativistic images created by photons passing very close to a Kerr black hole. We find out that, to the lowest order, all observables (including shape and shift of the black hole shadow, caustic drift and size, images position and magnification) depend on the projection of the spin on a plane orthogonal to the line of sight. In order to break the degeneracy between the black hole spin and its inclination relative to the observer, it is necessary to push the expansion to higher orders. In terms of future VLBI observations, this implies that very accurate measures are needed to determine these two parameters separately.Comment: 17 pages, 4 figures, one section added, to appear on Physical Review

On Fermat's principle for causal curves in time oriented Finsler spacetimes

In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the secondvariation of the {\it time arrival functional} along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of the critical points of the time arrival functional is investigated and a Morse index theorem in the context of Finsler spacetime is presented.Comment: 20 pages, minor corrections, references adde

Possible potentials responsible for stable circular relativistic orbits

Bertrand's theorem in classical mechanics of the central force fields attracts us because of its predictive power. It categorically proves that there can only be two types of forces which can produce stable, circular orbits. In the present article an attempt has been made to generalize Bertrand's theorem to the central force problem of relativistic systems. The stability criterion for potentials which can produce stable, circular orbits in the relativistic central force problem has been deduced and a general solution of it is presented in the article. It is seen that the inverse square law passes the relativistic test but the kind of force required for simple harmonic motion does not. Special relativistic effects do not allow stable, circular orbits in presence of a force which is proportional to the negative of the displacement of the particle from the potential center.Comment: 11 pages, Latex fil

Can we distinguish between black holes and wormholes by their Einstein-ring systems?

For the last decade, the gravitational lensing in the strong gravitational field has been studied eagerly. It is well known that, for the lensing by a black hole, infinite number of Einstein rings are formed by the light rays which wind around the black hole nearly on the photon sphere, which are called relativistic Einstein rings. This is also the case for the lensing by a wormhole. In this paper, we study the Einstein ring and relativistic Einstein rings for the Schwarzschild black hole and the Ellis wormhole, the latter of which is an example of traversable wormholes of the Morris-Thorne class. Given the configuration of the gravitational lensing and the radii of the Einstein ring and relativistic Einstein rings, we can distinguish between a black hole and a wormhole in principle. We conclude that we can detect the relativistic Einstein rings by wormholes which have the radii of the throat $a\simeq 0.5$pc at a galactic center with the distance 10Mpc and which have $a\simeq 10$AU in our galaxy using by the most powerful modern instruments which have the resolution of $10^{-2}$arcsecond such as a 10-meter optical-infrared telescope. The black holes which make the Einstein rings of the same size as the ones by the wormholes are galactic supermassive black holes and the relativistic Einstein rings by the black holes are too small to measure at this moment. We may test some hypotheses of astrophysical wormholes by using the Einstein ring and relativistic Einstein rings in the future.Comment: 13 pages, 2 figures, minor changes from v

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

Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.Comment: 25 pages, 4 figures. Invited review submitted for special issue of General Relativity and Gravitatio

Applications of the Gauss-Bonnet theorem to gravitational lensing

In this geometrical approach to gravitational lensing theory, we apply the Gauss-Bonnet theorem to the optical metric of a lens, modelled as a static, spherically symmetric, perfect non-relativistic fluid, in the weak deflection limit. We find that the focusing of the light rays emerges here as a topological effect, and we introduce a new method to calculate the deflection angle from the Gaussian curvature of the optical metric. As examples, the Schwarzschild lens, the Plummer sphere and the singular isothermal sphere are discussed within this framework.Comment: 10 pages, 1 figure, IoP styl

Gravitational lensing in the Kerr-Randers optical geometry

A new geometric method to determine the deflection of light in the equatorial plane of the Kerr solution is presented, whose optical geometry is a surface with a Finsler metric of Randers type. Applying the Gauss-Bonnet theorem to a suitable osculating Riemannian manifold, adapted from a construction by Naz\i m, it is shown explicitly how the two leading terms of the asymptotic deflection angle of gravitational lensing can be found in this way.Comment: 7 pages, 1 figure. Accepted by Gen. Rel. Grav. Version 2: change of notation in sec.