Lightlike Geodesics Research Articles

Generalized Dirac Operators on Lorentzian Manifolds and Propagation of Singularities

We survey the correct definition of a generalized Dirac operator on a Space-Time and the classicla result about propagation of singularities. This says that light travels along light-like geodesics. Finally we show this is also true for generalized Dirac operators.

Geodesic motion in Hořava-Lifshitz gravity

We propose an action for a free particle in Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity based on foliation preserving diffeomorphisms. The action reduces to the usual relativistic action in the low-energy limit and allows for subluminal and superluminal motions with upper and lower bounds on velocity, respectively. We find that deviation from general relativity is governed by a position-dependent coupling constant which also depends on the mass of the particle. As a result, lightlike geodesics are not affected whereas massive particles follow geodesics that become mass-dependent, hence the equivalence principle is violated. We make an exact study for geodesics in flat space and a qualitative analysis for those in a spherically symmetric curved background.

GeodesicViewer – A tool for exploring geodesics in the theory of relativity

The GeodesicViewer realizes exocentric two- and three-dimensional illustrations of lightlike and timelike geodesics in the general theory of relativity. By means of an intuitive graphical user interface, all parameters of a spacetime as well as the initial conditions of the geodesics can be modified interactively. New version program summary Program title: GeodesicViewer Catalogue identifier: AEFP_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEFP_v2_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 76 202 No. of bytes in distributed program, including test data, etc.: 1 722 290 Distribution format: tar.gz Programming language: C++, OpenGL Computer: All platforms with a C++ compiler, Qt, OpenGL Operating system: Linux, Mac OS X, Windows RAM: 24 MBytes Classification: 1.5 External routines: • Motion4D (included in the package) • Gnu Scientific Library (GSL) ( http://www.gnu.org/software/gsl/) • Qt ( http://qt.nokia.com/downloads) • OpenGL ( http://www.opengl.org/) Catalogue identifier of previous version: AEFP_v1_0 Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 413 Does the new version supersede the previous version?: Yes Nature of problem: Illustrate geodesics in four-dimensional Lorentzian spacetimes. Solution method: Integration of ordinary differential equations. 3D-Rendering via OpenGL. Reasons for new version: The main reason for the new version was to visualize the parallel transport of the Sachs legs and to show the influence of curved spacetime on a bundle of light rays as is realized in the new version of the Motion4D library ( http://cpc.cs.qub.ac.uk/summaries/AEEX_v3_0.html). Summary of revisions: • By choosing the new geodesic type “lightlike_sachs”, the parallel transport of the Sachs basis and the integration of the Jacobi equation can be visualized. • The 2D representation via Qwt was replaced by an OpenGL 2D implementation to speed up the visualization. • Viewing parameters can now be stored in a configuration file (.cfg). • Several new objects can be used in 3D and 2D representation. • Several predefined local tetrads can be choosen. • There are some minor modifications: new mouse control (rotate on sphere); line smoothing; current last point in coordinates is shown; mutual-coordinate representation extended; current cursor position in 2D; colors for 2D view. Running time: Interactive. The examples given take milliseconds.

Numerical modeling of a Global Navigation Satellite System in a general relativistic framework

In this article we model a Global Navigation Satellite System (GNSS) in a Schwarzschild space–time, as a first approximation of the relativistic geometry around the Earth. The closed time-like and scattering light-like geodesics are obtained analytically, describing respectively trajectories of satellites and electromagnetic signals. We implement an algorithm to calculate Schwarzschild coordinates of a GNSS user who receives proper times sent by four satellites, knowing their orbital parameters; the inverse procedure is implemented to check for consistency. The constellation of satellites therefore realizes a geocentric inertial reference system with no a priori realization of a terrestrial reference frame. We perform a simulation of position determination and show that the determination of the four coordinates with a 25–32 digit accuracy takes only around 60 ms. Effects of non-gravitational perturbations on positioning errors are assessed, and methods to reduce them are sketched. In particular, inter-links between satellites could greatly enhance stability and accuracy of the positioning system. Effects of gravitational perturbations are omitted in this paper in order to make a clearer comparison between the relativistic and non-relativistic scheme, but they will be included in subsequent work. We believe that the final algorithm will be a serious alternative to the usual post-Newtonian scheme.

Finsler geodesics in the presence of a convex function and their applications

In this paper, 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.

On non-normalizable orbits of isometric actions on Lorentz manifolds

For an isometric action of a Lie group G on a Lorentz manifold (M, g) we consider non-normalizable orbits, i.e. orbits which do not posses a G-invariant normal bundle. Orbits of this type are lightlike. It is shown, that such orbits contain lightlike homogeneous geodesics. Moreover, conditions are given, under which there exists a set of normalizable orbits having an open dense union.

Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime (M0×R,g) is equal to the index of its spatial projection as a geodesic of a Finsler metric F on M0 associated to (M0×R,g). Moreover we obtain the Morse relations of lightlike geodesics connecting a point p to a curve γ(s)=(q0,s) by using Morse theory on the Finsler manifold (M0,F). To this end, we prove a splitting lemma for the energy functional of a Finsler metric. Finally, we show that the reduction to Morse theory of a Finsler manifold can be done also for timelike geodesics.

Non-relativistic conformal symmetries and Newton–Cartan structures

This paper provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton–Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational ‘dynamical exponent’, z. The Schrödinger–Virasoro algebra of Henkel et al corresponds to z = 2. Viewed as projective Newton–Cartan symmetries, they yield, for timelike geodesics, the usual Schrödinger Lie algebra, for which z = 2. For lightlike geodesics, they yield, in turn, the Conformal Galilean Algebra (CGA) of Lukierski, Stichel and Zakrzewski (alias ‘’ of Henkel), with z = 1. Physical systems realizing these symmetries include, e.g. classical systems of massive and massless non-relativistic particles, and also hydrodynamics, as well as Galilean electromagnetism.

GeodesicViewer – A tool for exploring geodesics in the theory of relativity

The GeodesicViewer realizes exocentric two- and three-dimensional illustrations of lightlike and timelike geodesics in the general theory of relativity. By means of an intuitive graphical user interface, all parameters of a spacetime as well as the initial conditions of the geodesics can be modified interactively. This makes the GeodesicViewer a useful instrument for the exploration of geodesics in four-dimensional Lorentzian spacetimes. Program summary Program title: GeodesicViewer Catalogue identifier: AEFP_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEFP_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 168 868 No. of bytes in distributed program, including test data, etc.: 6 076 202 Distribution format: tar.gz Programming language: C++, Qt, Qwt, OpenGL Computer: All platforms with a C++ compiler, Qt, Qwt, OpenGL Operating system: Linux, Mac OS X RAM: 24 Mbytes Classification: 1.5 External routines: • Gnu Scientific Library (GSL) ( http://www.gnu.org/software/gsl/) • Motion4D (included in the package). The Motion4D library can also be downloaded from CPC. Catalogue identifier: AEEX • Qt ( http://qt.nokia.com/downloads) • Qwt ( http://qwt.sourceforge.net/) • OpenGL ( http://www.opengl.org/) Nature of problem: Illustrate geodesics in four-dimensional Lorentzian spacetimes. Solution method: Integration of ordinary differential equations. 3D-Rendering via OpenGL. Running time: Interactive. The examples given take milliseconds.

Pseudo-Riemannian geodesics and billiards

In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generalizations. We also introduce and study pseudo-Euclidean billiards, emphasizing their distinction from Euclidean ones. We present a pseudo-Euclidean version of the Clairaut theorem on geodesics on surfaces of revolution. We prove pseudo-Euclidean analogs of the Jacobi–Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.

Primary caustics and critical points behind a Kerr black hole

The primary optical caustic surface behind a Kerr black hole is a four-cusped tube displaced from the line of sight. We derive the caustic surface in the nearly asymptotic region far from the black hole through a Taylor expansion of the lightlike geodesics up to and including fourth-order terms in $m/b$ and $a/b$, where $m$ is the black hole mass, $a$ the spin, and $b$ the impact parameter. The corresponding critical locus in the observer's sky is elliptical and a pointlike source inside the caustics will be imaged as an Einstein cross. With regard to lensing near critical points, a Kerr lens is analogous to a circular lens perturbed by a dipole and a quadrupole potential. The caustic structure of the supermassive black hole in the Galactic center could be probed by lensing of low mass x-ray binaries in the galactic inner regions or by hot spots in the accretion disk.

Conservation laws and scattering for de Sitter classical particles

Starting from an intrinsic geometric characterization of de Sitter timelike and lightlike geodesics we give a new description of the conserved quantities associated with classical free particles on the de Sitter manifold. These quantities allow for a natural discussion of classical pointlike scattering and decay processes. We also provide an intrinsic definition of the energy of a classical de Sitter particle and discuss its different expressions in various local coordinate systems and their relations with earlier definitions found in the literature.

On the generalized Jacobi equation

The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to the coordinates, but not with respect to the velocities. The resulting equation has been studied by several authors in some detail for timelike geodesics in a Lorentzian manifold. Here we begin by briefly considering the generalized Jacobi equation on affine manifolds, without a metric; then we specify to lightlike geodesics in a Lorentzian manifold. We illustrate the latter case by considering particular lightlike geodesics (a) in Schwarzschild spacetime and (b) in a plane-wave spacetime.

Light-like homogeneous geodesics and the geodesic lemma for any signature

Light-like homogeneous geodesics and the geodesic lemma for any signature

Lightlike rays in stationary spacetimes with critical growth

The aim of this paper is to extend some previous results on the existence of lightlike geodesics joining a point to a line to the case of stationary Lorentzian manifolds whose metric coefficients have an optimal growth.

Analytical Kerr black hole lensing in the weak deflection limit

We present an analytical treatment of gravitational lensing by a Kerr black hole in the weak deflection limit. Lightlike geodesics are expanded as a Taylor series up to and including third-order terms in $m/b$ and $a/b$, where $m$ is the black hole mass, $a$ the angular momentum, and $b$ the impact parameter of the light ray. Positions and magnifications of individual images are computed with a perturbative analysis. At this order, the degeneracy with the translated Schwarzschild lens is broken. The critical curve is still a circle displaced from the black hole position in the equatorial direction and the corresponding caustic is pointlike. The degeneracy between the black hole spin and its inclination relative to the observer is broken through the angular coordinates of the perturbed images.

A Morse-theoretical analysis of gravitational lensing by a Kerr-Newman black hole

Consider, in the domain of outer communication M+ of a Kerr-Newman black hole, a point p (observation event) and a timelike curve γ (worldline of light source). Assume that γ (i) has no past endpoint, (ii) does not intersect the caustic of the past light cone of p, and (iii) goes neither to the horizon nor to infinity in the past. We prove that then for infinitely many positive integers k there is a past-pointing lightlike geodesic λk of (Morse) index k from p to γ, hence an observer at p sees infinitely many images of γ. Moreover, we demonstrate that all lightlike geodesics from an event to a timelike curve in M+ are confined to a certain spherical shell. Our characterization of this spherical shell shows that in the Kerr-Newman space-time the occurrence of infinitely many images is intimately related to the occurrence of centrifugal-plus-Coriolis force reversal.

Connecting Solutions of the Lorentz Force Equation do Exist

Recent results on the maximization of the charged-particle action I in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of I over a given causal homotopy class C of curves connecting two causally related events x_0 <= x_1. Action I is proved to admit a maximum on C, and also one in the adherence of each timelike homotopy class. Moreover, the maximum on C is timelike if C contains a timelike curve (and the degree of differentiability of all the elements is at least C^2). In particular, this last result yields a complete Avez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x_0 << x_1 can be connected by means of a timelike solution of the Lorentz force equation (LFE) corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m) in the non-exact case. As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied.

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.

Gravitational Lensing from a Spacetime Perspective.

The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others.