Geodesics on a supermanifold and projective equivalence of superconnections

We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field associated with the connection. Our (super) geodesics possess the same properties as in the classical case: there exists a unique (super) geodesic satisfying a given initial condition and when the connection is metric, our supergeodesics coincide with the trajectories of a free particle with unit mass. Moreover, using our definition, we are able to establish Weyl’s characterization of projective equivalence in the super context: two torsion-free (super) connections define the same geodesics (up to reparametrizations) if and only if their difference tensor can be expressed by means of a (smooth, even, super) 1-form.     

Homogeneous geodesics in homogeneous Finsler spaces

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on the compact semi-simple Lie group is established. We introduce the notion of a naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous geodesics. In particular, we prove that the S-curvature vanishes along the homogeneous geodesics.     

Geodesic focusing and space-time topology

Given a space-timeM and a pointp inM, it is shown that, if the locus of first conjugate points ofp along future-directed null geodesics consists of a single point, thenM admits a compact (S ³) spacelike hypersurface. If in addition the null geodesics do not intersect before focusing, then, in a simply connected space-time, the spacelike hypersurface is a partial Cauchy surface.     

Structure of geodesics in the regular Hayward black hole space-time

The regular Hayward model describes a non-singular black hole space-time. By analyzing the behaviors of effective potential and solving the equation of orbital motion, we investigate the time-like and null geodesics in the regular Hayward black hole space-time. Through detailed analyses of corresponding effective potentials for massive particles and photons, all possible orbits are numerically simulated. The results show that there may exist four orbital types in the time-like geodesics structure: planetary orbits, circular orbits, escape orbits and absorbing orbits. In addition, when \(\ell \), a convenient encoding of the central energy density \(3/8\pi \ell ^{2}\), is 0.6M, and b is 3.9512M as a specific value of angular momentum, escape orbits exist only under \(b>3.9512M\). The precession direction is also associated with values of b. With \(b=3.70M\) the bound orbits shift clockwise but counter-clockwise with \(b=5.00M\) in the regular Hayward black hole space-time. We also find that the structure of null geodesics is simpler than that of time-like geodesics. There only exist three kinds of orbits (unstable circle orbits, escape orbits and absorbing orbits).     

The regularity of geodesics in impulsive pp-waves

We consider the geodesic equation in impulsive pp-wave space-times in Rosen form, where the metric is of Lipschitz regularity. We prove that the geodesics (in the sense of Carathéodory) are actually continuously differentiable, thereby rigorously justifying the ${\mathcal C}^1$ -matching procedure which has been used in the literature to explicitly derive the geodesics in space-times of this form.     

The Monge Problem for Distance Cost in Geodesic Spaces

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and d _L is a geodesic Borel distance which makes (X, d _L ) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by d _L . It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting d _L -cyclical monotonicity is not sufficient for optimality.     

Multiloop amplitudes for the bosonic Polyakov string

We provide a simple formula for multiloop amplitudes of the bosonic, closed oriented Polyakov string (in d = 26) as integrals over moduli space with respect to the Weil-Petersson measure. The integrand consists of Green functions and the determinants of laplacians acting on functions and vectors. We compute these determinants in terms of the lengths of the closed geodesics on the surface. They are finite and different from zero. The one on functions equals the derivative at 1 of the Selberg zeta function. A discussion of lengths of closed geodesics and coordinates for Teichmueller space is given.     

Busemann Functions and Infinite Geodesics in Two-Dimensional First-Passage Percolation

We study first-passage percolation on ${\mathbb{Z}^2}$ , where the edge weights are given by a translation-ergodic distribution, addressing questions related to existence and coalescence of infinite geodesics. Some of these were studied in the late 1990s by C. Newman and collaborators under strong assumptions on the limiting shape and weight distribution. In this paper we develop a framework for working with distributional limits of Busemann functions and use it to prove forms of Newman’s results under minimal assumptions. For instance, we show a form of coalescence of long finite geodesics in any deterministic direction. We also introduce a purely directional condition which replaces Newman’s global curvature condition and whose assumption we show implies the existence of directional geodesics. Without this condition, we prove existence of infinite geodesics which are directed in sectors. Last, we analyze distributional limits of geodesic graphs, proving almost-sure coalescence and nonexistence of infinite backward paths. This result relates to the conjecture of nonexistence of “bigeodesics.”     

Renormalisation group flow and geodesics in the O(N) model for large N

A metric is introduced on the space of parameters (couplings) describing the large N limit of the O(N) model in Euclidean space. The geometry associated with this metric is analysed in the particular case of the infinite volume limit in three dimensions and it is shown that the Ricci curvature diverges at the ultra-violet (Gaussian) fixed point but is finite and tends to constant negative curvature at the infra-red (Wilson-Fisher) fixed point. The renormalisation group flow is examined in terms of geodesics of the metric. The critical line of cross-over from the Wilson-Fisher fixed point to the Gaussian fixed point is shown to be a geodesic but all other renormalisation group trajectories, which are repulsed from the Gaussian fixed point in the ultraviolet, are not geodesics. The geodesic flow is interpreted in terms of a maximisation principle for the relative entropy.     

Traffic noise and the hyperbolic plane

We consider the problem of sound propagation in a wind. We note that the rays, as in the absence of a wind, are given by Fermat’s principle and show how to map them to the trajectories of a charged particle moving in a magnetic field on a curved space. For the specific case of sound propagating in a stratified atmosphere with a small wind speed, we show that the corresponding particle moves in a constant magnetic field on the hyperbolic plane. In this way, we give a simple ‘straightedge and compass’ method to estimate the intensity of sound upwind and downwind. We construct Mach envelopes for moving sources. Finally, we relate the problem to that of finding null geodesics in a squashed anti-de Sitter spacetime and discuss the SO(3,1)×R symmetry of the problem from this point of view.     

Geodesics in open universes

We present the geodesics on homogeneous and isotropic negatively curved spaces in a simple form suitable for application to cosmological problems. The pattern of geodesics translates into a pattern on the microwave background radiation. Generalizing, we discuss how the patterns in the microwave sky of anisotropic homogeneous universes can be predicted qualitatively by looking at the invariances that generate their three-geometries and their geodesics.     

The Jacobi map

This paper defines nth order Jacobi fields to be solutions to a second-order nonlinear differential equation defined by the Jacobi map. nth order Jacobi fields arise naturally as acceleration vector fields of geodesic variations. As a main theorem we prove necessity and sufficiency conditions for an nth order Jacobi field to be the acceleration vector field of a variation of geodesics normal to a submanifold. An m geodesic, m ≥ 2, is a smooth curve whose mth covariant derivative vanishes. We prove an index theorem giving bounds for the total m focal multiplicity along an m geodesic m normal to a submanifold in a flat manifold.     

Gravitational waves and Papapetrou metric in the six-dimensional treatment of gravitation

Six-dimensional treatment of gravitation based on the principle of simplicity to which there corresponds motion of particles with the speed of light in the Compton neighborhood of the three-dimensional space along the geodesics complying with the Fermat principle is given to the Papapetrou metric and gravitational waves. The envelope of the geodesics has the form of a tubular surface with the Compton transverse sizes in the additional subspace where the radius and speed of light vary along the tube. Gravitational waves, which are perturbations of these radii and speed of light, turn out to attenuate exponentially here. Their amplitudes are considered in the near-field zone of the rotator with n Maltese cross lobes and calculated at n = 4.     

On the motion of particles in covariant Ho?ava-Lifshitz gravity and the meaning of the A-field

We studied the low energy motion of particles in the general covariant version of Ho?ava-Lifshitz gravity proposed by Ho?ava and Melby-Thompson. Using a scalar field coupled to gravity according to the minimal substitution recipe proposed by da Silva and taking the geometrical optics limit, we could write an effective relativistic metric for a general solution. As a result, we discovered that the equivalence principle is not in general recovered at low energies, unless the spatial Laplacian of A vanishes. Finally, we analyzed the motion on the spherical symmetric solution proposed by Ho?ava and Melby-Thompson, where we could find its effective line element and compute spin-0 geodesics. Using standard methods we have shown that such an effective metric cannot reproduce Newton?s gravity law even in the weak gravitational field approximation.     

Numerical calculation of bound geodesics in the Kerr metric

Timelike geodesics, especially bound orbits in the equatorial plane (?=π/2) and spherical orbits (r=const), are calculated numerically. We plot the orbits using the Kerr-Schild coordinate system. The periastron advance and the dragging of nodes have the same values in any coordinate system and can be directly measured by an observer at infinity.     

Some properties of the n-dimensional plane symmetric solution

An n-dimensional static plane symmetric solution of Einstein field equation, which is judged as the source of n-dimensional Taub solution, is presented in our previous work. The properties of geodesics of this solution are studied in this Letter. The essence of the source is also investigated. A phantom with dust and photon is suggested as the substance of the source matter.     

Schwarzschild black hole surrounded by quintessence: null geodesics

We have studied the null geodesics of the Schwarzschild black hole surrounded by quintessence matter. Quintessence matter is a candidate for dark energy. Here, we have done a detailed analysis of the geodesics and exact solutions are presented in terms of Jacobi-elliptic integrals for all possible energy and angular momentum of the photons. The circular orbits of the photons are studied in detail. As an application of the null geodesics, the angle of deflection of the photons are computed.     

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.     

Scattering off of an instanton

We consider the scattering of a classical colored particle off an instanton. That is, we investigate Wong's equations (or equivalently, the Kaluza-Klein geodesic equations) for a colorSU(2) particle under the influence of a Euclidean instanton. We solve the equations in the limit in which the instanton becomes singular. Our main result is that particles with head-on trajectories scatter off the instanton with a scattering angle of π/3. This angle is independent of the magnitude of the color charge and velocity of the particle as long as both are nonzero. The plane in which the scattering takes place is determined by the particle's initial position and color charge. We also solve for the geodesics for the corresponding (singular) Kaluza-Klein metric onS ⁷.     

Light rays joining two submanifolds in space-times

Let M = Mo × R be a stationary Lorentz metric and P₀, P₁ be two closed submanifolds of M0. By using the Ljusternik-Schnirelman theory and variational tools, we prove the influence of the topology of P₀ and P₁ on the number of lightlike geodesics in P₀ joining P₀ × {0} to P₁ × R.