Integrating the geodesic equations in the Schwarzschild and Kerr space-times using Beltrami's “geometrical” method

We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.     

Regular and irregular geodesics on spherical harmonic surfaces

The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using the Morales-Ramis theorem and Kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable, and we use Poincaré sections to demonstrate the breakdown of regular motion.     

On the twin paradox in static spacetimes: I. Schwarzschild metric

Motivated by a conjecture put forward by Abramowicz and Bajtlik we reconsider the twin paradox in static spacetimes. According to a well known theorem in Lorentzian geometry the longest timelike worldline between two given points is the unique geodesic line without points conjugate to the initial point on the segment joining the two points. We calculate the proper times for static twins, for twins moving on a circular orbit (if it is a geodesic) around a centre of symmetry and for twins travelling on outgoing and ingoing radial timelike geodesics. We show that the twins on the radial geodesic worldlines are always the oldest ones and we explicitly find the the conjugate points (if they exist) outside the relevant segments. As it is of its own mathematical interest, we find general Jacobi vector fields on the geodesic lines under consideration. In the first part of the work we investigate Schwarzschild geometry.     

Abstract Kelvin–Noether Theorems for Lie Group Extensions

We present an abstract Kelvin–Noether theorem for geodesic equations on abelian Lie group extensions with right invariant metrics and we apply it to equations of hydrodynamical type. Another Kelvin–Noether theorem for a class of central extensions of semidirect products is shown.     

The Egorov theorem for transverse Dirac-type operators on foliated manifolds

Egorov’s theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first-order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.     

Multimode geodesic branching components

Geodesic branching components are investigated for multimode guided wave optics. Geodesic structures with particular properties, e.g. focussing star couplers, are derived by a synthesis technique based on a theorem of Toraldo di Francia. Experimentally, the geodesic surfaces are printed on acrylic glass and are spin-coated with organic film waveguides.     

Using the Uncharged Kerr Black Hole as a Gravitational Mirror

We extend the study of the possible use of the Schwarzschild black hole as a gravitational mirror to the more general case of an uncharged Kerr black hole. We use the null geodesic equation in the equatorial plane to prove a theorem concerning the conditions the impact parameter has to satisfy if boomerang photons are to exist. We derive an equation for these boomerang photons and an equation for the emission angle. Finally, the radial null geodesic equation is integrated numerically in order to illustrate boomerang photons.     

A Brief Remark on Energy Conditions and the Geroch-Jang Theorem

The status of the geodesic principle in General Relativity has been a topic of some interest in the recent literature on the foundations of spacetime theories. Part of this discussion has focused on the role that a certain energy condition plays in the proof of a theorem due to Bob Geroch and Pong-Soo Jang [“Motion of a Body in General Relativity.” Journal of Mathematical Physics 16(1) (1975)] that can be taken to make precise the claim that the geodesic principle is a theorem, rather than a postulate, of General Relativity. In this brief note, I show, by explicit counterexample, that not only is a weaker energy condition than the one Geroch and Jang state insufficient to prove the theorem, but in fact a condition still stronger than the one that they assume is necessary.     

Singularities and conjugate points in FLRW spacetimes

Conjugate points play an important role in the proofs of the singularity theorems of Hawking and Penrose. We examine the relation between singularities and conjugate points in FLRW spacetimes with a singularity. In particular we prove a theorem that when a non-comoving, non-spacelike geodesic in a singular FLRW spacetime obeys conditions (39) and (40), every point on that geodesic is part of a pair of conjugate points. The proof is based on the Raychaudhuri equation. We find that the theorem is applicable to all non-comoving, non-spacelike geodesics in FLRW spacetimes with non-negative spatial curvature and scale factors that near the singularity have power law behavior or power law behavior times a logarithm. When the spatial curvature is negative, the theorem is applicable to a subset of these spacetimes.     

Egorov’s Theorem for Transversally Elliptic Operators on Foliated Manifolds and Noncommutative Geodesic Flow

The main result of the paper is Egorov’s theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations. Mathematics Subject Classifications (2000) 58J40, 58J42, 58B34.     

Affine symmetry,geodesics, and homogeneous spacetimes

We show that the conservation laws for the geodesic equation which are associated to affine symmetries can be obtained from symmetries of the Lagrangian for affinely parametrized geodesics according to Noether’s theorem, in contrast to claims found in the literature. In particular, using Aminova’s classification of affine motions of Lorentzian manifolds, we show in detail how affine motions define generalized symmetries of the geodesic Lagrangian. We compute all infinitesimal proper affine symmetries and the corresponding geodesic conservation laws for all homogeneous solutions to the Einstein field equations in four spacetime dimensions with each of the following energy–momentum contents: vacuum, cosmological constant, perfect fluid, pure radiation, and homogeneous electromagnetic fields.     

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.     

Equation of motion of small bodies in relativity

There is proven a theorem, to the effect that a material body in general relativity, in a certain limit of sufficiently small size and mass, moves along a geodesic.     

On Maximal Surfaces in Certain Non-Flat 3-Dimensional Robertson-Walker Spacetimes

An upper bound for the integral, on a geodesic disc, of the squared length of the gradient of a distinguished function on any maximal surface in certain non-flat 3-dimensional Robertson-Walker spacetimes is obtained. As an application, a new proof of a known Calabi-Bernstein??s theorem is given.     

Relativistic canonical formalism and the invariant single-particle distribution function in the general theory of relativity

The relativistic canonical formalism is used to construct an eight-dimensional phase space and an invariant distribution function, and integral and differential operations in the phase space and statistical averages, associated with the field of geodesic observers, are introduced. Liouville's theorem is proved.     

A focal index theorem for null geodesics

Employing techniques recently developed by D. Kalish for Riemannian manifolds, we obtain a focal Morse index theorem for a null geodesic segment initially and terminally perpendicular to spacelike submanifolds of arbitrary codimension in a general space-time.     

Local instability of orbits in polygonal and polyhedral billiards

We classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons, for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra. We sharpen a theorem of Boldrighini, Keane and Marchetti. We show that polygonal and polyhedral billiards have zero topological entropy. We also prove that billiards in polygons are positive expansive when restricted to the set of non-periodic points. The methods used are elementary geometry and symbolic dynamics.     

The Stability of Abstract Boundary Essential Singularities

The abstract boundary has, in recent years, proved a general and flexible way to define the singularities of space-time. In this approach an essential singularity is a non-regular boundary point of an embedding which is accessible by a chosen family of curves within finite parameter distance. Ashley and Scott proved the first theorem relating essential singularities in strongly causal space-times to causal geodesic incompleteness. Linking this with the work of Beem on the C ^r -stability of geodesic incompleteness allows proof of the stability of these singularities. Here I present this result stating the conditions under which essential singularities are C ¹-stable against perturbations of the metric.     

On the Robinson theorem and shearfree geodesic null congruences

Null electromagnetic fields and shearfree geodesic null congruences in curved and flat spacetimes are studied. We point out some mathematical problems connected with the validity of the Robinson theorem. The problem of finding nonanalytic twisting congruences in the Minkowski space is reduced to the construction of holomorphic functions with specific boundary conditions.Institute of Theoretical Physics, University of Warsaw Poland     

Persistent curvature and cosmic censorship

A definition is given which quantifies the strength of persistent Riemann curvature along a null geodesic. A numerical value thereof is identified which ensures the existence of conjugate points on null geodesics of infinite length. A class of examples show that no lesser value can suffice. One is led to a new theorem of cosmic censorship which identifies an upper bound on the persistent curvature strength with which any space-time may violate weak cosmic censorship. All previous theorems are superceded. Moreover an improved logical construction simplifies interpretation.