Showing papers on "Geodesic deviation published in 2007"

Measures for pathway analysis in brain white matter using diffusion tensor images

Abstract: In this paper we discuss new measures for connectivity analysis of brain white matter, using MR diffusion tensor imaging. Our approach is based on Riemannian geometry, the viability of which has been demonstrated by various researchers in foregoing work. In the Riemannian framework bundles of axons are represented by geodesics on the manifold. Here we do not discuss methods to compute these geodesics, nor do we rely on the availability of geodesics. Instead we propose local measures which are directly computable from the local DTI data, and which enable us to preselect viable or exclude uninteresting seed points for the potentially time consuming extraction of geodesics. If geodesics are available, our measures can be readily applied to these as well. We consider two types of geodesic measures. One pertains to the connectivity saliency of a geodesic, the second to its stability with respect to local spatial perturbations. For the first type of measure we consider both differential as well as integralmeasures for characterizing a geodesic's saliency either locally or globally. (In the latter case one needs to be in possession of the geodesic curve, in the former case a single tangent vector suffices.) The second type of measure is intrinsically local, and turns out to be related to a well known tensor in Riemannian geometry.

How far is 'infinity'?

Abstract: Our present knowledge of the distribution of matter in the local universe is reviewed. Appropriate boundaries isolating astrophysical systems are sought on the length scales of the solar system, the galaxy and the local group of galaxies. The influence of diffuse matter is compared to that of nearby objects using the geodesic deviation equation. The question of assigning realistic wave zones to some canonical sources of gravitational radiation is briefly reviewed. Taking our local environment as typical, it is found that compact systems of a size similar to that of the solar system can normally be considered isolated. Compact galactic nuclei with low matter flux can probably also be considered isolated.

Stringy Jacobi fields in Morse theory

Abstract: We consider the variation of the surface spanned by closed strings in a spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic surface equation and the geodesic surface deviation equation which yields a Jacobi field, and we define the index form of a geodesic surface as in the case of point particles to discuss conjugate strings on the geodesic surface.

Analysis of the transverse effect of Einstein's gravitational waves

Abstract: The investigation of the transverse effect of gravitational waves (GWs) could constitute a further tool to discriminate among several relativistic theories of gravity on the ground. After a review of the TT gauge, the transverse effect of GWs arising by standard general relativity (called Einstein's GWs in this paper) is reanalized with a different choice of coordinates. In the chosen gauge test masses have an apparent motion in the direction of propagation of the wave, while in the transverse direction they appear at rest. Of course, this is only a gauge artefact. In fact, from careful investigation of this particular gauge, it is shown that the tidal forces associated with GWs act along the directions orthogonal to the direction of propagation of waves. In the analysis it is also shown, in a heuristic way, that the transverse effect of Einstein's GWs arises from the propagation of the waves at the speed of the light, thus only massless GWs are purely transverse. But, because the physics of gravitational waves has to be investigated by studing the tidal forces as appearing in the geodesic deviation equation and directly in a laboratory environment on Earth, an analysis of these tidal forces and of the transverse effect in the frame of the local observer is also performed. After this, for a further better understanding of the transverse effect, an example of a wave, which arises from scalar tensor gravity, with both transverse and genuinely longitudinal modes is given and discussed. In the example the connection between the longitudinal component and the velocity of the wave will be mathematical shown. At the end of this paper the review of the TT gauge is completed, recovering the gauge invariance between the presented gauge and the TT one.

Analysis of the transverse effect of einstein's gravitational waves

Abstract: The investigation of the transverse effect of gravitational waves (GW's) could constitute a further tool to discriminate among several relativistic theories of gravity on the ground. After a review of the TT gauge, the transverse effect of GW's arising by standard general relativity (called Einstein's GW's in this paper) is reanalyzed with a different choice of coordinates. In the chosen gauge, test masses have an apparent motion in the direction of propagation of the wave, while in the transverse direction they appear at rest. Of course, this is only a gauge artefact. In fact, from careful investigation of this particular gauge, it is shown that the tidal forces associated with GW's act along the directions orthogonal to the direction of propagation of waves. In the analysis it is also shown, in a heuristic way, that the transverse effect of Einstein's GW's arises from the propagation of the waves at the speed of the light, thus only massless GW's are purely transverse. But, because the physics of gravitational waves has to be investigated by studying the tidal forces as appearing in the geodesic deviation equation and directly in a laboratory environment on Earth, an analysis of these tidal forces and of the transverse effect in the frame of the local observer is also performed. After this, for a further better understanding of the transverse effect, an example of a wave, which arises from scalar tensor gravity, with both transverse and genuinely longitudinal modes is given and discussed. In the example the connection between the longitudinal component and the velocity of the wave is mathematically shown. At the end of this paper, the review of the TT gauge is completed, recovering the gauge invariance between the presented gauge and the TT gauge.

Orbital period derivative of a binary system using an exact orbital energy equation

Abstract: It is proposed that the equations of motion in periodic relativity which yielded major predictions of general relativity without utilizing Riemannian geometry and geodesic trajectories are exact in nature and can be applied to pulsars and inspiraling compact binaries for analyzing orbital period derivative and two polarization gravitational wave forms. Exactness of these equations eliminates the need for higher order xPN corrections to the orbital energy part of the balance equation. This is mainly due to the introduction of dynamic WEP which states that the gravitational mass is equal to the relativistic mass.

Stability of Reissner-Nordström Black Hole

Abstract: The singularity of the solutions obtained before in the teleparallel theory of gravitation is studied. Also the stability of these solutions is studied using the equations of geodesic deviation. The condition of stability is obtained. From this condition the stability of the Schwarzschild solution can be obtained.

Geodesic deviation equation for relativistic tops and the detection of gravitational waves

Abstract: In this contribution, we review the derivation of the relativistic top geodesic deviation equation. This equation results in a generalization of the geodesic deviation equation for a pair of nearby point particles. This property is taken into account in investigating the detection of gravitational waves, and we show how our generalized formula for a relativistic top can be used to study the gravitational wave backgrounds. Besides of these facts, we argue that our formulation may be of special interest for detecting the inflationary gravitational waves via the polarization of the cosmic background radiation.

On the Jacobi-Metric Stability Criterion

Abstract: We investigate the exact relation existing between the stability equation for the solutions of a mechanical system and the geodesic deviation equation of the associated geodesic problem in the Jacobi metric constructed via the Maupertuis-Jacobi Principle. We conclude that the dynamical and geometrical approaches to the stability/instability problem are not equivalent.

The Theory of Vortical Gravitational Fields

Abstract: This paper treats of vortical gravitational fields, a tensor of which is the rotor of the general covariant gravitational inertial force. The field equations for a vortical gravitational field (the Lorentz condition, the Maxwell-like equations, and the continuity equation) are deduced in an analogous fashion to electrodynamics. From the equations it is concluded that the main kind of vortical gravitational fields is “electric”, determined by the non-stationarity of the acting gravitational inertial force. Such a field is a medium for traveling waves of the force (they are different to the weak deformation waves of the space metric considered in the theory of gravitational waves). Standing waves of the gravitational inertial force and their medium, a vortical gravitational field of the “magnetic” kind, are exotic, since a non-stationary rotation of a space body (the source of such a field) is a very rare phenomenon in the Universe. 1 The mathematical method There are currently two methods for deducing a formula for the Newtonian gravitational force in General Relativity. The first method, introduced by Albert Einstein himself, has its basis in an arbitrary interpretation of Christoffel’s symbols in the general covariant geodesic equations (the equation of motion of a free particle) in order to obtain a formula like that by Newton (see [1], for instance). The second method is due to Abraham Zelmanov, who developed it in the 1940’s [2, 3]. This method determines the gravitational force in an exact mathematical way, without any suppositions, as a part of the gravitational inertial force derived from the non-commutativity of the differential operators invariant in an observer’s spatial section. This formula results from Zelmanov’s mathematical apparatus of chronometric invariants (physical observable quantities in General Relativity). The essence of Zelmanov’s mathematical apparatus [4] is that if an observer accompanies his reference body, his observable quantities are the projections of four-dimensional quantities upon his time line and the spatial section — chronometrically invariant quantities, via the projecting operators b α = dx

Stability of Reissner Nordstr$\ddot{o}$m Black Hole

Abstract: The singularity of the solutions obtained before in the teleparallel theory of gravitation is studied. Also the stability of these solutions is studied using the equations of geodesic deviation. The condition of stability is obtained. From this condition the stability of Schwarzschild solution can be obtained.

On the generalized Jacobi equation

Abstract: 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.

Geodesics and Geodesic Deviation in a Stringy Charged Black Hole

Abstract: The radial motion along null geodesics in static charged black hole space-times, in particular, the Reissner-Nordstrom and stringy charged black holes are studied. We analyzed the properties of the effective potential. The circular photon orbits in these space-times are investigated. We found that the radius of circular photon orbits in both charged black holes are different and differ from that given in Schwarzschild space-time. We studied the physical effects of the gravitational field between two test particles in stringy charged black hole and compared the results with that given in Schwarzschild and Reissner-Nordstrom black holes.