Showing papers on "Geodesic deviation published in 2019"

Scalar modes in extended hybrid metric-Palatini gravity: Weak field phenomenology

Abstract: We investigate the nature of additional scalar degrees of freedom contained in extended hybrid metric-Palatini gravity, outlining the emergence of two coupled dynamical scalar modes. In particular, we discuss the weak field limit of the theory, both in the static case and from a gravitational waves perspective. In the first case, performing an analysis at the lowest order of the post parameterized Newtonian (PPN) structure of the model, we stress the settling of Yukawa corrections to the Newtonian potential. In this respect, we show that one scalar field can have long range interactions and used in the principle for mimicking dark matter effects. Concerning the gravitational waves propagation, instead, we demonstrate that is possible to have well-defined physical degrees of freedom, provided by suitable constraints on model parameters. Moreover, the study of the geodesic deviation points out the presence of breathing and longitudinal polarizations due to these novel scalar waves, which on peculiar assumptions can give rise to beating phenomena during their propagation.

Gauge invariant formulation of metric f (R ) gravity for gravitational waves

Abstract: We analyze the gravitational waves propagation in metric $f(R)$ theories of gravity. In particular, adopting a gauge invariant formalism we clearly determine the exact propagating degrees of freedom. Then, investigating their effects on test masses via geodesic deviation equation, we show that the additional dynamical degree contained in such extended formulations is actually responsible for two distinguished polarizations, corresponding to a breathing and a longitudinal mode, respectively.

On geometric objects, the non-existence of a gravitational stress-energy tensor, and the uniqueness of the Einstein field equation

Abstract: The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the inception of the theory Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given Most arguments to that effect invoke one version or another of the Principle of Equivalence I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the issue Based on a novel analysis of what it may mean for one tensor to depend in the proper way on another, which, en passant, provides a precise characterization of the idea of a “geometric object”, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity It follows that gravitational energy, such as it is in general relativity, is necessarily non-local Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime I conclude by showing that my results also imply that, under a few natural conditions, the Einstein field equation is the unique equation relating gravitational phenomena to spatiotemporal structure, and discuss how this relates to the non-localizability of gravitational stress-energy The main theorem proven underlying all the arguments is considerably stronger than the standard result in the literature used for the same purposes (Lovelock's theorem of 1972): it holds in all dimensions (not only in four); it does not require an assumption about the differential order of the desired concomitant of the metric; and it has a more natural physical interpretation

Memory effect in anti–de Sitter spacetime

Abstract: The geodesic deviation of a pair of test particles is a natural observable for the gravitational memory effect. Nevertheless in curved spacetime, this observable is plagued with various issues that need to be clarified before one can extract the essential part that is related to the gravitational radiation. In this paper we consider the anti--de Sitter (AdS) space as an example and analyze this observable carefully. We show that by employing the Fermi normal coordinates around the geodesic of one of the particles (i.e., the standard free falling frame attached to this particle), one can elegantly separate out the curvature contribution of the background spacetime to the geodesic deviation from the contribution of the gravitational wave. The gravitational wave memory obtained this way depends linearly and locally on the retarded metric perturbation caused by the gravitational wave, and, remarkably, it takes on exactly the same formula as in the flat case. To determine the memory, in addition to the standard tail contribution to the gravitational radiation, one needs to take into account the contribution from the reflected gravitational wave off the AdS boundary. For general curved spacetime, our analysis suggests that the use of a certain coordinate system adapted to the local geodesic (e.g., the Fermi normal coordinates system in the AdS case) would allow one to dissect the geodesic deviation of test particles and extract the relevant contribution to define the memory due to gravitational radiation.

Modified gravitational waves across galaxies from macroscopic gravity

Abstract: We analyze the propagation of gravitational waves in a medium containing bounded subsystems ("molecules"), able to induce significant Macroscopic Gravity effects. We establish a precise constitutive relation between the average quadrupole and the amplitudes of a vacuum gravitational wave, via the geodesic deviation equation. Then we determine the modified equation for the wave inside the medium and the associated dispersion relation. A phenomenological analysis shows that anomalous polarizations of the wave emerge with an appreciable experimental detectability if the medium is identified with a typical galaxy. Both the modified dispersion relation (wave velocity less than the speed of light) and anomalous oscillations modes could be detectable by the incoming LISA or pulsar timing arrays experiments, having the appropriate size to see the concerned wavelengths (larger than the molecular size) and the appropriate sensitivity to detect the expected deviation from vacuum General Relativity.

A contextual analysis of the early work of Andrzej Trautman and Ivor Robinson on equations of motion and gravitational radiation

Abstract: In a series of papers published in the course of his dissertation work in the mid 1950's, Andrzej Trautman drew upon the slow motion approximation developed by his advisor Infeld, the general covariance based strong conservation laws enunciated by Bergmann and Goldberg, the Riemann tensor attributes explored by Goldberg and related geodesic deviation exploited by Pirani, the permissible metric discontinuities identified by Lichnerowicz, O'Brien and Synge, and finally Petrov's classification of vacuum spacetimes. With several significant additions he produced a comprehensive overview of the state of research in equations of motion and gravitational waves that was presented in a widely cited series of lectures at King's College, London, in 1958. Fundamental new contributions were the formulation of boundary conditions representing outgoing gravitational radiation the deduction of its Petrov type, a covariant expression for null wave fronts, and a derivation of the correct mass loss formula due to radiation emission. Ivor Robinson had already in 1956 developed a bi-vector based technique that had resulted in his rediscovery of exact plane gravitational wave solutions of Einstein's equations. He was the first to characterize shear-free null geodesic congruences. He and Trautman met in London in 1958, and there resulted a long-term collaboration whose initial fruits were the Robinson-Trautman metric, examples of which were exact spherical gravitational waves.

Strong equivalence principle and gravitational wave polarizations in Horndeski theory

Abstract: The relative acceleration between two nearby particles moving along accelerated trajectories is studied, which generalizes the geodesic deviation equation. The polarization content of the gravitational wave in Horndeski theory is investigated by examining the relative acceleration between two self-gravitating particles. It is found out that the apparent longitudinal polarization exists no matter whether the scalar field is massive or not. It would be still very difficult to detect the enhanced/apparent longitudinal polarization with the interferometer, as the violation of the strong equivalence principle of mirrors used by interferometers is extremely small. However, the pulsar timing array is promised relatively easily to detect the effect of the violation as neutron stars have large self-gravitating energies. The advantage of using this method to test the violation of the strong equivalence principle is that neutron stars are not required to be present in the binary systems.

Geodesics in exact plane wave spacetimes and the memory effect

Abstract: Recently, in several articles [notably, Phys. Rev. D 96, 064013(2017)], it has been shown qualitatively, how the displacement and velocity memory effects can be analysed by studying geodesics in the exact, vacuum, plane gravitational wave line element. In particular, one may look for such memory effects by investigating the behaviour of geodesics using the geodesic equations themselves, the geodesic deviation equations or the Raychaudhuri equations for geodesic congruences. In our work here, we first demonstrate qualitatively, using geodesic deviation, how a memory effect may arise. Thereafter, we solve the geodesic equations explicitly to provide simple and largely analytical examples of such effects, using two different pulse profiles in the exact plane gravitational wave line element: (a) a square pulse and (b) a sech-squared pulse. We show the emergence of displacement and velocity memory effects for timelike geodesics, for each of the above-mentioned profiles. Finally, we demonstrate in qualitative terms, how the Raychaudhuri equations may be used to arrive at memory effects through the nature of the shear and expansion of geodesic congruences. As an illustration, we obtain the shear and expansion for timelike geodesic congruences in this line element, for the two pulse profiles mentioned earlier. The growth of shear in causing a focusing of an initially parallel congruence, after the pulse has left, is seen to be distinctly linked to the memory effect.

Modifications to Plane Gravitational Waves from Minimal Lorentz Violation

Abstract: General Relativity predicts two modes for plane gravitational waves. When a tiny violation of Lorentz invariance occurs, the two gravitational wave modes are modified. We use perturbation theory to study the detailed form of the modifications to the two gravitational wave modes from the minimal Lorentz-violation coupling. The perturbation solution for the metric fluctuation up to the first order in Lorentz violation is discussed. Then, we investigate the motions of test particles under the influence of the plane gravitational waves with Lorentz violation. First-order deviations from the usual motions are found.

Visualizing gravitational Bessel waves

Abstract: We explicitly derive a~vortex inspired solution for the metric perturbation within the linearized Einsteins general theory of relativity in arbitrary dimensions $D\geq 4$. We focus on $D=4$ where our solution is the gravitational analog of the well-known electromagnetic (or electron) Bessel vortex beams. Next we visualize the perturbed spacetime via tidal tendexes and frame-drag vortexes. We display and analyze mostly two-dimensional sections of the tendexes and the vortexes for different values of an angular momentum of the wave solution. Corresponding geodesic deviation equations are solved and the results are visualized. We show that the physically most important quadrupolelike case leads to a~wave with rotating polarization. We discuss asymptotical features of the found solution. We provide also several 3D plots of tendex lines. One of them concerns a~special cylindrical-like case and we utilize the topological classification of singularities of the depicted line fields as an approach to characterize the radiation field.

Dark Matter: the Problem of Motion

Abstract: Dark matter (DM) may be studied through the motion of objects following nongeodesic trajectories either due to the existence of an extra mass as a projection of higher dimensions onto lower ones or as motion of dipolar particles and fluids in the halos of spiral galaxies. The effect of DM has been extended nearby the core of the galaxy by means of the excess of mass appearing in the motion of fluids in the accretion disc. Nongeodesic equations and those of their deviation are derived in the presence of different classes of bimetric theories of gravity. The stability of these trajectories using the geodesic deviation technique is investigated.

Perihelion advance and trajectory of charged test particles in Reissner-Nordstrom field via the higher-order geodesic deviations

Abstract: By using the higher-order geodesic deviation equations for charged particles, we apply the method described by Kerner this http URL. to calculate the perihelion advance and trajectory of charged test particles in the Riessner-Nordstrom spacetime. The effect of charge on the perihelion advance is studied and compared the results with those obtained earlier via the perturbation method. The advantage of this approximation method is to provide a way to calculate the perihelion advance and orbit of planets in the vicinity of massive and compact objects without considering Newtonian and post-Newtonian approximation.

Perihelion Advance and Trajectory of Charged Test Particles in Reissner-Nordstrom Field via the Higher-Order Geodesic Deviations

Abstract: By using the higher-order geodesic deviation equations for charged particles, we apply the method described by Kerner et.al. to calculate the perihelion advance and trajectory of charged test particles in the Reissner-Nordstrom space-time. The effect of charge on the perihelion advance is studied and we compared the results with those obtained earlier via the perturbation method. The advantage of this approximation method is to provide a way to calculate the perihelion advance and orbit of planets in the vicinity of massive and compact objects without considering Newtonian and post-Newtonian approximations.

On the Applicability of the Geodesic Deviation Equation in General Relativity

Abstract: Within the theory of General Relativity, we study the solution and range of applicability of the standard geodesic deviation equation in highly symmetric spacetimes. In the Schwarzschild spacetime, the solution is used to model satellite orbit constellations and their deviations around a spherically symmetric Earth model. We investigate the spatial shape and orbital elements of perturbations of circular reference curves. In particular, we reconsider the deviation equation in Newtonian gravity and then determine relativistic effects within the theory of General Relativity by comparison. The deviation of nearby satellite orbits, as constructed from exact solutions of the underlying geodesic equation, is compared to the solution of the geodesic deviation equation to assess the accuracy of the latter. Furthermore, we comment on the so-called Shirokov effect in the Schwarzschild spacetime and limitations of the first order deviation approach.

Calculation of the amplitudes of elastic waves in anisotropic media in Cartesian or ray-centred coordinates

Abstract: We derive various expressions for the amplitude of the ray-theory approximation of elastic waves in heterogeneous anisotropic media, and show their mutual relations. The amplitude of a wavefield with general initial conditions can be expressed in terms of two paraxial vectors of geometrical spreading in Cartesian coordinates, and in terms of the 2×2 matrix of geometrical spreading in ray-centred coordinates. The amplitude of the Green tensor can be expressed in six different ways: (a) in terms of the paraxial vectors corresponding to two ray parameters in Cartesian coordinates, (b) in terms of the 2×2 paraxial matrices corresponding to two ray parameters in ray-centred coordinates, (c) in terms of the 3×3 upper right submatrix of the 6×6 propagator matrix of geodesic deviation in Cartesian coordinates, (d) in terms of the 2×2 upper right submatrix of the 4×4 propagator matrix of geodesic deviation in ray-centred coordinates, (e) in terms of the 3×3 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the source and receiver Cartesian coordinates, and (f) in terms of the 2×2 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the source and receiver ray-centred coordinates. The step-by-step derivation of various equivalent expressions, both known or novel, elucidates the mutual relations between these expressions.

Higher-order geodesic deviations and orbital precession in a Kerr-Newman space-time

Abstract: A novel approximation method in studying the perihelion precession and planetary orbits in general relativity is to use geodesic deviation equations of first and high-orders, proposed by Kerner this http URL. Using higher-order geodesic deviation approach, we generalize the calculation of orbital precession and the elliptical trajectory of neutral test particles to Kerr$-$Newman space-times. One of the advantage of this method is that, for small eccentricities, one obtains trajectories of planets without using Newtonian and post-Newtonian approximations for arbitrary values of quantity ${G M}/{R c^2}$.

Examining the Kerr metric through wave fronts of null geodesics

Abstract: We examine the singularities of the wave fronts of null geodesics from point sources in the Kerr metric. We find that the wave fronts develop a tube like structure that collapses non-symmetrically, leading to cusp features in the wave front singularities. As the wave front advances, the cusps trace out an astroidal shaped caustic tube, which had been discovered previously using lens mapping and geodesic deviation methods. Thus, the wave front approach in this study helps to complete a picture of caustics and gravitational lensing in the Kerr geometry.

Cosmic acceleration via space-time-matter theory

Abstract: We consider the space-time-matter theory (STM) in a five-dimensional vacuum space-time with a generalized FLRW metric to investigate the late-time acceleration of the universe. For this purpose, we derive the four-dimensional induced field equations and obtain the evolution of the state parameter with respect to the redshift. Then, we show that with consideration of the extra dimension scale factor to be a linear function of redshift, this leads to a model which gives an accelerating phase in the universe. Moreover, we derive the geodesic deviation equation in the STM theory to study the relative acceleration of the parallel geodesics of this space-time, and also, obtain the observer area-distance as a measurable quantity to compare this theory with two other models.

World-Line Perturbation Theory

Abstract: The motion of a compact body in space and time is commonly described by the world line of a point representing the instantaneous position of the body. In General Relativity such a world-line formalism is not quite straightforward because of the strict impossibility to accommodate point masses and rigid bodies. In many situations of practical interest it can still be made to work using an effective hamiltonian or energy-momentum tensor for a finite number of collective degrees of freedom of the compact object. Even so exact solutions of the equations of motion are often not available. In such cases families of world lines of compact bodies in curved space-times can be constructed by a perturbative procedure based on generalized geodesic deviation equations. Examples for simple test masses and for spinning test bodies are presented.

Axial symmetry cosmological constant vacuum solution of field equations with a curvature singularity, closed time-like curves and deviation of geodesics

Abstract: In this paper, we present a type D, non-vanishing cosmological constant, vacuum solution of the Einstein's field equations, extension of an axially symmetric, asymptotically flat vacuum metric with a curvature singularity. The space-time admits closed time-like curves (CTCs) that appear after a certain instant of time from an initial spacelike hypersurface, indicating it represents a time-machine space-time. We wish to discuss the physical properties and show that this solution can be interpreted as gravitational waves of Coulomb-type propagate on anti-de Sitter space backgrounds. Our treatment focuses on the analysis of the equation of geodesic deviation.

Recursion relations for gravitational lensing

Abstract: The weak gravitational lensing formalism can be extended to the strong lensing regime by integrating a nonlinear version of the geodesic deviation equation. The resulting "roulette" expansion generalises the notion of convergence, shear and flexion to arbitrary order. The independent coefficients of this expansion are screen space gradients of the optical tidal tensor which approximates to the usual lensing potential in the weak field limit. From lensed images, knowledge of the roulette coefficients can in principle be inverted to reconstruct the mass distribution of a lens. In this paper, we simplify the roulette expansion and derive a family of recursion relations between the various coefficients, generalising the Kaiser-Squires relations beyond the weak-lensing regime.

On the geodesic congruences in quantum improved spacetimes

Abstract: We investigate the geodesic deviation equation in the context of quantum improved spacetimes. The improved Raychaudhuri equation is derived, and it is shown that the classical strong energy condition does not necessarily lead to the convergence of geodesics in a congruence in the quantum improved spacetime.

Higher-order geodesic deviations and orbital precession in a Kerr–Newman space–time

Abstract: A novel approximation method in studying the perihelion precession and planetary orbits in general relativity is to use geodesic deviation equations of first and high-orders, proposed by Kerner et al. Using higher-order geodesic deviation approach, we generalize the calculation of orbital precession and the elliptical trajectory of neutral test particles to Kerr–Newman space–times. One of the advantage of this method is that, for small eccentricities, one obtains trajectories of planets without using Newtonian and post-Newtonian approximations for arbitrary values of quantity $${G M}/{R c^2}$$ .

Examining the Kerr Metric through Wave Fronts of Null Geodesics.

Abstract: We examine the singularities of the wave fronts of null geodesics from point sources in the Kerr Metric. We find that the wave fronts develop a tube like structure that collapses non-symmetrically, leading to cusp features in the wave front singularities. As the wave front advances, the cusps trace out an astroidal shaped caustic tube, which had been discovered previously using lens mapping and geodesic deviation methods. Thus, the wave front approach in this study helps to complete a picture of caustics and gravitational lensing in the Kerr geometry.