Showing papers on "Geodesic deviation published in 2018"

Gravitational waves in modified teleparallel theories

Abstract: We investigate the gravitational waves and their properties in various modified teleparallel theories, such as $f(T)$, $f(T,B)$, and $f(T,{T}_{G})$ gravities. We perform the perturbation analysis both around a Minkowski background and in the case where a cosmological constant is present, and for clarity we use both the metric and the tetrad languages. For $f(T)$ gravity we verify the result that no further polarization modes comparing to general relativity are present at first-order perturbation level, and we show that in order to see extra modes one should look at third-order perturbations. For nontrivial $f(T,B)$ gravity, by examining the geodesic deviation equations, we show that extra polarization models, namely the longitudinal and breathing modes, do appear at first-order perturbation level, and the reason for this behavior is the fact that although the first-order perturbation does not have any effect on $T$, it does affect the boundary term $B$. Finally, for $f(T,{T}_{G})$ gravity we show that at first-order perturbations the gravitational waves exhibit the same behavior as those of $f(T)$ gravity. Since different modified teleparallel theories exhibit different gravitational wave properties, the advancing gravitational-wave astronomy would help to alleviate the degeneracy not only between curvature and torsional modified gravity but also between different subclasses of modified teleparallel gravities.

Tidal stresses and energy gaps in microstate geometries

Abstract: We compute energy gaps and study infalling massive geodesic probes in the new families of scaling, microstate geometries that have been constructed recently and for which the holographic duals are known. We find that in the deepest geometries, which have the lowest energy gaps, the geodesic deviation shows that the stress reaches the Planck scale long before the probe reaches the cap of the geometry. Such probes must therefore undergo a stringy transition as they fall into microstate geometry. We discuss the scales associated with this transition and comment on the implications for scrambling in microstate geometries.

Inhomogeneous Jacobi equation for minimal surfaces and perturbative change in holographic entanglement entropy

Abstract: The change in holographic entanglement entropy (HEE) for small fluctuations about pure anti-de Sitter (AdS) is obtained by a perturbative expansion of the area functional in terms of the change in the bulk metric and the embedded extremal surface. However it is known that change in the embedding appears at second order or higher. It was shown that these changes in the embedding can be calculated in the $2+1$ dimensional case by solving a ``generalized geodesic deviation equation.'' We generalize this result to arbitrary dimensions by deriving an inhomogeneous form of the Jacobi equation for minimal surfaces. The solutions of this equation map a minimal surface in a given space time to a minimal surface in a space time which is a perturbation over the initial space time. Using this we perturbatively calculate the changes in HEE up to second order for boosted black brane like perturbations over ${\mathrm{AdS}}_{4}$.

Tidal Effects in Some Regular Black Holes

Abstract: This paper is aimed to study the tidal forces produced by a class of regular black holes. We consider the radial infall of test particle and find radial as well as angular components of tidal forces by taking geodesic deviation equations. We also compute geodesic deviation vector by solving geodesic deviation equation numerically. It is concluded that a particle undergos either compression or stretching in radial or angular direction due to tidal forces.

Gravitational Waves in Einstein-{\AE}ther Theory and Generalized T$e$V$e$S Theory after GW170817

Abstract: In this work, the polarization contents of Einstein-\ae ther theory and the Generalized T$e$V$e$S theory are discussed, as both theories have a normalized timelike vector field. The linearized equations of motion about the flat spacetime background are derived using the gauge invariant variables to easily separate physical degrees of freedom. The plane wave solutions are then found, and the polarizations are identified by examining the geodesic deviation equations. It shows that both theories contain six polarizations: plus, cross, vector-$x$, vector-$y$, transverse breathing and longitudinal polarizations. We also discuss the experimental tests of the extra polarizations in Einstein-\ae ther theory using pulsar timing arrays with the gravitational wave speed bound derived from the observations on GW 170817 and GRB 170817A. It turns out that it is difficult to use pulsar timing arrays to distinguish different polarizations in Einstein-\ae ther theory. The same speed bound also forces one of the propagating modes in the Generalized T$e$V$e$S theory to travel much faster than the speed of light, so the Generalized T$e$V$e$S theory is excluded.

Reduced phase space optics for general relativity: Symplectic ray bundle transfer

Abstract: In the paraxial regime of Newtonian optics, propagation of an ensemble of rays is represented by a symplectic ABCD transfer matrix defined on a reduced phase space. Here, we present its analogue for general relativity. Starting from simultaneously applied null geodesic actions for two curves, we obtain a geodesic deviation action up to quadratic order. We achieve this by following a preexisting method constructed via Synge's world function. We find the corresponding Hamiltonian function and the reduced phase space coordinates that are composed of the components of the Jacobi fields projected on an observational screen. Our thin ray bundle transfer matrix is then obtained through the matrix representation of the Lie operator associated with this quadratic Hamiltonian. Moreover, Etherington's distance reciprocity between any two points is shown to be equivalent to the symplecticity conditions of our ray bundle transfer matrix. We further interpret the bundle propagation as a free canonical transformation with a generating function that is equal to the geodesic deviation action. We present it in the form of matrix inner products. A phase space distribution function and the associated Liouville equation is also provided. Finally, we briefly sketch the potential applications of our construction. Those include reduced phase space and null bundle averaging; factorization of light propagation in any spacetime uniquely into its thin lens, pure magnifier and fractional Fourier transformer components; wavization of the ray bundle; reduced polarization optics and autonomization of the bundle propagation on the phase space to find its invariants and obtain the stability analysis.

Spacetime geometry fluctuations and geodesic deviation

Abstract: The quantum fluctuations of the geodesic deviation equation in a flat background spacetime are discussed. We calculate the resulting mean-squared fluctuations in the relative velocity and separation of test particles. The effect of these quantum fluctuations of the spacetime geometry is given in terms of the Riemann tensor correlation function. Three different sources of the Riemann tensor fluctuations are considered: a thermal bath of gravitons, gravitons in a squeezed state, and the graviton vacuum state.

On integrability of the geodesic deviation equation

Abstract: The Jacobi equation for geodesic deviation describes finite size effects due to the gravitational tidal forces. In this paper we show how one can integrate the Jacobi equation in any spacetime admitting completely integrable geodesics. Namely, by linearizing the geodesic equation and its conserved charges, we arrive at the invariant Wronskians for the Jacobi system that are linear in the ‘deviation momenta’ and thus yield a system of first-order differential equations that can be integrated. The procedure is illustrated on an example of a rotating black hole spacetime described by the Kerr geometry and its higher-dimensional generalizations. A number of related topics, including the phase space formulation of the theory and the derivation of the covariant Hamiltonian for the Jacobi system are also discussed.

On Integrability of the Geodesic Deviation Equation

Abstract: The Jacobi equation for geodesic deviation describes finite size effects due to the gravitational tidal forces. In this paper we show how one can integrate the Jacobi equation in any spacetime admitting completely integrable geodesics. Namely, by linearizing the geodesic equation and its conserved charges, we arrive at the invariant Wronskians for the Jacobi system that are linear in the `deviation momenta' and thus yield a system of first-order differential equations that can be integrated. The procedure is illustrated on an example of a rotating black hole spacetime described by the Kerr geometry and its higher-dimensional generalizations. A number of related topics, including the phase space formulation of the theory and the derivation of the covariant Hamiltonian for the Jacobi system are also discussed.

Comparing an analytical spacetime metric for a merging binary to a fully nonlinear numerical evolution using curvature scalars

Abstract: We introduce a new geometrically invariant prescription for comparing two different spacetimes based on geodesic deviation We use this method to compare a family of recently introduced analytical spacetime representing inspiraling black-hole binaries to fully nonlinear numerical solutions to the Einstein equations Our method can be used to improve analytical spacetime models by providing a local measure of the effects that violations of the Einstein equations will have on timelike geodesics, and indirectly, gas dynamics We also discuss the advantages and limitations of this method

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 longitudinal polarization exists no matter whether the scalar field is massive or not. It would be still very difficult to detect the enhanced 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 to relatively easily detect the effect of the violation as neutron stars have large self-energy. 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.

Formulations of General Relativity and their Applications to Quantum Mechanical Systems (with an emphasis on gravitational waves interacting with superconductors)

Abstract: Author(s): Inan, Nader | Advisor(s): Chiao, Raymond | Abstract: The linearization of General Relativity leads to various formulations of gravity often referred to as gravito-electromagnetism due to its resemblance to electromagnetism. Three methods are compared: (i) the harmonic gauge approach; (ii) the Parameterized Post-Newtonian (PPN) approach; and (iii) the Helmholtz Decomposition (HD) approach. New relationships are developed that are not generally found in the literature. These include the use of the linearized Bianchi identity, the Landau-Lifshitz pseudotensor, the Isaacson power formula, the geodesic equation of motion, and the geodesic deviation equation. The formalism is applied to examples such as a mass-solenoid and a gravitational mutual inductance system. The HD approach is shown to be the most favorable of the three methods due to being gauge-invariant (to linear order in the metric), and because it shows explicitly that the transverse-traceless part of the metric contains the only radiative degrees of freedom. This is similar to the transverse-traceless (TT) gauge except that the HD formulation is fully valid in matter. Therefore, unlike the TT gauge, the HD formulation can be used to describe how gravitational waves interact with various types of material. Traditionally, it is believed that all known materials are essentially transparent to gravitational waves. However, this conclusion relies on a classical treatment which describes how gravitational waves (originating from astrophysical sources) are passively detected with no affect on the wave itself. As an alternative, we consider how gravitational waves could be coupled to quantum systems which may be used for detection as well as reflection and even generation of gravitational waves. To investigate this possibility, a classical Hamiltonians is developed which describes the kinematics of charged, relativistic, massive particles in curved space-time. The coupling of quantum matter to gravitational fields is then described by quantizing the Hamiltonian. This leads to various gravitational quantum effects such gravitational Aharonov-Bohm effects, gravitational Casimir effects, and various time-holonomies. Furthermore, developing a quantized stress tensor and taking the expectation value allows the Einstein field equation to predict how quantum matter can produce classical gravitational fields. This semi-classical approach is used to describe how superconductors interact with gravitational waves. A London-like constitutive equation describes the response of the superconductor in terms of a "gravitational shear modulus" analogous to the standard shear modulus of elastic mechanics. Also using a "gravitational permeativity" (analogous to the magnetic permeability) leads to a gravitational plasma frequency, index of refraction, penetration depth, and impedance. The same analysis also done for a normal conductor using a gravitational Ohm-like constitutive equation, however, it is shown that a superconductor exhibits a gravitational Meissner-like effect, while a normal conductor does not. For the case of a superconductor, the Cooper pairs are described by the Ginzburg-Landau free energy density embedded in curved spacetime. This leads to a new gravito-London gauge condition and a predicted graviton mass within the superconductor. Next, the ionic lattice is modeled by an ensemble of quantum harmonic oscillators coupled to gravitational waves and characterized by quasi-energy eigenvalues for the phonon modes. This formulation predicts a gravitationally-induced dynamical Casimir effect within the ionic lattice since the zero-point energy of the phonon modes is modulated by the gravitational wave. Applying periodic thermodynamics and the Debye model in the low-temperature limit leads to a free energy density for the ionic lattice. From these results it is shown that the response to a gravitational wave is far less for the Cooper pair density than for the ionic lattice. This predicts a charge separation effect which can be used to detect the passage of a gravitational wave, and the possibility of reflection of gravitational waves by a superconductor. Lastly, a long-range communication system is proposed based on the coupling of gravitational and electromagnetic waves via ellipsoidal superconducting cavities.

Some clarifications about the Bohmian geodesic deviation equation and Raychaudhuri’s equation

Abstract: One of the important and famous topics in general theory of relativity and gravitation is the problem of geodesic deviation and its related singularity theorems. An interesting subject is the investigation of these concepts when quantum effects are considered. Since the definition of trajectory is not possible in the framework of standard quantum mechanics (SQM), we investigate the problem of geodesic equation and its related topics in the framework of Bohmian quantum mechanics in which the definition of trajectory is possible. We do this in a fixed background and we do not consider the backreaction effects of matter on the space–time metric.

On K-Jet Field Approximations to Geodesic Deviation Equations

Abstract: Let M be a smooth manifold and S a semi-spray defined on a sub-bundle C of the tangent bundle TM . In this work it is proved that the only non-trivial k-jet approximation to the exact geodesic deviation equation of S, linear on the deviation functions and invariant under an specific class of local coordinate transformations is the Jacobi equation. However, if the linearity property on the dependence in the deviation functions is not imposed, then there are differential equations whose solutions admit k-jet approximations and are invariant under arbitrary coordinate transformations. As an example of higher order geodesic deviation equations we study the first and second order geodesic deviation equations for a Finsler spray.

Temperature oscillations of a gas moving close to circular geodesic in Reissner-Nordstr\"om spacetime

Abstract: The objective of this work is to analyze the temperature oscillations that occur in a gas in a circular motion under the action of a Reissner-Nordstrom gravitational field, verifying the effect of the charge term of the metric on the oscillations. The expression for temperature oscillations follows from Tolman's law written in Fermi normal coordinates for a comoving observer. The motion of the gas is close to geodesic so the equation of geodesic deviation was used to obtain the expression for temperature oscillations. Then these oscillations are calculated for some compact stars, quark stars, black holes and white dwarfs, using values of electric charge and mass from models found in the literature. Comparing the various models analyzed, it is possible to verify that the role of the charge is the opposite of the mass. While the increase of the mass produces a reduction in the frequencies, amplitude and in the ratio between the frequencies, the increase of the electric charge produces the inverse effect. In addition, it is shown that if the electric charge is proportional to the mass, the ratio between the frequencies does not depend on the mass, but only on the proportionality factor between charge and mass. The ratios between the frequencies for all the models analyzed (except for supermassive black holes in the extreme limit situations) are close to the $3/2$ ratio for twin peak quasi-periodic oscillation frequencies, observed in many galactic black holes and neutron star sources in low-mass X-ray binaries.

The Possibility of Introducing of Metric Structure in Vortex Hydrodynamic Systems

Abstract: Geometrization of the description of vortex hydrodynamic systems can be made on the basis of the introduction of the Monge –Clebsch potentials, which leads to the Hamiltonian form of the original Euler equations. For this, we construct the kinetic Lagrange potential with the help of the flow velocity field, which is preliminarily determined through a set of scalar Monge potentials, and thermodynamic relations. The next step is to transform the resulting Lagrangian by means of the Legendre transformation to the Hamiltonian function and correctly introduce the generalized impulses canonically conjugate to the configuration variables in the new phase space of the dynamical system. Next, using the Hamiltonian function obtained, we define the Hamiltonian space on the cotangent bundle over the Monge potential manifold. Calculating the Hessian of the Hamiltonian, we obtain the coefficients of the fundamental tensor of the Hamiltonian space defining its metric. Next, we determine analogs of the Christoffel coefficients for the N -linear connection. Considering the Euler –Lagrange equations with the connectivity coefficients obtained, we arrive at the geodesic equations in the form of horizontal and vertical paths in the Hamiltonian space. In the case under study, nontrivial solutions can have only differential equations for vertical paths. Analyzing the resulting system of equations of geodesic motion from the point of view of the stability of solutions, it is possible to obtain important physical conclusions regarding the initial hydrodynamic system. To do this, we investigate a possible increase or decrease in the infinitesimal distance between the geodesic vertical paths (solutions of the corresponding system of Jacobi – Cartan equations). As a result, we can formulate very general criterions for the decay and collapse of a vortex continual system.

BGV theorem and Geodesic deviation

Abstract: I point out a simple expression for the "Hubble" parameter $H$ defined by Borde, Guth and Vilenkin (BGV) in their proof of past incompleteness of inflationary spacetimes. I show that $H$ is equal to the fractional rate of change of the magnitude of the Jacobi field $\xi^i$ of the congruence $u^i$ used by BGV, measured along the points of intersection of an arbitrary observer $O$ with $u^i$.

BGV theorem, Geodesic deviation, and Quantum fluctuations

Abstract: I point out a simple expression for the "Hubble" parameter $\mathscr{H}$, defined by Borde, Guth and Vilenkin (BGV) in their proof of past incompleteness of inflationary spacetimes. I show that the parameter $\mathscr{H}$ which an observer $O$ with four-velocity $\bf v$ will associate with a congruence $\bf u$ is equal to the fractional rate of change of the magnitude $\xi$ of the Jacobi field associated with $\bf u$, measured along the points of intersection of $O$ with $\bf u$, with its direction determined by $\bf v$. I then analyse the time dependence of $\mathscr{H}$ and $\xi$ using the geodesic deviation equation, computing these exactly for some simple spacetimes, and perturbatively for spacetimes close to maximally symmetric ones. The perturbative solutions are used to characterise the rms fluctuations in these quantities arising due to possible fluctuations in the curvature tensor.