Fedosov Quantization of Fractional Lagrange Spaces

The main goal of this work is to perform a nonholonomic deformation (Fedosov type) quantization of fractional Lagrange–Finsler geometries. The constructions are provided for a fractional almost K?hler model encoding equivalently all data for fractional Euler–Lagrange equations with Caputo fractional derivative.     

Modified dispersion relations in Hořava–Lifshitz gravity and Finsler brane models

We study possible links between quantum gravity phenomenology encoding Lorentz violations as nonlinear dispersions, the Einstein–Finsler gravity models, EFG, and nonholonomic (non-integrable) deformations to Hořava–Lifshitz, HL, and/or Einstein’s general relativity, GR, theories. EFG and its scaling anisotropic versions formulated as Hořava–Finsler models, HF, are constructed as covariant metric compatible theories on (co) tangent bundle to Lorentz manifolds and respective anisotropic deformations. Such theories are integrable in general form and can be quantized following standard methods of deformation quantization, A-brane formalism and/or (perturbatively) as a nonholonomic gauge like model with bi-connection structure. There are natural warping/trapping mechanisms, defined by the maximal velocity of light and locally anisotropic gravitational interactions in a (pseudo) Finsler bulk spacetime, to four dimensional (pseudo) Riemannian spacetimes. In this approach, the HL theory and scenarios of recovering GR at large distances are generated by imposing nonholonomic constraints on the dynamics of HF, or EFG, fields.     

Spin–gravity coupling and gravity-induced quantum phases

External gravitational fields induce phase factors in the wave functions of particles. The phases are exact to first order in the background gravitational field, are manifestly covariant and gauge invariant and provide a useful tool for the study of spin–gravity coupling and of the optics of particles in gravitational or inertial fields. We discuss the role that spin–gravity coupling plays in particular problems.     

Gravimagnetism,causality, and aberration of gravity in the gravitational light-ray deflection experiments

Experimental verification of the existence of gravimagnetic fields generated by currents of matter is important for a complete understanding and formulation of gravitational physics. Although the rotational (intrinsic) gravimagnetic field has been extensively studied and is now being measured by the Gravity Probe B, the extrinsic gravimagnetic field generated by the translational current of matter is less well studied. The present paper uses the post-Newtonian parametrized Einstein and light geodesics equations to show that the extrinsic gravimagnetic field generated by the translational current of matter can be measured by observing the relativistic time delay and/or light deflection caused by the moving mass. We prove that the extrinsic gravimagnetic field is generated by the relativistic effect of the aberration of the gravity force caused by the Lorentz transformation of the metric tensor and the Levi–Civita connection. We show that the Lorentz transformation of the gravity field variables is equivalent to the technique of the retarded Lienard–Wiechert gravitational potentials predicting that a light particle is deflected by gravitational field of a moving body from its retarded position so that both general-relativistic phenomena—the aberration and the retardation of gravity—are tightly connected and observing the aberration of gravity proves that gravity has a causal nature. We explain in this framework the 2002 deflection experiment of a quasar by Jupiter where the aberration of gravity from its orbital motion was measured with accuracy 20%. We describe a theory of VLBI experiment to measure the gravitational deflection of radio waves from a quasar by the Sun, as viewed by a moving observer from the geocentric frame, to improve the measurement accuracy of the aberration of gravity to a few percent.     

Nonholonomic Black Ring and Solitonic Solutions in Finsler and Extra Dimension Gravity Theories

We study stationary configurations mimicking nonholonomic locally anisotropic black rings (for instance, with ellipsoidal polarizations and/or imbedded into solitonic backgrounds) in three/six dimensional pseudo-Finsler/Riemannian spacetimes. In the asymptotically flat limit, for holonomic configurations, a subclass of such spacetimes contains the set of five dimensional black ring solutions with regular rotating event horizon. For corresponding parameterizations, the metrics and connections define Finsler–Einstein geometries modeled on tangent bundles, or on nonholonomic (pseudo) Riemannian manifolds. In general, there are vacuum nonholonomic gravitational configurations which can not be generated in the limit of zero cosmological constant.     

Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

We investigate statistics of occupation times for an over-damped Brownian particle in an external force field, using a backward Fokker–Planck equation introduced by Majumdar and Comtet. For an arbitrary potential field the distribution of occupation times is expressed in terms of solutions of the corresponding first passage time problem. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker–Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding cases, rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.     

Energy and Momentum Densities of Cosmological Models,with Equation of State <Emphasis Type="Italic">ρ</Emphasis>=<Emphasis Type="Italic">μ</Emphasis>, in General Relativity and Teleparallel Gravity

We calculated the energy and momentum densities of stiff fluid solutions, using Einstein, Bergmann–Thomson and Landau–Lifshitz energy-momentum complexes, in both general relativity and teleparallel gravity. In our analysis we get different results comparing the aforementioned complexes with each other when calculated in the same gravitational theory, either this is in general relativity and teleparallel gravity. However, interestingly enough, each complex’s value is the same either in general relativity or teleparallel gravity. Our results sustain that (i) general relativity or teleparallel gravity are equivalent theories (ii) different energy-momentum complexes do not provide the same energy and momentum densities neither in general relativity nor in teleparallel gravity. In the context of the theory of teleparallel gravity, the vector and axial-vector parts of the torsion are obtained. We show that the axial-vector torsion vanishes for the space-time under study.     

A naturally light dilaton and a small cosmological constant

We explore off-diagonal deformations of ‘prime’ metrics in Einstein gravity (for instance, for wormhole configurations) into ‘target’ exact solutions in $f(R,T)$ -modified and massive/bi-metric gravity theories. The new classes of solutions may, or may not, possess Killing symmetries and can be characterized by effective induced masses, anisotropic polarized interactions, and cosmological constants. For nonholonomic deformations with (conformal) ellipsoid/ toroid and/or solitonic symmetries and, in particular, for small eccentricity rotoid configurations, we can generate wormhole-like objects matching an external black ellipsoid—de Sitter geometries. We conclude that there are nonholonomic transforms and/or non-trivial limits to exact solutions in general relativity when modified/massive gravity effects are modeled by off-diagonal and/or nonholonomic parametric interactions.     

Early and late-time cosmic acceleration in non-minimal Yang–Mills-<Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity

In this paper we show that power-law inflation can be realized in non-minimal gravitational coupling of Yang–Mills field with a general function of the Gauss–Bonnet invariant in the framework of Einstein gravity. Such a non-minimal coupling may appear due to quantum corrections. We also discuss the non-minimal Yang–Mills-f(G) gravity in the framework of modified Gauss–Bonnet action which is widely studied recently. It is shown that both inflation and late-time cosmic acceleration are possible in such a theory.     

Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics

We present a study of fractional configurations in gravity theories and Lagrange mechanics. The approach is based on a Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists of a proof that, for corresponding classes of nonholonomic distributions, a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.     

Classes of exact Einstein–Maxwell solutions

We find new classes of exact solutions to the Einstein–Maxwell system of equations for a charged sphere with a particular choice of the electric field intensity and one of the gravitational potentials. The condition of pressure isotropy is reduced to a linear, second order differential equation which can be solved in general. Consequently we can find exact solutions to the Einstein–Maxwell field equations corresponding to a static spherically symmetric gravitational potential in terms of hypergeometric functions. It is possible to find exact solutions which can be written explicitly in terms of elementary functions, namely polynomials and product of polynomials and algebraic functions. Uncharged solutions are regainable with our choice of electric field intensity; in particular we generate the Einstein universe for particular parameter values.     

Aberration and the Fundamental Speed of Gravity in the Jovian Deflection Experiment

We describe our explicit Lorentz-invariant solution of the Einstein and null geodesic equations for the deflection experiment of 2002 September 8 when a massive moving body, Jupiter, passed within 3.7’ of a line-of-sight to a distant quasar. We develop a general relativistic framework which shows that our measurement of the retarded position of a moving light-ray deflecting body (Jupiter) by making use of the gravitational time delay of quasar’s radio wave is equivalent to comparison of the relativistic laws of the Lorentz transformation for gravity and light. Because, according to Einstein, the Lorentz transformation of gravity field variables must depend on a fundamental speed c, its measurement through the retarded position of Jupiter in the gravitational time delay allows us to study the causal nature of gravity and to set an upper limit on the speed of propagation of gravity in the near zone of the solar system as contrasted to the speed of the radio waves. In particular, the v/c term beyond of the standard Einstein’s deflection, which we measured to 20% accuracy, is associated with the aberration of the null direction of the gravity force (“aberration of gravity”) caused by the Lorentz transformation of the Christoffel symbols from the static frame of Jupiter to the moving frame of observer. General relativistic formulation of the experiment identifies the aberration of gravity with the retardation of gravity because the speed of gravitational waves in Einstein’s theory is equal to the speed of propagation of the gravity force. We discuss the misconceptions which have inhibited the acceptance of this interpretation of the experiment. We also comment on other interpretations of this experiment by Asada, Will, Samuel, Pascual–Sánchez, and Carlip and show that their “speed of light” interpretations confuse the Lorentz transformation for gravity with that for light, and the fundamental speed of gravity with the physical speed of light from the quasar. For this reason, the “speed of light” interpretations are not entirely consistent with a retarded Liénard–Wiechert solution of the Einstein equations, and do not properly incorporate how the phase of the radio waves from the quasar is perturbed by the retarded gravitational field of Jupiter. Although all of the formulations predict the same deflection to the order of v/c, our formulation shows that the underlying cause of this deflection term is associated with the aberration of gravity and not of light, and that the interpretations predict different deflections at higher orders of v/c beyond the Shapiro delay, thus, making their measurement highly desirable for deeper testing of general relativity in future astrometric experiments like Gaia, SIM, and SKA.     

Non-gravitating scalar field in the FRW background

We study interacting scalar field theory non-minimally coupled to gravity in the FRW background. We show that for a specific choice of interaction terms, the energy–momentum tensor of the scalar field ϕ vanishes, and as a result the scalar field does not gravitate. The naive space dependent solution to equations of motion gives rise to singular field profile. We carefully analyze the energy–momentum tensor for such a solution and show that the singularity of the solution gives a subtle contribution to the energy–momentum tensor. The space dependent solution therefore is not non-gravitating. Our conclusion is applicable to other space–time dependent non-gravitating solutions as well. We study hybrid inflation scenario in this model when purely time dependent non-gravitating field is coupled to another scalar field χ.     

Non-minimal Maxwell-modified Gauss–Bonnet cosmologies: inflation and dark energy

In this paper we show that power-law inflation can be realized in non-minimal gravitational coupling of electromagnetic field with a general function of the Gauss–Bonnet invariant. Such a non-minimal coupling may appear due to quantum corrections. We also consider modified Maxwell-F(G) gravity in which non-minimal coupling between electromagnetic field and f(G) occurs in the framework of modified Gauss–Bonnet gravity. It is shown that inflationary cosmology and late-time accelerated expansion of the universe are possible in such a theory.     

On the <Emphasis Type="Italic">Λ</Emphasis>CDM Universe in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity

In the context of the so-called Gauss–Bonnet gravity, where the gravitational action includes function of the Gauss–Bonnet invariant, we study cosmological solutions, especially the well-known ΛCDM model. It is shown that the dark energy contribution and even the inflationary epoch can be explained in the frame of this kind of theories with no need of any other kind of component. Other cosmological solutions are constructed and the rich properties that this kind of theories provide are explored.     

Stationary Dyonic regular and black hole solutions

We consider globally regular and black hole solutions in SU(2) Einstein–Yang–Mills–Higgs theory, coupled to a dilaton field. The basic solutions represent magnetic monopoles, monopole–antimonopole systems or black holes with monopole or dipole hair. When the globally regular solutions carry additionally electric charge, an angular momentum density results, except in the simplest spherically symmetric case. We evaluate the global charges of the solutions and their effective action, and analyze their dependence on the gravitational coupling strength. We show, that in the presence of a dilaton field, the black hole solutions satisfy a generalized Smarr type mass formula. B. Kleihaus gratefully acknowledges support by the German Aerospace Center. F. Navarro-Lérida gratefully acknowledges support by the Ministerio de Educación y Ciencia under grant EX2005-0078.     

Periodic relativity: basic framework of the theory

An alternative gravity theory is proposed which does not rely on Riemannian geometry and geodesic trajectories. The theory named periodic relativity (PR) does not use the weak field approximation and allows every two body system to deviate differently from the flat Minkowski metric. PR differs from general relativity (GR) in predictions of the proper time intervals of distant objects. PR proposes a definite connection between the proper time interval of an object and gravitational frequency shift of its constituent particles as the object travels through the gravitational field. PR is based on the dynamic weak equivalence principle which equates the gravitational mass with the relativistic mass. PR provides very accurate solutions for the Pioneer anomaly and the rotation curves of galaxies outside the framework of general relativity. PR satisfies Einstein’s field equations with respect to the three major GR tests within the solar system and with respect to the derivation of Friedmann equation in cosmology. This article defines the underlying framework of the theory.     

Measurement of the space–time interval between two events using the retarded and advanced times of each event with respect to a time-like world-line

Several recent studies have been devoted to investigating the limitations that standard quantum mechanics and/or quantum gravity might impose on the measurability of space–time observables. These analyses are often confined to the simplified context of 2D flat space–time and rely on a simple procedure for the measurement of space-like distances based on the exchange of light signals. We present a generalization of this measurement procedure applicable to all three types of space–time intervals between two events in space–times of any number of dimensions. We also present a preliminary account of an alternative measurement procedure that can be applied taking into account the gravitational field of the macroscopic measuring apparatus.     

Creep,relaxation and viscosity properties for basic fractional models in rheology

The purpose of this paper is twofold: from one side we provide a general survey to the viscoelastic models constructed via fractional calculus and from the other side we intend to analyze the basic fractional models as far as their creep, relaxation and viscosity properties are considered. The basic models are those that generalize via derivatives of fractional order the classical mechanical models characterized by two, three and four parameters, that we refer to as Kelvin–Voigt, Maxwell, Zener, anti–Zener and Burgers. For each fractional model we provide plots of the creep compliance, relaxation modulus and effective viscosity in non dimensional form in terms of a suitable time scale for different values of the order of fractional derivative. We also discuss the role of the order of fractional derivative in modifying the properties of the classical models.     

Cylindrically symmetric Brans–Dicke–Maxwell solutions

We investigate cylindrically symmetric vacuum solutions with both null and non-null electromagnetic fields in the framework of the Brans–Dicke theory and compare these solutions with some of the well-known solutions of general relativity for special values of the parameters of the resulting field functions. We see that, unlike general relativity where the gravitational force of an infinite and charged line mass acting on a test particle is always repulsive, it can be attractive or repulsive for Brans–Dicke theory depending on the values of the parameters as well as the radial distance from the symmetry axis.