Radial asymptotics of Lemaître–Tolman–Bondi dust models

We examine the radial asymptotic behavior of spherically symmetric Lemaître–Tolman–Bondi dust models by looking at their covariant scalars along radial rays, which are spacelike geodesics parametrized by proper length ?, orthogonal to the 4-velocity and to the orbits of SO(3). By introducing quasi-local scalars defined as integral functions along the rays, we obtain a complete and covariant representation of the models, leading to an initial value parametrization in which all scalars can be given by scaling laws depending on two metric scale factors and two basic initial value functions. Considering regular “open” LTB models whose space slices allow for a diverging ?, we provide the conditions on the radial coordinate so that its asymptotic limit corresponds to the limit as ? → ∞. The “asymptotic state” is then defined as this limit, together with asymptotic series expansion around it, evaluated for all metric functions, covariant scalars (local and quasi-local) and their fluctuations. By looking at different sets of initial conditions, we examine and classify the asymptotic states of parabolic, hyperbolic and open elliptic models admitting a symmetry center. We show that in the radial direction the models can be asymptotic to any one of the following spacetimes: FLRW dust cosmologies with zero or negative spatial curvature, sections of Minkowski flat space (including Milne’s space), sections of the Schwarzschild–Kruskal manifold or self-similar dust solutions.     

Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space–time

Gravitational radiation of binary systems can be studied by using the adiabatic approximation in General Relativity. In this approach a small astrophysical object follows a trajectory consisting of a chained series of bounded geodesics (orbits) in the outer region of a Kerr Black Hole, representing the space time created by a bigger object. In our paper, we study the entire class of orbits, both of constant radius (spherical orbits), as well as non-null eccentricity orbits, showing a number of properties on the physical parameters and trajectories. The main result is the determination of the geometrical locus of all the orbits in the space of physical parameters in Kerr space–time. This becomes a powerful tool to know if different orbits can be connected by a continuous change of their physical parameters. A discussion on the influence of different values of the angular momentum of the hole is given. Main results have been obtained by analytical methods.     

Measurement of the space–time interval in modified gravity theories in Palatini formalism

There are strong motivations that lead cosmologists to consider alternatives to Einstein’s theory of gravity in Palatini formalism. In addition, there are two distinguishable local frames in this formalism in which one of them is local inertial frame and the equivalence principle is satisfied. Different features of speed of light such as the causal structure constant, electromagnetic and gravitational wave velocities and Einstein velocity will not coincide in this local inertial frame for extended gravity theories in Palatini formalism. On the other hand, both the measurement of time and exchange of a signal between the distant points are required to determine spatial distances. In a particular situation where these aspects of the speed of light do not coincide, the distance determination will become more demanding because light will follow a time-like geodesic of the metric. In modified gravity theories in Palatini approach, theories involve a varying speed of photon. Therefore these kinds of theories must be based on some other technique of measuring spatial distances than radar or some terms should be corrected in the line element in the proposed model. We found out we should consider a coefficient which is proportional to energy density in each era, in the line element in order to be able to use radar for measuring distance in modified gravity theories in Palatini formalism. Analysis of some observational data will be affected by considering this coefficient in the line element.     

Exact solutions to the space–time fractional Schrödinger–Hirota equation and the space–time modified KDV–Zakharov–Kuznetsov equation

In this paper, the first integral method and the functional variable method are used to establish exact traveling wave solutions of the space–time fractional Schrödinger–Hirota equation and the space–time fractional modified KDV–Zakharov–Kuznetsov equation in the sense of conformable fractional derivative. The results obtained confirm that proposed methods are efficient techniques for analytic treatment of a wide variety of the space–time fractional partial differential equations.     

On the motion of a quantum particle in the spinning cosmic string space–time

We analyze the energy spectrum and the wave function of a particle subjected to magnetic field in the spinning cosmic string space–time and investigate the influence of the spinning reference frame and topological defect on the system. To do this we solve Schrödinger equation in the spinning cosmic string background. In our work, instead of using an approximation in the calculations, we use the quasi-exact ansatz approach which gives the exact solutions for some primary levels.     

Analytic solutions of the space–time conformable fractional Klein–Gordon equation in general form

ABSTRACT

The Klein–Gordon equation plays an important role in mathematical physics. In this paper, a direct method which is very effective, simple, and convenient, is presented for solving the conformable fractional Klein–Gordon equation. Using this analytic method, the exact solutions of this equation are found in terms of the Jacobi elliptic functions. This method is applied to both time and space fractional equations. Some solutions are also illustrated by the graphics.     

Parametrization of Bose–Einstein correlations and reconstruction of the space–time evolution of pion production in ee annihilation

A parametrization of the Bose–Einstein correlation function of pairs of identical pions produced in hadronic e⁺e⁻ annihilation is proposed within the framework of a model (the τ-model) in which space–time and momentum space are very strongly correlated. Using information from the Bose–Einstein correlations as well as from single-pion spectra, it is then possible to reconstruct the space–time evolution of pion production.     

Spinning particles in Schwarzschild–de Sitter space–time

After considering the reference case of the motion of spinning test bodies in the equatorial plane of the Schwarzschild space–time, we generalize the results to the case of the motion of a spinning particle in the equatorial plane of the Schwarzschild–de Sitter space–time. Specifically, we obtain the loci of turning points of the particle in this plane. We show that the cosmological constant affect the particle motion when the particle distance from the black hole is of the order of the inverse square root of the cosmological constant.     

Spin 3/2 Field Equation: Separation and Solution in a Class of Lema?tre–Tolman–Bondi Cosmologies

The spin 3/2 field equation is studied in the general Lema?tre–Tolman–Bondi (LTB) space-time. The equation is separated by variable separation. The angular dependence factors out at the level of the general LTB metric. Due to spherical symmetry the separated angular equations coincide with those, previously integrated, relative to the Robertson–Walker and Schwarzschild metric. Separation of time and radial dependence is possible within a class of LTB cosmological models for which the physical radius is a product of a time and a radial function, the last one being further selected by the consistency condition of the radial equations. The separated time dependence, that can be integrated by series, results essentially unique. Instead the radial dependence can be reduced to two independent second order ordinary differential equations that still depend on an arbitrary radial function that is an integration function of the cosmological model. The generalization of the scheme to arbitrary spin field equation is suggested.     

Sobolev gradient approach for the time evolution related to energy minimization of Ginzburg–Landau functionals

The Sobolev gradient technique has been discussed previously in this journal as an efficient method for finding energy minima of certain Ginzburg–Landau type functionals [S. Sial, J. Neuberger, T. Lookman, A. Saxena, Energy minimization using Sobolev gradients: application to phase separation and ordering, J. Comput. Phys. 189 (2003) 88–97]. In this article a Sobolev gradient method for the related time evolution is discussed.     

Comment on “Strength and genericity of singularities in Tolman–Bondi–de Sitter collapse” and a note on central singularities

It has been claimed in Phys. Lett. A 287 (2001) 53 that the Lemaitre–Tolman–Bondi–de Sitter solution always admits future-pointing radial time-like geodesics emerging from the shell-focussing singularity, regardless of the nature of the (regular) initial data. This is despite the fact that some data rule out the emergence of future pointing radial null geodesics. We correct this claim and show that, in general in spherical symmetry, the absence of radial null geodesics emerging from a central singularity is sufficient to prove that the singularity is censored.     

A stochastic approach to the Euler-Poincaré number of the loop space of a developable orbifold

We define a regularised version of the de Rham operator over the free loop space. We perform a semi-classical approximation of it, such that the index of the limit operator is equal to the “orbit Euler characteristic” of physicists.     

Gravitational collapse in higher dimensional Husain space–time

We investigate exact solution in higher dimensional Husain model for a null fluid source with pressure p and density ρ are related by the following relations (i) p  =  kρ, (ii) (variable modified Chaplygin) and (iii) p  =  kρ^α (polytropic). We have studied the nature of singularity in gravitational collapse for the above equations of state and also for different choices of the of the parameters k and B namely, (i) k  =  0, B  =  constant (generalized Chaplygin), (ii) B  =  constant (modified Chaplygin). It is found that the nature of singularity is independent of these choices of different equation of state except for variable Chaplygin model. Choices of various parameters are shown in tabular form. Finally, matching of Szekeres model with exterior Husain space–time is done.     

The Riemann tensor and the Bianchi identity in 5D space–time

The initial assumption of theories with extra dimension is based on the efforts to yield a geometrical interpretation of the gravitation field. In this paper, using an infinitesimal parallel transportation of a vector, we generalize the obtained results in four dimensions to five-dimensional space–time. For this purpose, we first consider the effect of the geometrical structure of 4D space–time on a vector in a round trip of a closed path, which is basically quoted from chapter three of Ref. [5]. If the vector field is a gravitational field, then the required round trip will lead us to an equation which is dynamically governed by the Riemann tensor. We extend this idea to five-dimensional space–time and derive an improved version of Bianchi's identity. By doing tensor contraction on this identity, we obtain field equations in 5D space–time that are compatible with Einstein's field equations in 4D space–time. As an interesting result, we find that when one generalizes the results to 5D space–time, the new field equations imply a constraint on Ricci scalar equations, which might be containing a new physical insight.