Spinning solutions in general relativity with infinite central density

This paper presents general relativistic numerical simulations of uniformly rotating polytropes. Equations are developed using MSQI coordinates, but taking a logarithm of the radial coordinate. The result is relatively simple elliptical differential equations. Due to the logarithmic scale, we can resolve solutions with near-singular mass distributions near their center, while the solution domain extends many orders of magnitude larger than the radius of the distribution (to connect with flat space–time). Rotating solutions are found with very high central energy densities for a range of adiabatic exponents. Analytically, assuming the pressure is proportional to the energy density (which is true for polytropes in the limit of large energy density), we determine the small radius behavior of the metric potentials and energy density. This small radius behavior agrees well with the small radius behavior of large central density numerical results, lending confidence to our numerical approach. We compare results with rotating solutions available in the literature, which show good agreement. We study the stability of spherical solutions: instability sets in at the first maximum in mass versus central energy density; this is also consistent with results in the literature, and further lends confidence to the numerical approach.     

Anisotropic static solutions in modelling highly compact bodies

Einstein field equations for static anisotropic spheres are solved and exact interior solutions obtained. This paper extends earlier treatments to include anisotropic models which accommodate a wider variety of physically viable energy densities. Two classes of solutions are possible. The first class contains the limiting caseμ,∝ r^-2 for the energy density which arises in many astrophysical applications. In the second class the singularity at the centre of the star is not present in the energy density.     

Regular and Black Hole Skyrmions with Axisymmetry

It has been known that a B=2 skyrmion is axially symmetric. We consider the Skyrme model coupled to gravity and obtain static axially symmetric regular and black hole solutions numerically. Computing the energy density of the skyrmion, we discuss the effect of gravity to the energy density and baryon density of the skyrmion.     

Photon Stars

We discuss numerical solutions of Einstein's field equation describing static, spherically symmetric conglomerations of a photon gas. These equations imply a back reaction of the metric on the energy density of the photon gas according to Tolman's equation. The 3–fold of solutions corresponds to a class of physically different solutions which is parameterized by only two quantities, i.e. mass and surface temperature. The energy density is typically concentrated on a shell because the center contains a repelling singularity, which can, however, not be reached by timelike geodesics and only by radial null geodesics. The physical relevance of these solutions is completely open, although their existence may raise some doubts w.r.t. the stability of black holes.     

Bianchi Type I Viscous Universe with Variable G and Λ

Exact solutions for an anisotropic Bianchi type I model with bulk viscosity and variable G and are obtained. We have found some solutions that correspond to our earlier work for the isotropic one. Unlike Kalligas et al., an inflationary solution with a variable energy density has been found where the anisotropy energy decreases exponentially with time. There is a period of hyper-inflation during which the energy density remains constant.     

Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term that depends on theS 2−P 2 invariant

Exact plane-symmetric solutions of the spinor-field equation with zero mass parameter and nonlinear term that depends arbitrarily on the S²−P² invariant are derived with consideration of an intrinsic gravitational field. The existence of regular solutions with localized energy density among the solutions obtained is investigated. Equations with powerlaw and polynomial nonlinearity types are examined in detail. For the power-law nonlinearity, when the nonlinear term entering into the Lagrangian has the form L_N=γIⁿ, where γ is the nonlinearity parameter and n=const, it is shown that the initial system of Einstein and spinor-field equations has regular solutions with localized energy density only under the conditions λ=−Λ² < 0, n > 1. In this case, the examined field configuration posesses a negative energy. In the case of polynomial nonlinearity, regular solutions with localized energy density T ₀ ⁰ (x), positive energy (upon integration over y and z between finite limits), and an everywhere regular metric that transforms into a two-dimensional space-time metric at spatial infinity are obtained. It is shown that the initial nonlinear spinor-field equations in two-dimensional space-time have no solutions with localized energy density. Thus, it is established that the intrinsic gravitational field plays a regularizing role in the frmation of regular localized solutions to the examined nonlinear spinor-field equations. Russian University of People's Friendship. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 12–19, November, 1999.     

Vibrational energy flow models for the Rayleigh–Love and Rayleigh–Bishop rods

Energy Flow Analysis (EFA) has been developed to predict the vibrational energy density of the system structures in the medium-to-high frequency range. The elementary longitudinal wave theory is often used to describe the longitudinal vibration of a slender rod. However, for relatively large diameter rods or high frequency ranges, the elementary longitudinal wave theory is inaccurate because the lateral motions are not taken into account. In this paper, vibrational energy flow models are developed to analyze the longitudinally vibrating Rayleigh–Love rod considering the effect of lateral inertia, and the Rayleigh–Bishop rod considering the effect not only of the lateral inertia but also of the shear stiffness. The derived energy governing equations are second-order differential equations which predict the time and space averaged energy density and active intensity distributions in a rod. To verify the accuracy of the developed energy flow models, various numerical analyses are performed for a rod and coupled rods. Also, the EFA results for the Rayleigh–Love and Rayleigh–Bishop rods are compared with the analytical solutions for these models, the traditional energy flow solutions, and the analytical solutions for the classical rod.     

Physical properties of scalar and spinor field states with the Rindler-Milne (hyperbolic) symmetry

It is shown that right and left combinations of the positive-and negative-frequency hyperbolically symmetric solutions of the Klein-Fock-Gordon equation possess an everywhere timelike current density vector with a definite Lorentz-invariant sign of the charge density, and similar combinations of solutions to the Dirac equation possess the energy-momentum tensor with everywhere real eigenvalues and a definite Lorentz-invariant sign of the energy density. These right and left modes, just as their ±-frequency components, are eigenfunctions of the Lorentz boost generator with the eigenvalue к. The sign of the charge (energy) density coincides with the sign of к for the right scalar (spinor) modes and is opposite to it for the left modes. It is then reasonable to assume that the particles (antiparticles) are precisely described by the right modes with к>0(к<0) and by the left modes with к<0(к>0).     

Dark energy interacting with dark matter and a third fluid: Possible EoS for this component

A cosmological model of dark energy interacting with dark matter and another general component of the universe is considered. The equations for the coincidence parameters r and s, which represent the ratios between dark energy and dark matter and the other cosmic fluid respectively, are analyzed in terms of the stability of stationary solutions. The obtained general results allow to shed some light on the equations of state of the three interacting fluids, due to the constraints imposed by the stability of the solutions. We found that for an interaction proportional to the sum of the dark energy density and the third fluid density, the hypothetical fluid must have positive pressure, which leads naturally to a cosmological scenario with radiation, unparticle or even some form of warm dark matter as the third interacting fluid.     

The Lynden-Bell and Katz definition of gravitational energy: Applications to singular solutions

We apply the Lynden-Bell and Katz (LK) definition of gravitational energy to static and spherically symmetric space-times which admit a curvature singularity. These are the Tolman V, Tolman VI and the interior Schwarzschild solutions, the latter with the boundary limit of 9/8th of the gravitational radius. We show that the LK definition can still be applied to these solutions despite the presence of a singularity which nonetheless appears to carry no energy in the LK sense. While in the solutions that we mentioned the KL gravitational energy is positive definite everywhere in space time, this is not the case for the overcharged Reissner-Nordström space-time. In the latter case in fact the LK energy density becomes negative sufficiently close to the singularity hence we use the positivity criterion to impose a more stringent limit of validity to the Reissner-Nordström solution.     

Dynamical Casimir Effect in a Cavity with a Resonantly Oscillating Boundary

We present analytical solutions describing quantized vacuum field in a one-dimensional cavity with one of its two mirrors fixed and another vibrating in simple harmonic form.These solutions are accurate up to the second order of the oscillating magnitude and they are uniformly valid for all time.We obtain the simple analytical expression for the energy density of the field which explicitly manifests that for a cavity vibrating at its q-th (q≥2) eigenfrequency, q traveling wave packets emerge in the finite part of the field energy density,and their amplitudes grow their widths shrink in time,representing a large concentration of energy.The finite part of the field energy density originating from the oscillations is shown to be proportional to the factor(q^2-1).     

Kinetic Thomas-Fermi solutions of the Gross-Pitaevskii equation

Approximate solutions of the Gross-Pitaevskii (GP) equation, obtained upon neglection of the kinetic energy, are well known as Thomas-Fermi solutions. They are characterized by the compensation of the local potential by the collisional energy. In this article we consider exact solutions of the GP-equation with this property and definite values of the kinetic energy, which suggests the term “kinetic Thomas-Fermi” (KTF) solutions. Despite their formal simplicity, KTF-solutions can possess complex current density fields with unconventional topology. We point out that a large class of light-shift potentials gives rise to KTF-solutions. As elementary examples, we consider one-dimensional and two-dimensional optical lattice scenarios, obtained by means of the superposition of two, three and four laser beams, and discuss the stability properties of the corresponding KTF-solutions. A general method is proposed to excite two-dimensional KTF-solutions in experiments by means of time-modulated light-shift potentials.     

Relativistic stellar models

We obtain a class of solutions to the Einstein–Maxwell equations describing charged static spheres. Upon specifying particular forms for one of the gravitational potentials and the electric field intensity, the condition for pressure isotropy is transformed into a hypergeometric equation with two free parameters. For particular parameter values we recover uncharged solutions corresponding to specific neutron star models. We find two charged solutions in terms of elementary functions for particular parameter values. The first charged model is physically reasonable and the metric functions and thermodynamic variables are well behaved. The second charged model admits a negative energy density and violates the energy conditions.     

Non-minimal coupling of torsion–matter satisfying null energy condition for wormhole solutions

We explore wormhole solutions in a non-minimal torsion–matter coupled gravity by taking an explicit non-minimal coupling between the matter Lagrangian density and an arbitrary function of the torsion scalar. This coupling describes the transfer of energy and momentum between matter and torsion scalar terms. The violation of the null energy condition occurred through an effective energy-momentum tensor incorporating the torsion–matter non-minimal coupling, while normal matter is responsible for supporting the respective wormhole geometries. We consider the energy density in the form of non-monotonically decreasing function along with two types of models. The first model is analogous to the curvature–matter coupling scenario, that is, the torsion scalar with T-matter coupling, while the second one involves a quadratic torsion term. In both cases, we obtain wormhole solutions satisfying the null energy condition. Also, we find that the increasing value of the coupling constant minimizes or vanishes on the violation of the null energy condition through matter.     

Amplification of Gravitational Waves in Scalar-Tensor Inflationary Lambda-Model

After reviewing the scalar-tensor inflationary solutions by Berman and Trevisan (Int. J. Theor. Phys. 29, 1411–1414, 2009), we obtain solutions for the amplification of gravitational waves in the models. The solutions consider a perfect gas equation of state, with cosmic pressure proportional to the energy density, the proportionality constant being smaller than −2/3, and a cosmological term.     

New family of simple solutions of relativistic perfect fluid hydrodynamics

A new class of accelerating, exact and explicit solutions of relativistic hydrodynamics is found—more than 50 years after the previous similar result, the Landau–Khalatnikov solution. Surprisingly, the new solutions have a simple form, that generalizes the renowned, but accelerationless, Hwa–Bjorken solution. These new solutions take into account the work done by the fluid elements on each other, and work not only in one temporal and one spatial dimensions, but also in arbitrary number of spatial dimensions. They are applied here for an advanced estimation of initial energy density and life-time of the reaction in ultra-relativistic heavy ion collisions. New formulas are also conjectured, that yield further important increase of the initial energy density estimate and the measured life-time of the reaction if the value of the speed of sound is in the realistic range.     

Homogeneous Statistical Solutions and Local Energy Inequality for 3D Navier-Stokes Equations

We are interested in space-time spatially homogeneous statistical solutions of Navier-Stokes equations in space dimension three. We first review the construction of such solutions, and introduce convenient tools to study the pressure gradient. Then we show that given a spatially homogeneous initial measure with finite energy density, one can construct a homogeneous statistical solution concentrated on weak solutions which satisfy the local energy inequality.     

New exact cylindrical solutions of the Einstein-Maxwell equations of general relativity

We find new exact cylindrical solutions of the Einstein-Maxwell equations, which are cylindrical waves. In some of these solutions Painlevé transcendental functions of type III and V appear. There are maxima in the electromagnetic energy density.     

Wannier–Stark chaos of a Bose–Einstein condensate in 1D optical lattices

Using the idea of the macroscopic quantum wave function and the definition of the Melnikov chaos, we investigate the spatially chaotic features of a Bose–Einstein condensate (BEC) in a Wannier–Stark potential for the trivial phase and the non-trivial phase cases. The perturbed chaotic solutions are constructed, and the chaotic and unstable regions on the parameter space are illustrated. Numerical calculations to the spatial evolutions of the atomic number density and the energy density demonstrate the analytical results and exhibit the chaotic spatial distribution and energy distribution of the BEC atoms.