On the Reynolds stresses in mean-field hydrodynamics II. Two-dimensional turbulence and the problem of negative viscosity

The methods proposed in a foregoing paper are used for the derivation of the eddy viscosity of a two-dimensional homogeneous isotropic turbulence. In contrast to three-dimensional isotropic turbulence is shown that for the two-dimensional case the eddy viscosity (i) has no definite sign, (ii) tends to zero if the molecular viscosity tends to zero, (iii) is negative for special cases. However, the modes with large wave numbers decay in any case as is shown by investigating a dispersion relation.     

The different transport regimes of pitch-angle scattering of energetic particles

Recently an advanced nonlinear diffusion theory for particle transport across the mean magnetic field has been developed. The method used in the derivation of the latter theory is based on the cosmic ray Fokker-Planck equation. In the present article we use the same approach to describe pitch-angle scattering and parallel spatial diffusion nonlinearly. Furthermore, we derive the quasilinear transport theory, the weakly nonlinear theory as well as the Bohm limit as special cases from our more general approach.     

Spacetime is not just space and time

Our physical intuition usually separates space from time, ignoring the spacetime character of the physical reality. In strongly curved spacetimes this may lead to confusion and paradoxes. I present here two examples: (1) in the non-static cosmological spacetimes with flat space sections, the cosmological expansion of space is a true physical effect. Contrary to what the intuition imagines, it cannot be explained as a motion of matter in a non-expanding flat space. (2) Contrary to intuition, for static spacetimes the mathematically simplest 3+1 split is not given by the direct projection as in the standard ADM scheme. The simplest split is defined by a counter intuitive “optical geometry” that redshifts both space and time by the same conformal factor.     

Black holes and rotation

In this article, we first consider briefly the basic properties of the non-rotating Schwarzschild black hole and the rotating Kerr black hole Rotational effects are then described in static and stationary spacetimes with arial symmetry by studying inertial forces, gyroscopic precession and gravi-electromagnetism. The results are applied to the black hole spacetimes.     

The plane symmetric vacuum solutions of modified field equations in metric f(R) gravity

The f(R) theories of gravity have been interested in recent years. A considerable amount of work has been devoted to the study of modified field equations with the assumption of constant Ricci scalar which may be zero or nonzero. In this paper, the exact vacuum solutions of plane symmetric spacetime are analyzed in f(R) theory of gravity. The modified field equations are studied not only for R=constant but also for general case R≠constant. In particular, we show that the Novotný-Horský and anti-de Sitter spacetimes are the exact solutions of the field equations with the non-zero constant Ricci scalar. Finally, the family of solutions with R≠constant is obtained explicitly which includes the Novotný-Horský, Kottler-Whittaker, Taub and conformally flat spacetimes.     

Solutions to the field equations and the deceleration parameter

We utilise a form for the Hubble parameter to generate a number of solutions to the Einstein field equations with variable cosmological constant and variable gravitational constant. The Hubble law utilised yields a constant value for the deceleration parameter. A variety of solutions is presented in the Robertson-Walker spacetimes. A generalisation of the cosmic scale factor is utilised in the anisotropic Bianchi I spacetime to illustrate that new solutions may also be found in spacetimes with less symmetry than Robertson-Walker. We also show that the constant deceleration parameter used is consistent with alternate theories of gravity by considering the scalar-tensor theory of Lau and Prokhovnik with ak = 0 Robertson-Walker background.     

A prediction method for ground-based stellar occultations by ellipsoidal solar system bodies and its application

A new programmable prediction method is developed to refine the occultation band by taking into consideration the triaxiality of an occulting body, as well as two more factors, namely, the barycenter offset of an occulting planet from the relevant planetary satellite system and the gravitational deflection of light rays due to an occulting planet. Although these factors can be neglected in most cases, it is shown that there are cases when these factors can cause a variation ranging from several tens to thousands of kilometers in the boundaries of occultation bands. Knowledge of analytic geometry simplifies the process of derivation and computation. This method is applied to long-term predictions of Jovian and Saturnian events.     

Synchrotron emission from anisotropic disordered magnetic fields

We derive expressions for the total and linearly polarized synchrotron emissivity of an element of plasma containing relativistic particles and disordered magnetic field that has been sheared or compressed along three independent directions. Our treatment follows that given by Matthews & Scheuer in the special case of a power-law electron energy spectrum. We show that the emissivity integrals depend on a single parameter, making it straightforward to generate one-dimensional look-up tables. We also demonstrate that our formulae give identical results to those in the literature in special cases.     

FRW and Bianchi type I cosmology of f-essence

F-essence is a generalization of the usual Dirac model with the nonstandard kinetic term. In this paper, we introduce a new model of spinor cosmology containing both Ricci scalar and the non minimally coupled spinor fields in its action. We have investigated the cosmology with both isotropy and anisotropy, where the equations of motion of FRW and Bianchi type-I spacetimes have been derived and solved numerically. Finally the quantization of these models through Wheeler-De Witt (WD) wave function has been discussed.     

Exact spatially self-similar cosmological models of Bianchi type-fV

We investigate the Einstein field equations for tilted spatially self-similar Bianchi type-_fV spacetimes. The solutions represent anisotropic models which can be made consistent with the observed isotropy.     

Gravitational waves: The future of black hole physics

The new millennium will witness the operation of several long-baseline ground-based interferometric detectors, possibly a space-based detector too, which will make it possible to directly observe black holes by catching gravitational waves emitted by them during their formation or when they are perturbed or when a binary consisting of black holes in-spirals due to radiation reaction. Such observations will help us not only to test some of the fundamental predictions of Einstein's general relativity but will also give us the unique opportunity to map black hole spacetimes, to measure the masses and spins of black holes and their population, etc.     

Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque

New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

An amended magnetohydrodynamic equation which predicts field-aligned current sheets

A derivation of the physical conditions for magnetohydrodynamic equilibrium from first principles establishes a set of equations which differ slightly from the usual ones. The difference is only relevant in the special case when the current is parallel to the magnetic field. What is noteworthy is that these modified equations predict that the simplest magnetohydrodynamic equilibria can only exist in regions of field-aligned current sheets. This prediction is entirely consistent with the observed large-scale structures in the Earth's magnetosphere.     

Magnetic field in a cavity of a perfect diamagnetic material

A new method using an integral equation has been proposed in this paper for calculating the magnetic field enclosed within a perfect diamagnetic material. A magnetic field is assumed to exist inside an arbitrary cavity in a perfect diamagnetic material. Self-consistent magnetic fields inside the cavity and currents on its surface are calculated, resulting as it should in zero magnetic field outside the cavity. This method has been tested in a special case and the results have been compared to the analytical solutions. This method should be also applicable to any case concerning the interaction between magnetic fields and perfect diamagnetic materials, e.g., to cases where a perfect diamagnetic material is surrounded by a magnetic field.     

Concerning the regularizing KS-transformation

In this note we give by means of quaternions in vector notation a new derivation of the KS-transformation acting from a four-dimensional parameter space into the three-dimensional physical space. Using quaternions in vector notation each step in the derivation has an immediate geometrical interpretation. In particular, the KS-transformation appears as the Levi-Civita transformation, formulated in a rotated coordinate system.Dedicated to Professor Otto Volk.     

Anisotropic and inhomogeneous solutions of the Einstein-Boltzmann equations for Friedmann-Robertson-Walker spacetimes

It is shown how the techniques that yield anisotropic and inhomogeneous solutions of the Einstein-Liouville equations for Friedmann-Robertson-Walker (FRW) spacetimes can be used to demonstrate the existence of anisotropic and inhomogeneous FRW solutions of the Einstein-Boltzmann equations, thereby providing a possible relativistic kinetic theory basis (and consequently a more self-consistent basis) for tilting imperfect fluid FRW cosmological models (based on a set of phenomenological laws of relativistic thermodynamics).     

Light bending in Reissner-Nordstrom-de Sitter black hole by Rindler-Ishak method

We investigate the influence of the cosmological constant, Λ, on the bending of light by a charged black hole in a de Sitter spacetime. Despite vanishing of the cosmological constant in the second order null geodesic equation, considering the method introduced by Rindler and Ishak (2007), we obtain an expression for the deflection angle, consistent with previous results for Schwarzschild, Schwarzschild-de Sitter (SdS), and Reissner-Nordstrom (RN) spacetimes.     

Simple impact crater shape determination from shadows

Crater depths, often obtained from shadow measurements, have long been used for several purposes in planetary science. However, the usual method for obtaining depth from shadow length suffers from several drawbacks and limitations. Chappelow and Sharpton (2002) introduced a much improved shadow method, which has the advantages of giving some shape information (as well as depth), and is not limited to shadows that cross the crater bottom. However, it is not general, in that it only gives very approximate crater shape information, in terms of three special cases (parabolic, conical, or flat‐floored). Here, I present a completely generalized method, valid for any conic section shaped crater, and give a proof of concept and demonstration of its use, using Linne crater as a test case. In the process, I find that Linne is neither parabolic nor conical, and that it contains approximately 20 m of bottom fill, which forms a flat floor. I also conclude that the long‐used parabolic paradigm for the shapes of simple craters may need to be revised.     

Wave propagation in protoplanetary disks: Formation of twin planets by ‘disk-brown dwarf’ collisions?

In this paper we derive a special linear non-vortical wave propagation solution in the shearing sheet, a model of a compressible two-dimensional fluid system with constant density, constant shear and constant Coriolis force, but without self-gravity. The linear analysis of the shearing sheet leads to a single differential equation for the azimuthal velocity perturbation. A detailed derivation of a special solution with a prescribed azimuthal wavenumber k is presented. More general wave solutions, eventually excited by large local ‘impacts’, can be derived by superimposing all k-modes. The special wave functions so obtained describe the formation of two independent spiral wave arms originating out of a ring-shaped structure. The motivation for this investigation lies in the fact that similar wave propagators can be excited by the transit of a solid or ‘clumpy’ object through a protoplanetary disk. We speculate that a disk-brown dwarf collision can produce in the disk a pair of two spiral density wave fragments triggering the rapid accretion of two giant planets by a gravitational shear instability simultaneously (Hypothesis of a mechanism for the production of giant planets in pairs).