Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann–Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine–Gordon equation in light–cone coordinates, the nonlinear Schrodinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrodinger equation is derived.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1997.0077

A unified transform method for solving linear and certain nonlinear PDEs

Using the concept of cracking we explore the influence that density fluctuations and local anisotropy have on the stability of local and non-local anisotropic matter configurations in general relativity. This concept, conceived to describe the behavior of a fluid distribution just after its departure from equilibrium, provides an alternative approach to consider the stability of self-gravitating compact objects. We show that potentially unstable regions within a configuration can be identified as a function of the difference of propagations of sound along tangential and radial directions. In fact, it is found that these regions could occur when, at a particular point within the distribution, the tangential speed of sound is greater than the radial one.

https://iopscience.iop.org/article/10.1088/0264-9381/24/18/005/pdf

Sound speeds, cracking and the stability of self-gravitating anisotropic compact objects

Atomic polarization phenomena impinge upon a number of areas and processes in physics. The dielectric constant and refractive index of any gas are examples of macroscopic properties that are largely determined by the dipole polarizability. When it comes to microscopic phenomena, the existence of alkaline-earth anions and the recently discovered ability of positrons to bind to many atoms are predominantly due to the polarization interaction. An imperfect knowledge of atomic polarizabilities is presently looming as the largest source of uncertainty in the new generation of optical frequency standards. Accurate polarizabilities for the group I and II atoms and ions of the periodic table have recently become available by a variety of techniques. These include refined many-body perturbation theory and coupled-cluster calculations sometimes combined with precise experimental data for selected transitions, microwave spectroscopy of Rydberg atoms and ions, refractive index measurements in microwave cavities, ab initio calculations of atomic structures using explicitly correlated wavefunctions, interferometry with atom beams and velocity changes of laser cooled atoms induced by an electric field. This review examines existing theoretical methods of determining atomic and ionic polarizabilities, and discusses their relevance to various applications with particular emphasis on cold-atom physics and the metrology of atomic frequency standards.

https://iopscience.iop.org/article/10.1088/0953-4075/43/20/202001/pdf

Theory and applications of atomic and ionic polarizabilities

The variational theory of an ideal spinning fluid is developed. This is described by original Weyssenhoff-Raabe tensors of spin and energy-momentum. Both neutral and charged fluids are considered, and the equations of motion and conservation laws for the Weyssenhoff neutral fluid in the Riemann-Cartan spacetime are discussed. The results are applied to the study of cosmological models with rotation, shear and expansion in the framework of the Einstein-Cartan theory of gravity.

https://iopscience.iop.org/article/10.1088/0264-9381/4/6/021/pdf

The Weyssenhoff fluid in Einstein-Cartan theory

The weak, dominant and strong energy conditions are investigated for various kinds of imperfect fluids. In this context, attention has been given to the model of a collapsing or expanding sphere of shear-free fluid which conducts heat and radiates energy to infinity.

/pdf/energy-conditions-for-an-imperfect-fluid-53grgvd8zp.pdf

Energy conditions for an imperfect fluid

This paper considers a spherical body consisting of a fluid with heat flow which radiates in its exterior a null fluid described by the outgoing Vaidya's metric. A Friedmann-like exact solution of the interior Einstein field equations is given. It is proved that this solution, matched with the outgoing Vaidya matric, represents a physically reasonble collapsing model which, when the heat flow is switched off, reduces to the well-known collapsing model with dust. The proposed model has the remarkable property that even if the heat flow is small, the horizon will never be formed because, before this happens, the collapsing body will be destroyed by opposite gradients of pressure. 6 references.

Friedmann-like collapsing model of a radiating sphere with heat flow

On the systematic approach to the classification of differential equations by group theoretical methods

We consider lattice equations on which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the generalized three- and five-point symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain non-degeneracy condition. Finally, symmetry reductions of certain lattice equations to discrete analogs of the Painleve equations are considered.

Affine linear and D4 symmetric lattice equations : symmetry analysis and reductions

The motion of test particles in polar orbit about the source of the Kerr field of gravity is studied, using Carter's first integrals for timelike geodesies in the Kerr space-time. Expressions giving the angular coordinates of such particles as functions of the radial one are derived, both for the case of a rotating black hole as well as for that of a naked singularity.

/pdf/polar-orbits-in-the-kerr-space-time-4uj8wfc73u.pdf

D. Tsoubelis

Papers

Energy conditions for an imperfect fluid

Friedmann-like collapsing model of a radiating sphere with heat flow

On the systematic approach to the classification of differential equations by group theoretical methods

Affine linear and D4 symmetric lattice equations : symmetry analysis and reductions

Polar Orbits in the Kerr Space-Time