Some problems of classical and quantum gravitational collapse

Various aspects of the statics and dynamics of spherical gravitational distributions of an ideal fluid with distributed sources of zero-mass and massive vector fields are investigated. Both unrestricted configurations and isolated configurations with the appropriate matching to the external geometry are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 57–62, February, 1978.In conclusion the author considers it his pleasant duty to thank his colleagues V. N. Ponomarev and V. A. Berezin for stimulating discussions and analysis of the results.     

Cosmological Friedman models with the charged vacuum

Exact solutions corresponding to the stable charged vacuum are obtained for cosmological Friedman models with a charged scalar field. It is demonstrated that the presence of a massive vector field for the vacuum changes the law of expansion at early stages of evolution of the universe. A method for construction of solutions with the charged vacuum for an O(N) invariant scalar multiplet is given.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 19–24, December, 1985.     

Electrodynamics in an expanding universe

An algebraic form of the energy momentum tensor of the electromagnetic field is derived in terms of two scalars and two mutually orthogonal vector fields. Upon inserting this tensor into the field equations, solutions of the co-determined Einstein-Maxwell equations are obtained. The line element used is that corresponding to a conformal flat universe, whose form is then uniquely determined by the field equations. The case of a charged fluid is also considered and it is found that the particular form of the velocity field chosen limits the choice of the possible equation of state connecting the pressure and density distributions.     

Problems of classical and quantum gravitational collapse

An exact solution of Einstein's equations is obtained in the case of a spherically symmetric distribution of a perfect fluid with maximally hard equation of state. A sufficient condition of halting of the collapse of charged dust is obtained. The dynamics and statics of a vector massive configuration with massless source are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 7–12, September, 1977.We are very grateful to N. V. Pavlov and K. A. Bronnikov for valuable discussions and helpful comments.     

Euler's fluid equations: Optimal control vs optimization

An optimization method used in image-processing (metamorphosis) is found to imply Euler's equations for incompressible flow of an inviscid fluid, without requiring that the Lagrangian particle labels exactly follow the flow lines of the Eulerian velocity vector field. Thus, an optimal control problem and an optimization problem for incompressible ideal fluid flow both yield the same Euler fluid equations, although their Lagrangian parcel dynamics are different. This is a result of the gauge freedom in the definition of the fluid pressure for an incompressible flow, in combination with the symmetry of fluid dynamics under relabeling of their Lagrangian coordinates. Similar ideas are also illustrated for SO(N) rigid body motion.     

Models for the dynamics of dust-like matter in the self-gravity field: The method of hydrodynamic substitutions

Models for the dynamics of a dust-like medium in the self-gravity field are investigated. Solutions of the corresponding problems are constructed by the method of hydrodynamic substitutions generalizing the Cole–Hopf substitutions. The method is extended to multidimensional ideal and viscous fluid flows with cylindrical and spherical symmetries for which exact solutions are constructed. Solutions for the dynamics of self-gravitating dust with arbitrary initial distributions of both fluid density and velocity are constructed using special coordinate transformations. In particular, the problem of cosmological expansion is considered in terms of Newton’s gravity theory. Models of a one-dimensional viscous dust fluid flow and some problems of gas hydrodynamics are considered. Examples of exact solutions and their brief analysis are provided.     

An approach to introducing fractional integro-differentiation in classical electrodynamics

Analogs for Maxwell’s equations with fractional derivatives are obtained using the concepts of an effective current and the velocity of a charged particle in a medium. The calibration invariance is considered and a diffusion-wave equation is found and analyzed for scalar and vector potentials. It is shown that the stochastic nature of charged particle motion in a medium influences the dynamics of an electromagnetic field.     

Effective quantities and effective rules in two-dimensional ferromagnetic antidot lattices

In this paper the metamaterial properties of two-dimensional arrays of circular antidots (holes) embedded into a ferromagnetic medium of Permalloy are studied according to both micromagnetic and analytical calculations. The periodicity of the arrays and the diameters of the antidots are in the nanometric range. The collective mode dynamics is described by means of effective physical quantities for the scattering geometry with the external magnetic field applied perpendicularly to the Bloch wave vector in the antidot plane. As an example, the definition of an effective field, incorporating the demagnetizing effects due to the holes, permits to describe the dynamical properties of collective modes in terms of effective properties in the travelling regime. An effective wavelength and a small wave vector are introduced both for extended and localized magnonic modes. By means of these effective quantities it is shown that holes play the role of point defects affecting the spin dynamics in the microwave range. Relations between the effective wavelength and the Bloch wavelength and between the corresponding small wave vector and the Bloch wave vector are found. Some effective rules on the dynamic magnetization, based upon the effective wavelength and the corresponding small wave vector, are derived. An application that exploits the definition of the small wave vector is proposed and an experiment based upon the notion of effective wavelength and small wave vector is suggested.     

Self-energy operator of a massive neutrino in an external magnetic field

The effect of the magnetic field on the properties of a massive neutrino is analyzed. A general expression is derived in terms of the self-energy operator of the neutrino in an external magnetic field of arbitrary strength. This expression is valid for any relationship between the masses of the neutrino, a charged lepton, and a W-boson. An anomalous magnetic moment of a standard neutrino is investigated. The probability of massive neutrino decay into a W-boson and a charged lepton is calculated for various values of the magnetic field strength.     

Asymptotic analysis of ultra-relativistic charge

This article offers a new approach for analysing the dynamic behaviour of distributions of charged particles in an electromagnetic field. After discussing the limitations inherent in the Lorentz-Dirac equation for a single point particle a simple model is proposed for a charged continuum interacting self-consistently with the Maxwell field in vacuo. The model is developed using intrinsic tensor field theory and exploits to the full the symmetry and light-cone structure of Minkowski spacetime. This permits the construction of a regular stress-energy tensor whose vanishing divergence determines a system of non-linear partial differential equations for the velocity and self-fields of accelerated charge. Within this covariant framework a particular perturbation scheme is motivated by an exact class of solutions to this system describing the evolution of a charged fluid under the combined effects of both self and external electromagnetic fields. The scheme yields an asymptotic approximation in terms of inhomogeneous linear equations for the self-consistent Maxwell field, charge current and time-like velocity field of the charged fluid and is defined as an ultra-relativistic configuration. To facilitate comparisons with existing accounts of beam dynamics an appendix translates the tensor formulation of the perturbation scheme into the language involving electric and magnetic fields observed in a laboratory (inertial) frame.     

Massive tensor multiplets in N = 1 supersymmetry

We derive the action for a massive tensor multiplet coupled to chiral and vector multiplets as it can appear in orientifold compactifications of type IIB string theory. We compute the potential of the theory and show its consistency with the corresponding Kaluza‐Klein reduction of N = 1 orientifold compactifications. The potential contains an explicit mass term for the scalar in the tensor multiplet which does not arise from eliminating an auxiliary field. A dual action with an additional massive vector multiplet is derived at the level of superfields.     

Orthonormal tetrads and the charged fluid in general relativity

A space-time filled with the self-gravitating charged fluid with constant electric permitivity and constant magnetic permeability is investigated. On expressing the stress-energy tensor in terms of an orthonormal tetrad, the equations of motion and Maxwell equations are formulated. In case of the geodesic flow, the conditions for divergence-free electric and magnetic fields are obtained. It is shown that the space-time permeated by the charged fluid with the electric field orthogonal to the magnetic one is embedded in 5-dimensional flat class-one space-time if and only if the electromagnetic energy flux vector vanishes.     

Topological and physical effects of rotation and spin in the general relativistic theory of gravitation

Interrelations of the intrinsic momentum (spin), rotation of material distributions, and intrinsic momentum of the gravitational field are investigated in the context of the general relativistic theory of gravitation involving the general relativity theory (GRT) and the Einstein-Cartan theory. It is demonstrated that the spin density vector of the gravitational field s _g ⁱ is equal to the rotor of the tetrad reference point ωⁱ=ɛ^iklm e _k ^(a) e_(a)l,m/2 to within the factor 1/κ (s _g ⁱ =ω/κc). It is demonstrated that the vector s _g ⁱ is proportional to the spin density vector of the gravitating field sⁱ (ω)=jc(Ψγⁱγ₅Ψ)/2 as well as the pseudovector of space-time torsion Qⁱ in the Einstein-Cartan theory, which in both cases induces a cubic nonlinearity of the spinor field. An expression for the energy-momentum density tensor of the eddy gravitational field is derived. It is also demonstrated that the free eddy gravitational field with polarized spin can form “mole holes.” An ideal fast-rotating self-gravitating fluid can cause a similar effect. The corresponding exact solutions of joint systems of the Einstein and rotating ideal fluid equations are presented. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 57–60, October, 2007.     

Links between microscopic and macroscopic fluid mechanics

The microscopic and macroscopic versions of fluid mechanics differ qualitatively. Microscopic particles obey time-reversible ordinary differential equations. The resulting particle trajectories {q(t)} may be time-averaged or ensemble-averaged so as to generate field quantities corresponding to macroscopic variables. On the other hand, the macroscopic continuum fields described by fluid mechanics follow irreversible partial differential equations. Smooth particle methods bridge the gap separating these two views of fluids by solving the macroscopic field equations with particle dynamics that resemble molecular dynamics. Recently, nonlinear dynamics have provided some useful tools for understanding the relationship between the microscopic and macroscopic points of view. Chaos and fractals play key roles in this new understanding. Non-equilibrium phase-space averages look very different from their equilibrium counterparts. Away from equilibrium the smooth phase-space distributions are replaced by fractional-dimensional singular distributions that exhibit time irreversibility.     

SU(2) × SU(2) Electroweak Theory II: Chiral and Vector Fields on the Physical Vacuum

In this letter a discussion is offered on how symmetry breaking of a theory with twisted bundle of two chiral SU(2) bundles leads to a set of gauge potentials from each group on the physical vacuum that are vector and chiral. The result is that symmetry breaking of this theory leads to massive A ³ transverse modes of the 3-photon along with electromagnetic photons plus the massive neutral and charged weakly interacting bosons. The electromagnetic sector is demonstrated to be a massless vector field and the remainder is a broken chiral field theory.     

Equilibrium distribution of an ideal fluid in space-times permitting groups of concircular motions

A space — time with an ideal fluid as a source that permits groups of concircular motions is investigated. It is assumed that the velocity vector of the fluid is directed along a time-like group vector. A theorem on the properties of such a fluid and its relation with symmetries of the space — time is proven. Kazan’ State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 7–13, June, 1997.     

Homogeneous solutions of the Einstein equations with an ideal charged fluid

The space-times V₄ with an ideal charged fluid as source allowing the motion group G_r, r 4, are investigated. It is assumed that the fluid velocity vector is directed along the timelike vector of the Killing group. In the case of groups G₄, acting on V₄, as well as groups of higher mobility, a complete investigation is performed of the space-times by using the system of Einstein-Maxwell equations. Exact solutions are found with fourth- and fifth-order groups.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 74–79, December, 1984.     

Dynamics of a thin shell in the Reissner—Nordström metric

The dynamics of a thin spherically symmetric gravitating shell around an electrically charged Reissner—Nordström black hole is considered. The energy—momentum tensor of an electrically neutral shell is modeled by an ideal fluid with a polytropic equation of state. The dynamics of a shell with a dust equation of state can be traced completely analytically. The Carter—Penrose diagrams that describe the global geometry and all possible types of motions of a gravitating shell in the case of an eternal black hole have been constructed.The conditions have been found under which stable oscillatory motions of the shell take place. These transfer it successively from one universe to the next in an infinite series of identical universes. Such stable oscillatory shell motions are shown to be possible for an arbitrary polytropic equation of state of the shell.