Static,Spherically Symmetric Solutions of Yang-Mills-Dilaton Theory

Static, spherically symmetric solutions of the Yang-Mills-Dilaton theory are studied. It is shown that these solutions fall into three different classes. The generic solutions are singular. Besides there is a discrete set of globally regular solutions further distinguished by the number of nodes of their Yang-Mills potential. The third class consists of oscillating solutions playing the role of limits of regular solutions, when the number of nodes tends to infinity. We show that all three sets of solutions are non-empty. Furthermore we give asymptotic formulae for the parameters of regular solutions and confront them with numerical results.     

Regular black holes in quadratic gravity

The first-order correction of the perturbative solution of the coupled equations of the quadratic gravity and nonlinear electrodynamics is constructed, with the zeroth-order solution coinciding with the ones given by Ayón-Beato and Garcí a and by Bronnikov. It is shown that a simple generalization of the Bronnikov's electromagnetic Lagrangian leads to the solution expressible in terms of the polylogarithm functions. The solution is parametrized by two integration constants and depends on two free parameters. By the boundary conditions the integration constants are related to the charge and total mass of the system as seen by a distant observer, whereas the free parameters are adjusted to make the resultant line element regular at the center. It is argued that various curvature invariants are also regular there that strongly suggests the regularity of the spacetime. Despite the complexity of the problem the obtained solution can be studied analytically. The location of the event horizon of the black hole, its asymptotics and temperature are calculated. Special emphasis is put on the extremal configuration.     

Eleven Spherically Symmetric Constant Density Solutions with Cosmological Constant

Einstein's field equations with cosmological constant are analysed for a static, spherically symmetric perfect fluid having constant density. Five new global solutions are described. One of these solutions has the Nariai solution joined on as an exterior field. Another solution describes a decreasing pressure model with exterior Schwarzschild-de Sitter spacetime having decreasing group orbits at the boundary. Two further types generalise the Einstein static universe. The other new solution is unphysical, it is an increasing pressure model with a geometric singularity.     

Standstill Electric Charge Generates Magnetostatic Field under Born-Infeld Electrodynamics

The Abelian Born-Infeld classical non-linear Electrodynamic has been used to investigate the electric and magnetostatic fields generated by a point-like electric charge at rest in an inertial frame. The results show a rich internal structure for the charge. Analytical solutions have also been found. Such fields configurations have been interpreted in terms of vacuum polarization and magnetic-like charges produced by the very high strengths of the electric field considered. Apparently non-linearity is responsible for the emergence of an anomalous magnetostatic field suggesting a possible connection to that created by a magnetic dipole composed of two magnetic charges with opposite signs. Consistently in situations where the Born-Infeld field strength parameter is free to become infinite, Maxwell’s regime takes over, the magnetic sector vanishes and the electric field assumes a Coulomb behavior with no trace of a magnetic component. The connection to other monopole solutions, like Dirac’s and ’tHooft’s Poliakov’s types are also discussed. Finally, some speculative remarks are presented in an attempt to explain such fields.     

Self-Interacting Electromagnetic Fields and a Classical Discussion on the Stability of the Electric Charge

The present work proposes a discussion on the self-energy of charged particles in the framework of nonlinear electrodynamics. We seek magnetically stable solutions generated by purely electric charges whose electric and magnetic fields are computed as solutions to the Born-Infeld equations. The approach yields rich internal structures that can be described in terms of the physical fields with explicit analytic solutions. This suggests that the anomalous field probably originates from a magnetic excitation in the vacuum due to the presence of the very intense electric field. In addition, the magnetic contribution has been found to exert a negative pressure on the charge. This, in turn, balances the electric repulsion, in such a way that the self-interaction of the field appears as a simple and natural classical mechanism that is able to account for the stability of the electron charge.     

Spherically Symmetric Vacuum Brane and Wormhole Solutions

In this letter, we have obtained static, spherically symmetric solutions of the effective vacuum Einstein field equations on a brane embedded in a five dimensional space time. The effective stress tensor is induced by the interaction with the bulk gravitational field and is given by the electric part of the five dimensional Weyl tensor. Due to traceless nature of this non-local effect of the bulk, any solution of  ⁽⁴⁾ R=0 is a possible solution of the vacuum brane. We have derived a class of solutions, which corresponds to wormhole solution. Physical properties and characteristics of the wormhole are studied.     

Classification of Static, Spherically Symmetric Solutions of the Einstein-Yang-Mills Theory with Positive Cosmological Constant

We give a complete classification of all static, spherically symmetric solutions of the SU(2) Einstein-Yang-Mills theory with a positive cosmological constant. Our classification proceeds in two steps. We first extend solutions of the radial field equations to their maximal interval of existence. In a second step we determine the Carter-Penrose diagrams of all 4-dimensional space-times constructible from such radial pieces. Based on numerical studies we sketch a complete phase space picture of all solutions with a regular origin.     

Spherically Symmetric Solutions and Dark Matter on the Brane

It has recently been suggested that our universe is a three-brane embedded in a higher dimensional spacetime. In this paper I examine static, spherically symmetric solutions that satisfy the effective Einstein field equations on a brane embedded in a five dimensional spacetime. The field equations involve a term depending on the five dimensional Weyl tensor, so that the solutions will not be Schwarzschild in general. This Weyl term is traceless so that any solution of ⁽⁴⁾ R = 0 is a possible four dimensional spacetime. Different solutions correspond to different five dimensional spacetimes and to different induced energy-momentum tensors on the brane. One interesting possibility is that the Weyl term could be responsible for the observed dark matter in the universe. It is shown that there are solutions of the equation ⁽⁴⁾ R = 0 that can account for the observed rotation curves of spiral galaxies.     

The Dynamical Instability of Static, Spherically Symmetric Solutions in Nonsymmetric Gravitational Theories

We consider the dynamical stability of a class of static, spherically symmetric solutions of the nonsymmetric gravitational theory. We numerically reproduce the Wyman solution and generate new solutions for the case where the theory has a nontrivial fundamental length scale ^-1. By considering spherically symmetric perturbations of these solutions we show that the Wyman solutions are generically unstable.     

Lagrange Structure and Dynamics for Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations

The compressible Navier-Stokes system (CNS) with density-dependent viscosity coefficients is considered in multi-dimension, the prototype of the system is the viscous Saint-Venat model for the motion of shallow water. A spherically symmetric weak solution to the free boundary value problem for CNS with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path, which is continuous away from the symmetry center. Detailed regularity and Lagrangian structure of this solution have been obtained. In particular, it is shown that the particle path is uniquely defined starting from any non-vacuum region away from the symmetry center, along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved. Starting from any non-vacuum point at a later-on time, a particle path is also uniquely defined backward in time, which either reaches at some initial non-vacuum point, or stops at a small middle time and connects continuously with vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time, and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up. This finally leads to the formation of vacuum state almost everywhere as the time goes to infinity.     

Local Existence and Continuation Criterion for Solutions of the Spherically Symmetric Einstein-Vlasov-Maxwell System

Using the iterative scheme we prove the local existence and uniqueness of solutions of the spherically symmetric Einstein-Vlasov-Maxwell system with small initial data. We prove a continuation criterion to global in-time solutions.     

Inconsistence of Non-Singular Spherically Symmetric Solution and a Satisfactory Energy-Momentum Complex

Two spherically symmetric non-singular black hole solutions in Moiler tetrad theory of gravitation have been obtained. Although the two solutions have the same form of metric （spherically symmetric nonsingular black hole）, their energy contents are different. We use another method given by Gibbons and Hawking to calculate the energy content of these solutions. We also obtained different value of energy. Study the requirements of a satisfactory energymomentum complex given by Moiler we find that the second solution, which behaves as 1/√r, is not transformed as a four-vector under Lorentz transformation.     

Inconsistence of Non-Singular Spherically Symmetric Solution and a Satisfactory Energy-Momentum Complex

Two spherically symmetric non-singular black hole solutions in MØller tetrad theory of gravitation have been obtained. Although the two solutions have the same form of metric (spherically symmetric nonsingular black hole), their energy contents are different. We use another method given by Gibbons andHawking to calculate the energy content of these solutions. We alsoobtained different value of energy. Study the requirements of a satisfactory energy-momentum complex given by MØller we find that the second solution, which behaves as 1/r^1/2, is not transformed as a four-vector under Lorentz transformation.     

Editor's Note: On the Gravitational Field of a Mass Point According to Einstein's Theory by K. Schwarzschild

We comment on the famous paper by Karl Schwarzschild from 1916, which is reprinted in the present issue.     

New Exact Solutions of the (2+1)-Dimensional Asymmetric Nizhnik-Novikov-Veselov System

Using symbolic and algebra computation, the extended tanh-function method (ETM) based on mapping method is further extended. The new variable separation solutions of the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov (ANNV) system are derived.     

Huygens' Principle,Dirac Operators,and Rational Solutions of the AKNS Hierarchy

We prove that rational solutions of the AKNS hierarchy of the form q=σ/τ and r=ρ/τ, where (σ,τ,ρ) are certain Schur functions, naturally yield Dirac operators of strict Huygens' type, i.e., the support of their fundamental solutions is the surface of the light-cone. This strengthens the connection between the theory of completely integrable systems and Huygens' principle by extending to the Dirac operators and the rational solutions of the AKNS hierarchy a classical result of Lagnese and Stellmacher concerning perturbations of wave operators. Mathematics Subject Classifications (2000) 37K10, 35Qxx, 35B40.     

Exact solutions and symmetry analysis for the limiting probability distribution of quantum walks

In the literature, there are numerous studies of one-dimensional discrete-time quantum walks (DTQWs) using a moving shift operator. However, there is no exact solution for the limiting probability distributions of DTQWs on cycles using a general coin or swapping shift operator. In this paper, we derive exact solutions for the limiting probability distribution of quantum walks using a general coin and swapping shift operator on cycles for the first time. Based on the exact solutions, we show how to generate symmetric quantum walks and determine the condition under which a symmetric quantum walk appears. Our results suggest that choosing various coin and initial state parameters can achieve a symmetric quantum walk. By defining a quantity to measure the variation of symmetry, deviation and mixing time of symmetric quantum walks are also investigated.