Charge transport in one dimension: Dissipative and non-dissipative space-charge-limited currents

We consider charge transport in a pore where the dielectric constant inside the pore is much greater than that in the surrounding material, so that the flux of the electric fields due to the charges is almost entirely confined to the pore. We develop exact solutions for the one component case for the Dirichlet and Neumann boundary conditions using a Hopf-Cole transformation, Fourier series, and periodic solutions of the Burgers equation. These are compared with a simpler model in which the scaled diffusivity is zero so that all charge motion is driven by the electric field. In this non-dissipative case, recourse to an admissibility condition is used to obtain the physically relevant weak solution of a Riemann problem concerning the electric field. It is shown that the admissibility condition is Poynting’s theorem.     

Are all static black hole solutions spherically symmetric?

The static black hole solutions to the Einstein-Maxwell equations are all spherically symmetric, as are many of the recently discovered black hole solutions in theories of gravity coupled to other forms of matter. However, counterexamples demonstrating that static black holes need not be spherically symmetric exist in theories, such as the standard electroweak model, with electrically charged massive vector fields. In such theories, a magnetically charged Reissner-Nordström solution with sufficiently small horizon radius is unstable against the development of a nonzero vector field outside the horizon. General arguments show that, for generic values of the magnetic charge, this field cannot be spherically symmetric. Explicit construction of the solution shows that it in fact has no rotational symmetry at all.This essay received the second award from the Gravity Research Foundation, 1995-Ed.     

Axially symmetric stationary solutions of Einstein-Maxwell equations

A scheme is presented by which we can obtain static electrovac exterior solutions from the stationary gravitational fields given by Kerr and Tomimatsu and Sato. A five-parameter solution of the combined Einstein-Maxwell equations is given that describes a source containing mass, electric charge, magnetic dipole, higher multipole moments of all three kinds and angular momentum. These are generated by using Kinnersley's method from Wang's electrovac solutions, which, in turn, were generated from the gravitational fields of Tomimatsu and Sato for =2–3. All the solutions are asymptotically flat.     

A case of dual interpretation of Einstein-Maxwell fields

It is proved that the stationary semicylindrically symmetric Einstein-Maxwell solutions given in Reference 1 also allow of an interpretation as charge without mass fields. The system of equations is simplified and a previously unknown solution is given.     

Color dielectric model with two scalar fields

SU(2) Yang-Mills theory coupled in a non-minimal way to two scalar fields is discussed. For the massless scalar fields a family of finite energy solutions generated by an external, static electric charge is found. Additionally, there is a single solution which can be interpreted as a confining one. Similar solutions have been obtained in the magnetic sector. In the case of massive scalar fields the Coulomb problem is investigated. We find that asymptotic behavior of the fields can also, for some values of the parameters of the model, give confinement of the electric charge. Quite interestingly one glueball-meson coupling gives the linear confining potential. Finally, it is shown how, for one non-dynamical scalar field, we can derive the color dielectric generalization of the Pagels-Tomboulis model.Received: 22 April 2003, Published online: 12 September 2003     

Static finite-energy solutions of gauge fields with separated radial variable

For arbitrary compact gauge group G and real representations of the Higgs fields, we seek static finite-energy solutions for which the radial dependence of the fields is factorized. We find that the gauge fields vanish outside a fixed SO(3) subgroup of G, and that inside SO(3) they reduce to the 't Hooft-Polyakov solution with unit magnetic charge. The Higgs fields may belong to any integer representation of this SO(3).     

Solitons of nonlinear scalar electrodynamics in general relativity

Solitons with spherical and/or cylindrical symmetry in the interacting system of scalar, electromagnetic, and gravitational fields have been obtained. As a particular case it is shown that the equations of motion admit a special kind of solution with a sharp boundary, known as droplets. For these solutions, the physical fields vanish and the space-time is flat outside of the critical sphere or cylinder. Therefore, the mass and the electric charge of these configurations are zero.     

Axially symmetric stationary electrovac solutions

Perjes and Israel and Wilson have given independently a new class of solutions of the sourcefree Einstein-Maxwell equations, which can be interpreted as the external gravitational and electromagnetic fields of a spinning source with unit specific charge. Starting from Zipoy's solutions in oblate and prolate spheroidal coordinates for the source-free gravitational field we generate some axially symmetric stationary solutions of the source-free Einstein-Maxwell equations by using Perjes' method. All these solutions become Euclidean at infinity. The asymptotic behavior and the singularity of the solutions are studied in order to gain some insight into the nature of the source. The solution in prolate spheroidal coordinates is found to contain closed timelike lines.     

Further properties of classical solutions in grassmannian σ models

We calculate the action and the topological charge of classical solutions in two-dimensional euclidean grassmannian σ models (for both generic and special solutions). We show that all the above non-instanton or non-anti-instanton solutions are unstable. We also find solutions of the Dirac equation describing fermions in the background of these grassmannian fields, and we exhibit relations between these fermionic solutions and those of the associated Hilbert-Riemann problem.     

Dehydration process in NaCl solutions under various external electric fields

Ionic motions at solid-liquid interface in supersaturated NaCl solutions have been investigated by molecular dynamics (MD) simulation for understanding crystal growth processes. The density profile in the vicinity of the interfaces between NaCl(100) and the supersaturated NaCl solution was calculated. Diffusion coefficients of water molecules in the solution were estimated as a function of distance from the crystal interface. It turned out that the structure and dynamics of the solution in the interfaces was different from those of bulk solution owing to electric fields depending on the surface charge. Therefore, the electric field was applied to the supersaturated solutions and dehydration phenomenon occurring in the process of the crystal growth was discussed. As the electric field increased, it was observed that the Na⁺ keeping strongly hydration structure broke out by the electric force. In supersaturated concentration, the solution structure is significantly different from that of dilution and has a complicated structure with hydration ions and clusters of NaCl. If the electric fields were applied to the solutions, the breakout of hydration structure was not affected with increasing the supersaturated ratio. This reason is that the cluster structures are destroyed by the electric force. The situation depends on the electric field or crystal surface structure.     

New exact solutions to Dirac's equation. I

This study is a continuation of [1]. New exact solutions have been found to the Dirac and Klein-Gordon equations for the charge in orthogonal fields equal in magnitude but inhomogeneous and of special form. Here we consider six different types of field that allow exact solution.     

Some new radiating Kerr-Newman solutions

Three exact non-static solutions of Einstein-Maxwell equations corresponding to a field of flowing null radiation plus an electromagnetic field are presented. These solutions are non-static generalizations of the well known Kerr-Newman solution. The current vector is null in all the three solutions. These solutions are the electromagnetic generalizations of the three generalized radiating Kerr solutions discussed by Vaidya and Patel. The solutions discussed by us describe the exterior gravitational fields of rotating radiating charged bodies. Many known solutions are derived as particular cases.     

Yang-Mills fields due to an infinite charge cylinder

The problem of determining time-independent solutions of the classical Yang-Mills equations for infinitely long charge cylinders is studied. A useful expression for the total energy in the field in terms of just the sources is derived. Numerical solutions have been found in the special cases of a small charge cylinder with a magnetic field B that either lies along the axis of symmetry or encircles the axis. It is as if these two solutions were due to currents encircling the axis or paralleling it, respectively. The condition that the solution behave well at infinity implies an exponential fall off for the fields in the azimuthal B field case and a fall off more rapid than 1/R in the axial B field case, so that in both cases the existence of a B field requires the charge on the axis to be shielded. Consequently, these solutions do not behave at infinity at all like the Maxwell solution for a charge cylinder, and they have a lower energy per unit length. They show that in Yang-Mills theories the source does not determine a unique field. A classical interpretation of this is that the field remembers how the charges were transported during the construction of the cylinder. It also suggests that a quantum mechanical version of this problem would exhibit a “spontaneous symmetry breaking” to a less symmetric, lower energy vacuum.These solutions exhibit a twofold degeneracy, as the magnetic field may be either left- or right-handed in the azimuthal B field case, or point along the +z or ?z axis in the axial B field case.     

New exact solutions of Einstein's field equations: Gravitational force can also be repulsive!

This article has not been written for specialists of exact solutions of Einstein's field equations but for physicists who are interested in nontrivial information on this topic. We recall the history and some basic properties of exact solutions of Einstein's vacuum equations. We show that the field equations for stationary axisymmetric vacuum gravitational fields can be expressed by only one nonlinear differential equation for a complex function. This compact form of the field equations allows the generation of almost all stationary axisymmetric vacuum gravitational fields. We present a new stationary two-body solution of Einstein's equations as an application of this generation technique. This new solution proves the existence of a macroscopic, repulsive spin-spin interaction in general relativity. Some estimates that are related to this new two-body solution are given.     

Replica-symmetric solutions of a dilute Ising ferromagnet in a random field

We use the replica method in order to obtain an expression for the variational free energy of an Ising ferromagnet on a Viana-Bray lattice in the presence of random external fields. Introducing a global order parameter, in the replica-symmetric context, the problem is reduced to the analysis of the solutions of a nonlinear integral equation. At zero temperature, and under some restrictions on the form of the random fields, we are able to perform a detailed analysis of stability of the replica-symmetric solutions. In contrast to the behaviour of the Sherrington-Kirkpatrick model for a spin glass in a uniform field, the paramagnetic solution is fully stable in a sufficiently large random field.     

New exact Einstein-Maxwell fields for a massless charge

New solutions of the Einstein-Maxwell equations are presented. The solutions describe the gravitational fields of a massless charge.     

Ann monopole solution with 4n — 1 degrees of freedom

An exact static monopole solution, possessingn units of magnetic charge and (4n-1) degrees of freedom, is constructed, generalising the recent work of Ward on two monopole solutions. The equations solved are those of anSU(2) gauge theory with adjoint representation Higgs field in the (BPS) limit of vanishing Higgs potential. The number of degrees of freedom is maximal for self-dual solutions. The construction is described in a deductive way, within the framework of the Atiyah-Ward formalism for self-dual gauge fields.     

Gauge transformations and quantum mechanics II. Physical interpretation of classical gauge transformations

New gauges are introduced. The potentials, vector and scalar, in these gauges are obtained in closed forms by the Green's function method. These closed form solutions are explicity expressed only in terms of the charge and current densities. The physical interpretation is on how potentials propagate from the charge and current densities. The Coulomb gauge and the Lorentz gauge are special cases of a new gauge defined in this paper. It is called the complete α-Lorentz gauge. The scalar potential propagates at speed αc from the charge density for any positive α. When α is one, the usual solutions for the Lorentz gauge are recovered. When α is not one, our results show that, in order to satisfy the requirement that electromagnetic fields be gauge invariant and in order to conform to Maxwell's interpretation that electromagnetic fields propagate at speed c from the charge and current densities (we only consider the vacuum), the vector potential must contain two mathematically and physically independent gradient components. Furthermore, one such component must propagate at speed αc while the other must at speed c from charge and current densities. Our discussions on the Coulomb gauge are based on the results obtained by letting α go to (positive) infinity. Guided by Maxwell's interpretation, we introduce a new decomposition of the vector potential in the Lorentz gauge into a longitudinal and a transverse component. For an arbitrary charge and current distribution, it is shown that the transverse component will generate all the fields only in the radiation zone. However, for a point charged particle, the transverse component only generates the “free fields”everywhere in the instantaneous rest frame of the charged particle.     

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.