Antonio F. Rañada

A Topological Theory of the Electromagnetic Field

It is shown that Maxwell equations in vacuum derive from an underlying topological structure given by a scalar field ϕ which represents a map S3×R→S2 and determines the electromagnetic field through a certain transformation, which also linearizes the highly nonlinear field equations to the Maxwell equations. As a consequence, Maxwell equations in vacuum have topological solutions, characterized by a Hopf index equal to the linking number of any pair of magnetic lines. This allows the classification of the electromagnetic fields into homotopy classes, labeled by the value of the helicity. Although the model makes use of only c-number fields, the helicity always verifies ∫ A·Bd3r=nα, n being an integer and α an action constant, which necessarily appears in the theory, because of reasons of dimensionality.

The electromagnetic helicity

It is shown that the electromagnetic helicity, defined as the sum of the helicities of the electric and magnetic fields of an electromagnetic field in vacuum, is a constant of the motion. Furthermore, it is equal to twice the classical limit of the difference between the numbers of right- and left-handed photons, which establishes a close relationship between the wave and the particle meanings of the word helicity. Resumen. Se demuestra que la helicidad electromagnetica, definida como la suma de las helicidades electrica y magnetica de un campo electromagnetico en el vacio, es una constante del movimiento. Ademas, es igual a dos veces el limite clasico de la diferencia entre los numeros de fotones polarizados a la derecha y a la izquierda, lo cual establece una relacion cercana entre los significados ondulatorio y corpuscular de la palabra helicidad.

The Pioneer Anomaly as Acceleration of the Clocks

This work proposes an explanation of the Pioneer anomaly, the unmodelled and as yet unexplained blueshift detected in the microwave signal of the Pioneer 10 and other spaceships by Anderson et al. in 1998. What they observed is similar to the effect that would have either (i) an anomalous acceleration aP the ship towards the Sun, or (ii) an acceleration of the clocks at=aP/c. The second alternative is investigated here, with a phenomenological model in which the anomaly is an effect of the background gravitational potential Ψ (t) that pervades all the universe and is increasing because of the expansion. It is shown that 2at=dΨ /dt=d2τclocks/dt2, evaluated at present time t0, where t and τclocks are the coordinate time and the time measured by the atomic clocks, respectively. The result of a simple estimate gives the value at⋍ 1.8× 10−18 s−1, while Anderson et al. suggested at=(2.9±0.4) × 10−18 s−1 on the basis of their observations. The calculation are performed near the Newtonian limit but in the frame of general relativity.

The weak‐Painlevé property as a criterion for the integrability of dynamical systems

We investigate the validity of the weak‐Painleve property as an integrability criterion. We present an example of a time‐dependent Hamiltonian system which possesses a weak‐Painleve type expansion, while presenting a chaotic behavior. However, this system presents also critical fixed singularities. The importance of the latter, as far as integrability is concerned, is discussed here.

A topological mechanism of discretization for the electric charge

We present a topological mechanism of discretization, which gives for the fundamental electric charge the value q0 = √ ¯ = 5.29 ×10 19 C, about 3.3 times the electron charge. Its basis is the following recently proved property of the standard linear classical Maxwell equations: they can be obtained by change of variables from an underlying

A model of ball lightning as a magnetic knot with linked streamers

We develop a topological model of ball lightning which explains its stability by the coupling of an air ball to a magnetic knot, a magnetic field with linked magnetic lines. Assuming that currents flow inside the ball, along short-circuited linked streamers following the lines of ∇ × B, the lifetime, energy, and radiated power of the average ball are correctly accounted for. The model explains why some witnesses do not feel heat, while others are burnt, and why filaments are seen to trail the ball in some cases.

TWO PROPERTIES OF ELECTROMAGNETIC KNOTS

Abstract We prove two properties of the electromagnetic knots in empty space. First, that any standard radiation fields (i.e. satisfying E · B = 0 ) coincide locally with an electromagnetic knot. Second, that the electric and magnetic helicities of any knot are equal. These results can be used as a basis for a topological model of electromagnetism.

On the magnetic helicity

The helicity of a divergenceless vector field B(r) in a region D, defined as the integral integral DA.Bd3r, where B= Del xA, characterizes the topological configuration of the integral lines of B, being equal to zero if they are not linked. An expression for its value in terms of the linking number of pairs of lines is given. Although the paper is devoted to the cases in which B is the magnetic field, some other applications are indicated.

Elementary Spinorial Excitations in a Model Universe

The generalized covariant Dirac equation for a certain five‐dimensional universe is studied. If a torsion invariant is included in the free Lagrangian, it is shown that particlelike stable solutions exist having definite positive rest energy, spin, and corresponding antiparticles. The treatment is throughout classical.

The Pioneer riddle, the quantum vacuum and the variation of the light velocity

It is shown that the same phenomenological Newtonian model recently proposed by the author to explain the cosmological evolution of the fine structure constant suggests furthermore an explanation of the unmodelled acceleration