arXiv subjects

Waldyr A. Rodrigues Jr.

Publications and source records attributed to Waldyr A. Rodrigues Jr..

11 recordsLinked to original sources

arXiv2017

Life in the Rindler Reference Frame: Does an Uniformly Accelerated Charge Radiates? Is there a Bell `Paradox'? Is Unruh Effect Real?

The determination of the electromagnetic field generated by a charge in hyperbolic motion is a classical problem for which the majority view is that the Liénard-Wiechert solution which implies that the charge radiates) is the correct one. However we analyze in this paper a less known solution due to Turakulov that differs from the Liénard-Wiechert one and which according to him does not radiate. We prove his conclusion to be wrong. We analyze the implications of both solutions concerning the validity of the Equivalence Principle. We analyze also two other issues related to hyperbolic motion, the so-called Bell's "paradox" which is as yet source of misunderstandings and the Unruh effect, which according to its standard derivation in the majority of the texts, is a correct prediction of quantum field theory. We recall that the standard derivation of the Unruh effect does not resist any tentative of any rigorous mathematical investigation, in particular the one based in the algebraic approach to field theory which we also recall. These results make us to align with some researchers that also conclude that the Unruh effect does not exist.

physics.gen-ph↗

arXiv2016

Notes on Conservation Laws, Equations of Motion of Matter and Particle Fields in Lorentzian and Teleparallel de Sitter Spacetime Structures

We discuss the physics of interacting tensor fields and particles living in $M=\mathrm{S0}(1,4)/\mathrm{S0} (1,3)\simeq\mathbb{R}\times S^{3}$ a submanifold of $\mathring{M}=(\mathbb{R}^{5},\boldsymbol{\mathring{g}})$, where $\boldsymbol{\mathring {g}}$ has signature $(1,4)$. Structure $(M,\boldsymbol{g})$ where $(\boldsymbol{g=i}^{\ast}\boldsymbol{\mathring{g}})$ is a Lorentzian manifold. Structure $(M,\boldsymbol{g,}τ_{\boldsymbol{g}},\uparrow)$ is primely used to study the energy-momentum conservation law (for a system of physical fields (and particles) living in $M$ and to get the respective equations of motion. We construct two different de Sitter spacetime structures $M^{dSL}=(M,\boldsymbol{g,D},τ_{\boldsymbol{g}},\uparrow)$ and $M^{dSTP}=(M,\boldsymbol{g,\nabla},τ_{\boldsymbol{g} },\uparrow)$. Both (metrical compatible) connections are used only as mathematical devices. In particular $M^{dSL}$ is not supposed to be the model of any gravitational field in the(\textbf{GRT}). We clarify some misconceptions appearing in the literature. We use the Clifford and spin-Clifford bundles formalism and gives a thoughtful presentation of the concept of a Komar current $\mathcal{J}_{A}$ (in GRT}) associated to any vector field $\mathbf{A}$. A formula for the Komar current and its physical meaning are given. We show also how $F=dA$ satisfy in the Clifford bundle a Maxwell like equation encoding the contents of Einstein equation. We show that in GRT there are infinitely many conserved currents,independently of the fact that the Lorentzian spacetime possess or not Killing vector fields and that even when the appropriate Killing vector fields exist there does not exist a conserved energy-momentum covector (not a covector field) as in SRT.

math-ph↗

arXiv2016

The Dirac-Hestenes Equation and its Relation with the Relativistic de Broglie-Bohm Theory

In this paper we provide using the Clifford and spin-Clifford formalism and some few results of the extensor calculus a derivation of the conservation laws that follow directly from the Dirac-Hestenes equation (DHE) describing a Dirac-Hestenes spinor field (DHSF) in interaction with an external electromagnetic field without using the Lagrangian formalism. In particular, we show that the energy-momentum and total angular momentum extensors of a DHSF is not conserved in spacetime regions permitting the existence of a null electromagnetic field F but a non null electromagnetic potential A. These results have been used together with some others recently obtained (e.g., that the classical relativistic Hamilton-Jacobi equation is equivalent to a DHE satisfied by a particular class of DHSF) to obtain the correct relativistic quantum potential when the Dirac theory is interpreted as a de Broglie-Bohm theory. Some results appearing in the literature on this issue are criticized and the origin of some misconceptions is detailed with a rigorous mathematical analysis.

math-ph↗

arXiv2016

Equations of Motion and Energy-Momentum 1-Forms for the Coupled Gravitational, Maxwell and Dirac Fields

A theory where the gravitational, Maxwell and Dirac fields (mathematically represented as particular sections of a convenient Clifford bundle) are supposed fields in Faraday's sense living in Minkowski spacetime is presented. In our theory there exist a genuine energy-momentum tensor for the gravitational field and a genuine energy-momentum conservation law for the system of the interacting gravitational, Maxwell and Dirac fields. Moreover, the energy-mometum tensors of the Maxwell and Dirac fields are symmetric, and it is shown that the equations of motion for the gravitational potentials is equivalent to Einstein equation of General Relativity (where the second member is the sum of the energy-momentum tensors of the Maxwell, Dirac and interaction Maxwell-Dirac fields) defined in an effective Lorentzian spacetime, whose use is eventually no more than a question of mathematical convenience.

math-ph↗

arXiv2016

On the Motion of a Free Particle in the de Sitter Manifold

Let $M=SO(1,4)/SO(1,3)\simeq S^{3}\times\mathbb{R}$ (a parallelizable manifold) be a submanifold in the structure $(\mathring{M}% ,\boldsymbol{\mathring{g}})$ (hereafter called the bulk) where $\mathring {M}\simeq\mathbb{R}^{5}$ and $\boldsymbol{\mathring{g}}$ is a pseudo Euclidian metric of signature $(1,4)$. Let $\boldsymbol{i}:M\rightarrow\mathbb{R}^{5}$ be the inclusion map and let \ $\boldsymbol{g}=\boldsymbol{i}^{\ast }\boldsymbol{\mathring{g}}$ be the pullback metric on $M$. It has signature $(1,3)$ Let $\boldsymbol{D}$ be the Levi-Civita connection of $\boldsymbol{g}% $. We call the structure $(M,\boldsymbol{g})$ a de Sitter manifold and $M^{dSL}=(M=\mathbb{R\times}S^{3},\boldsymbol{g},\boldsymbol{D},τ_{\boldsymbol{g}},\uparrow)$ a de Sitter spacetime structure, which is \ of course orientable by $τ_{\boldsymbol{g}}\in\sec% %TCIMACRO{\tbigwedge \nolimits^{4}}% %BeginExpansion {\textstyle\bigwedge\nolimits^{4}} %EndExpansion T^{\ast}M$ and time orientable (by $\uparrow$).\ Under these conditions we prove that if the motion of a free particle moving on $M$ happens with constant \emph{bulk} angular momentum then its motion in the structure $M^{dSL}$ is a timelike geodesic. Also any geodesic motion in the structure $M^{dSL}$ implies that the particle has constant angular momentum in the bulk.

math-ph↗

arXiv2014

A Clifford Bundle Approach to the Differential Geometry of Branes

The Clifford bundle formalism (CBF) of differential forms and the theory of extensors acting on $\mathcal{C\ell}(M,g)$ is first used for a fomulation of the intrinsic geometry of a differential manifold $M$ equipped with a metric field $\boldsymbol{g}$ of signature $(p,q)$ and an arbitrary metric compatible connection $\nabla$ introducing the torsion (2-1)-extensor field $τ$, the curvature $(2-2)$ extensor field $\mathfrak{R}$ and (once fixing a gauge) the connection $(1-2)$-extensor $ω$ and the Ricci operator $\boldsymbol{\partial}\wedge\boldsymbol{\partial}$ (where $\boldsymbol{\partial}$ is the Dirac operator acting on sections of $\mathcal{C\ell}(M,g)$) which plays an important role in this paper. Next, using the CBF we give a thoughtful presentation the Riemann or the Lorentzian geometry of an orientable submanifold $M$ ($\dim M=m$) living in a manifold $\mathring{M}$ (such that $\mathring{M}\simeq\mathbb{R}^{n}$ is equipped with a semi-Riemannian metric $\boldsymbol{\mathring{g}}$ with signature $(\mathring{p},\mathring{q})$ and \ $\mathring{p}+\mathring{q}=n$ and its Levi-Civita connection $\mathring{D}$) and where there is defined a metric $\boldsymbol{g=i}^{\ast}\mathring{g}$, where $\boldsymbol{i}:$ $M\rightarrow \mathring{M}$ is the inclusion map. We prove several equivalent forms for the curvature operator $\mathfrak{R}$ of $M$. It is shown that the Ricci operator of $M$ is the (negative) square of the shape operator $\mathbf{S}$ of $M$. Also we disclose the relationship between the connection (1-2%)-extensor $ω$ and the shape biform $\mathcal{S}$ (an object related to $\mathbf{S}$). We hope that our presentation will be useful for differential geometers and theoretical physists interested, e.g, in string and brane theories and relativity theory.

math-ph↗

arXiv2012

Extracting Energy from an External Magnetic Field

In this paper we describe the theory of a device that is able to extract energy from an external magnetic field. The device is a cylindrical magnetic insulator that once put in rotation makes electromagnetic angular momentum to be stored in the electromagnetic field in contrary direction to the mechanical angular momentum of the device. As a consequence due to total angular momentum conservation the device increases its angular velocity (when εμ>1) Natural units are used in the paper and all conservation laws are rigorously satisfied. The voltage generated by the device is found solving explicitly Maxwell equations for rotating magnetic insulators in external fields, a subject that have provoked lots of polemics in the literature and which we hope to be here clarified due to our pedagogical presentation.

math-ph↗

arXiv2008

An Ambiguous Statement Called 'Tetrad Postulate' and the Correct Field Equations Satisfied by the Tetrad Fields

The names tetrad, tetrads, cotetrads, have been used with many different meanings in the physical literature, not all of them, equivalent from the mathematical point of view. In this paper we introduce unambiguous definitions for each one of those terms, and show how the old miscellanea made many authors to introduce in their formalism an ambiguous statement called `tetrad postulate', which has been source of many misunderstandings, as we show explicitly examining examples found in the literature. Since formulating Einstein's field equations intrinsically in terms of cotetrad fields theta^{a}, a = 0,1,2,3 is an worth enterprise, we derive the equation of motion of each theta^{a} using modern mathematical tools (the Clifford bundle formalism and the theory of the square of the Dirac operator). Indeed, we identify (giving all details and theorems) from the square of the Dirac operator some noticeable mathematical objects, namely, the Ricci, Einstein, covariant D'Alembertian and the Hodge Laplacian operators, which permit to show that each theta^{a} satisfies a well defined wave equation. Also, we present for completeness a detailed derivation of the cotetrad wave equations from a variational principal. We compare the cotetrad wave equation satisfied by each theta^{a} with some others appearing in the literature, and which are unfortunately in error.

math-ph↗

arXiv2007

Clifford and Extensor Calculus and the Riemann and Ricci Extensor Fields in of Deformed Structures

Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection del) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of the the triple (M,del,g)is investigated to for each particular open set U of M through the introduction of a geometric structure on U, i.e., a triple (U,gamma,g) where gamma is a general connection field on U and g is a metric extensor field associated to g. The relation between geometrical structures related by gauge extensor fields is clarified. These geometries may be said to be deformations one of each other. Moreover we study the important case of a class of deformed Levi-Civita geometrical structures and prove key theorems about them that are important in the formulation of geometric theories of the gravitational field.

math.DG↗

arXiv2002

Spin and Electron Structure

The recent literature shows a renewed interest, with various independent approaches, in the classical models for spin. Considering the possible interest of those results, at least for the electron case, we purpose in this paper to explore their physical and mathematical meaning, by the natural and powerful language of Clifford algebras (which, incidentally, will allow us to unify those different approaches). In such models, the ordinary electron is in general associated to the mean motion of a point--like "constituent" Q, whose trajectory is a cylindrical helix. We find, in particular, that the object Q obeys a new, non-linear Dirac--like equation, such that --when averaging over an internal cycle (which corresponds to a linearization)-- it transforms into the ordinary Dirac equation (valid, of course, for the electron as a whole).

quant-ph↗

arXiv1997

A unified theory for construction of arbitrary speeds (0 < = v < \infty) solutions of the relativistic wave equations

Representing the relativistic physical fields as sections of the Clifford Bundle (or of the Spin-Clifford Bundle) of Minkowski spacetime we show that all the relativistic wave equations satisfied by these fields possess solutions traveling with arbitrary speeds $0 \leq v < \infty$. By giving rigorous mathematical definitions of reference frames and of the Principle of Relativity (PR) we prove that physical realizations of the $v > 1$ solutions of, e.g., the Maxwell equations imply in a breakdown of the PR, but in no contradiction at all with known physical facts.

physics.class-ph↗