Point-particles and energy tensors in special relativity

Summary In order to study the interactions of point-particles in terms of classical concepts, it is necessary to calculate the flux of 4-momentum across a tube enclosing the world line of a particle. Since the field is assumed to be retarded, this may involve complicated and unpleasant work unless the tube is properly chosen. In this paper it is proposed to use a tube defined by w=const. where w is the retarded distance of an event from the world line, this retarded distance being simply defined in terms of Minkowskian geometry. Then, no matter what value w may have, the required flux is given by a triple integral in which the differential element is ds dw, where ds is an element of proper time on the world line and dw an elementary solid angle. When this is applied to the self-field of a charged particle, we get a very simple formula which gives the usual flux at infinity when w tends to infinity and a singular term when w tends to zero. A similar formula emerges for a particle producing a scalar field. Being now in possession of a technique of reasonable simplicity, we can explore the question whether the infinite self-energy of a charged particle can be eliminated by modifying the energy tensor by a term which falls off rapidly as the distance from the particle is increased. It is shown that this can be done. A Bruno Finzi nel suo 70^mo compleanno. Entrata in Redazione il 4 novembre 1969.     

Nonclassical symmetries as special solutions of heir-equations

In Phys. D 78 (1994) 124, we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations, which inherit the Lie point symmetry algebra of the given equation. In the present paper, we show that special solutions of the right-order heir-equation correspond to classical and nonclassical symmetries of the original equations. An infinite number of nonlinear equations which possess nonclassical symmetries are derived.     

Alternative-algebra methods in the special theory of relativity

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 86, No. 2, pp. 294–299, February, 1991.     

About the foundations of the special theory of relativity

The principle of the constancy of the velocity of light propagation is used to introduce a differentiable structure on the universe of events E. Namely, using a theorem proved in [2] and some axioms imposed toE it is shown howE can be endowed with an atlas s.t. the coordinate transformations between the charts of this atlas are given by conformal (or Weyl) transformations.     

An analogy between dispersive wave propagation and special relativity

Summary It is shown that, with an appropriate definition of variables, the dynamics of monochromatic wave packets in a dispersive medium become, unexpectedly, identical in form to those describing the relativistic dynamics of classical mass particles. An example is provided by the analysis of wave propagation on a taut, massless string on which discrete masses are attached periodically at intervals that may be slowly dependent on position. An experiment illustrating the latter case shows the motion and the reflection of a wave packet by a gradient in the analog of a relativistic potential.

---

Sommaire Il est montré que par une définition appropriée de variables la dynamique de paquets d'ondes monochromatiques dans un milieu dispersif prend, de facon inattendue, une forme identique à celle de la dynamique relativiste de masses ponctuelles classiques. Un example est fourni par l'analyse de la propagation d'ondes sur une corde tendue dépourvue de masse propre à laquelle des masses ponctuelles sont attachées périodiquement à des intervalles qui peuvent dépendre lentement de la position. Une expérience illustrant ce dernier cas montre le mouvement et la réflection d'un paquet d'ondes par l'analogue d'un gradient de potentiel relativiste.

     

The application of the solid-angle theorem in the special theory of relativity

Ishlinskii's theorem, well known in classical mechanics, asserts that if an axis, selected in a rigid body, having zero projection of the angular velocity onto this axis, described a closed conical surface during the motion of the body, then, after the axis has returned to its initial position the body will have described an angle around it numerically equal to solid angle of the described cone. It is shown that the same relation also exists in the Special Theory of Relativity—the angle of rotation described by a rigid body during motion along a curvilinear trajectory due to the Thomas precession effect, is numerically equal to the solid angle observed in a fixed frame of reference described by an axis connected with the body due to a change in the rotation of the image of the rigid body. The latter phenomenon is due to the Lorentz contraction of the length and the retardation of light radiated by different parts of the body [10–13].     

Thermal molecule and atom test of the modified special relativity theory

The correction to Maxwell–Boltzmann's velocity distribution law is obtained in the framework of the modified special relativity theory. The detection of velocity and velocity rate distributions for thermal molecules or atoms can serve as a test of the modified special relativity theory.     

Material symmetries versus wavefront symmetries

We show that, in general, wavefronts are more symmetric thanthe medium in which they propagate. This means that we cannotdetermine the symmetries of the medium based solely on the symmetriesof the wavefronts. However, we show that we can determine thesymmetries of the medium from the symmetries of wavefronts andpolarizations together.     

Modification of special relativity and local structures of gravity-free space and time

Besides two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the one-way speed of light in all inertial frames of reference, the special theory of relativity uses the assumption about the Euclidean structure of gravity-free space and the homogeneity of gravity-free time in the usual inertial coordinate system. Introducing the so-called primed inertial coordinate system, in addition to the usual inertial coordinate system, for each inertial frame of reference, we assume the flat structures of gravity-free space and time in the primed inertial coordinate system and their generalized Finslerian structures in the usual inertial coordinate system. We combine this assumption with the two postulates (i) and (ii) to modify the special theory of relativity. The modified special relativity theory involves two versions of the light speed, infinite speed c^′ in the primed inertial coordinate system and finite speed c in the usual inertial coordinate system. It also involves the c^′-type Galilean transformation between any two primed inertial coordinate systems and the localized Lorentz transformation between any two usual inertial coordinate systems. The physical principle is: the c^′-type Galilean invariance in the primed inertial coordinate system plus the transformation from the primed to the usual inertial coordinate systems. Evidently, the modified special relativity theory and the quantum mechanics theory together found a convergent and invariant quantum field theory.     

Modification of special relativity and formulation of convergent and invariant quantum field theory

Besides two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the speed of light in all inertial frames of reference, the special theory of relativity uses another assumption. This other assumption concerns the Euclidean structure of gravity-free space and the homogeneity of gravity-free time in the usual inertial coordinate system. Introducing the primed inertial coordinate system, in addition to the usual inertial coordinate system, for each inertial frame of reference, we assume the Euclidean structures of gravity-free space and time in the primed inertial coordinate system and their generalized Finslerian structures in the usual inertial coordinate system. We combine the alternative assumption with the two postulates (i) and (ii) to modify the special theory of relativity. The modified special relativity theory involves two versions of the light speed, infinite c′ in the primed inertial coordinate system and finite c in the usual inertial coordinate system. It also involves the c′-type Galilean transformation between any two primed inertial coordinate systems and the localized Lorentz transformation between two corresponding usual inertial coordinate systems. Since all our experimental data are collected and expressed in the usual inertial coordinate system, the physical principle is: the c′-type Galilean invariance in the primed inertial coordinate system plus the transformation from the primed inertial coordinate system to the usual inertial coordinate system. This principle is applied to a reformulation of mechanics, field theory and quantum field theory. Relativistic mechanics in the usual inertial coordinate system is unchanged, while field theory is developed and divergence-free. Any c′-type Galilean-invariant field system can be quantized by using the canonical quantization method in the primed inertial coordinate system. We establish a transformation law for quantized field systems as they are transformed from the primed to the usual inertial coordinate system. It is shown that the modified special relativity theory, together with quantum mechanics, leads to a convergent and invariant quantum field theory, in full agreement with experimental facts. The formulation of this quantum field theory does not demand departures from the concepts such as local Lorentz invariance in the usual inertial coordinate system, locality of interactions, and local or global gauge symmetries.     

Computing ODE symmetries as abnormal variational symmetries

We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.     

Computing ODE symmetries as abnormal variational symmetries

We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387–409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.     

Approximate symmetries in viscoelasticity

In the framework of approximate symmetries, we investigate a perturbed system of partial differential equations for viscoelastic media with nonlinear dissipation. We completely classify the approximate symmetries and prove a theorem on the relation between the symmetries of two related models. In some physical cases, we find approximate solutions using the generator of the group of transformations taken in the first-order approximation.