Radiation from a charged particle and radiation reaction reexamined

Abstract

We study the electromagnetic fields of an arbitrarily moving charged particle and the radiation reaction on the charged particle using a novel approach. We first show that the fields of an arbitrarily moving charged particle in an inertial frame can be related in a simple manner to the fields of a uniformly accelerated charged particle in its rest frame. Since the latter field is static and easily obtainable, it is possible to derive the fields of an arbitrarily moving charged particle by a coordinate transformation. More importantly, this formalism allows us to calculate the self-force on a charged particle in a remarkably simple manner. We show that the original expression for this force, obtained by Dirac, can be rederived with much less computation and in an intuitively simple manner using our formalism. PACS number(s):The field of a charged particle at rest in an inertial frame is a static Coulomb field which falls as (1/r 2 ) in the standard spherical coordinate system. The field of a charge, moving with uniform velocity, can be obtained by Lorentz transforming the Coulomb field; this field also falls as inverse square of the distance. The situation changes dramatically for a charged particle which is moving with non zero acceleration. The field now has a piece which falls only as (1/r), usually called the radiation field. For a field which decreases as (1/r), the energy flux varies as (1/r 2 ) implying that the same amount of energy flows through spheres of different radii at sufficiently large distances from the charge. Because of this reason, the radiation fields acquire a life of their own and the entire phenomena of electromagnetic radiation hinges on this feature. Due to the continuous transfer of energy from the charged particle to large distances, there will be a damping force acting on the charged particle which is usually called the radiation reaction force. The derivation of radiation reaction force is conceptually and operationally quite complicated and the expression -obtained originally by Dirac (see [4])-has no simple intuitive description.We analyse these issues from a novel point of view in this paper which throws light on the conceptual and mathematical issues involved in this problem. The analysis is motivated by the following issue: Maxwell's equations are not only Lorentz invariant but can also be written in a generally covariant manner. Given a charged particle moving in some arbitrary trajectory, it is always possible to construct a proper coordinate system for such a charged particle. In such a coordinate system, the charge will be at rest for all times but the background metric will be non Minkowskian and -in general -time dependent. The Maxwell's equations in this coordinate system will correspond to that of a stationary charge located in a non trivial (and in general time dependent) metric. The solution to Maxwell's equation in this frame receives time dependent contributions not because of the motion of charged particles but because of the non trivial nature of the background me...

Reduced-order Abraham-Lorentz-Dirac equation and the consistency of classical electromagnetism

“…One can use the theory to obtain an equation that describes, approximately, the motion of a rigid spherical shell of charge subject to an arbitrary force. This is the Abraham-Lorentz-Dirac (ALD) equation [1][2][3][4][5][6]. This equation, and others like it, is problematic because it is higher than second order in the time derivatives of position, which can lead to problems with causality, and because it has pathological or 'runaway' solutions in which, for example, an object accelerates when no force is applied.…”

A Coaccelerated Observer

“…These expressions were found by Fulton and Rohrlich by using the retarded potential method (the use of the retarded potentials actually drops the basis for the first point of Boulware). The same expressions for the electromagnetic fields were also found by Gupta and Padmanabhan, (16) who calculated first the electromagnetic fields in the rest system of the charge, and then transformed them to a system moving in a hyperbolic motion relative to the charge. They also show that by using appropriate transformations to the system of reference of the retarded variables, they recover the equations given in the textbooks (11,13,15) for retarded potentials.…”

A “Fine Structure Constant” for Inertia

“…Such a motion is called "a hyperbolic motion", as it describes a hyperbola in Rindler coordinates (Rindler [5]). We shall use the method suggested by Gupta and Padmanabhan [6], where they calculate first the field equations of a particle (an electric charge) in its own system of reference, and then transform these equations to the system of free space, in which the particle is accelerated. In using this method for the calculation of electric field of an accelerated charge, they recovered the field equations as calculated by Schott [7], and by Fulton and Rohrlich [8].…”

“…The gravitational field is calculated first in the rest system of the particle, and then, these equations are transformed to the system in which the particle is accelerated in. These equations, given in cylindrical coordinates (z, ρ, Ɵ), are: (6) where ᶍ ( ) ( 2 2 2 2 2 2 2 is the mass location at t = 0 (the turning point in Rindler coordinates. (Here a is the acceleration).…”