Asymptotic Analysis of Ultra-relativistic Charge

A summary of a new approach is given for analysing the dynamic behaviour of distributions of charged particles in an electromagnetic field. Noting the limitations inherent in the Lorentz-Dirac equation for a single point particle, a simple model is proposed for a charged continuum interacting self-consistently with the Maxwell field in vacuo. The model is developed using intrinsic tensor field theory and exploits to the full the symmetry and light-cone structure of Minkowski spacetime. A covariant perturbation scheme is motivated by an exact class of solutions to this system describing the evolution of a charged fluid under the combined effects of both self and external electromagnetic fields. The scheme yields an asymptotic approximation involving inhomogeneous linear equations for the self-consistent Maxwell field, charge current and time-like velocity field of the charged fluid INTRODUCTION The intense international activity involved in probing the structure of matter on all scales, with particle beams and radiation, owes much to recent advances in accelerator science and technology. Developments in the production of high power laser radiation also offer new avenues for accelerator design and new diagnostic tools of relevance to medical science, engineering and the communications industry. A common theme in these developments is the interaction between charged particles and the electromagnetic field in domains where relativistic effects cannot be ignored. It is remarkable that many of the challenges that must be addressed in order to develop and control devices that accelerate charged particles have their origin in the interaction of particles with their own electromagnetic field. Despite the fact that the classical laws of electromagnetism were essentially formulated over a century and a half ago the subject of electromagnetic interactions with matter remains incomplete. This incompleteness has had concomitant effects on the development of quantum electrodynamics and renormalisation theory. At root, the difficulties reside in the recognition that the quantum structure of matter at some scale is beyond observation. Furthermore, the classical description of the electron as a point particle leads to singularities in the Maxwell self-fields that inevitably create ambiguities in its interaction with the Maxwell field. The general consensus is that a useful domain of validity of the Lorentz-Dirac equation [1, 2, 3], describing covariantly the radiation reaction on a point electron, can be accommodated by performing a “reduction of order” that effectively ∗[email protected] replaces the equation by a perturbative second order system for the particle world line. One must then decide whether higher order terms in this expansion should be maintained given the neglect of terms associated with the regularisation scheme. This approach is behind many of the successful applications of approximate radiation reaction dynamics, despite the somewhat delicate and unsatisfactory nature of the arguments that purport to support it. However, an understanding of the interaction of charged particle bunches with their electromagnetic fields is vital for the success of proposed future accelerators. In this article a summary is given of a new approach for analysing the behaviour of distributions of ultra-relativistic charged particles in a coupled electromagnetic field environment [4]. Rather than mixing particle and field concepts, our model is formulated entirely in the language of smooth tensor fields. Here, the non-linear field equations describing a thermodynamically inert (pressureless) charged fluid are adopted as a dynamical model of charged particle bunches. They are analysed using an ultrarelativistic asymptotic approximation scheme that conserves total energy, momentum and charge at all orders in the expansion parameter. EQUATIONS OF MOTION FOR A PRESSURELESS CHARGED FLUID The partial differential system of equations governing the motion of a pressureless charged fluid in MKS form is ∇ · e = 1 0 , ∇× b = μ0 v + 1 c2 ∂e ∂t , ∇× e + ∂b ∂t = 0, ∇ · b = 0, ∂p ∂t + (v ·∇)p = q0 (e + v × b) , p = m0 √ 1−v·v/c2 v (1) where e is the electric field, b is the magnetic induction, v is the fluid’s velocity field, is the fluid’s charge density, p is the fluid’s momentum field, m0 is the rest mass of the particles comprising the fluid, q0 is the fundamental charge of the particles comprising the fluid, 0 is the permittivity of the vacuum and μ0 is the permeability of the vacuum. Although (1) is the most familiar statement of the fluid’s equations of motion, their most succinct and analytically powerful form is expressed without reference to a particular inertial frame i.e. where the inertial time t does not appear explicitly. Let M be Minkowski spacetime and let g be its flat Lorentzian metric with ∇ the Levi-Civita connection. WEPCH073 Proceedings of EPAC 2006, Edinburgh, Scotland 2086 05 Beam Dynamics and Electromagnetic Fields D02 Non-linear Dynamics Resonances, Tracking, Higher Order The system (1) may be recast in the coordinate-free form dF = 0, d F = −ρ ̃ V , ∇V ̃ V = iV F, g(V, V ) = −1 (2) where V is the fluid’s 4-velocity field, F is an electromagnetic field 2-form, ρ is proportional to the proper charge density1 and is called the reduced proper charge density, d is the exterior derivative, is the Hodge map associated with g, iV is the interior operator and the 1-form ̃ V is the metric dual of V . For more information on the precise relationship between (2) and (1) see [4]. APPROXIMATION SCHEME In general situations, exact solutions to the non-linear partial differential system (2) are extremely difficult to find. A self-consistent method for obtaining approximate solutions is to let V , F and ρ depend on a running parameter ε and to solve the hierarchy of equations generated by (2) for the 1-parameter families of fields V , F ε and ρ order by order in ε. Whether or not the hierarchy of equations so obtained is easier to solve than the original system depends on the ε dependences of the fields. Although there are many possibilities, a viable choice is strongly suggested by the behaviour of a class of highly symmetric exact solutions relevant to accelerator physics. These solutions describe “walls of charge” of infinite extent accelerating due to their self-field (“space-charge”) and an applied constant electric field. Let (t, x, y, z) be an inertial coordinate system where t is the laboratory time and (x, y, z) are Cartesian coordinates. The spacetime metric tensor g has the form2 g = −dt⊗ dt + dx⊗ dx + dy ⊗ dy + dz ⊗ dz. (3) The “wall of charge” solutions have the following properties in the (dt, dx, dy, dz) frame : the magnetic field vanishes, the electric field has only a z component, the 4velocity field has only t and z components and all field components depend only on t and z. The z component of the reduced electric field has the form E(t, z) = ζ(σ̂(t, z)) (4) where σ = σ̂(t, z) is a solution to the implicit equation σ = z − 1 ζ(σ) (