In the consecutive sections we study the mechanism of particle acceleration during col- lisionless magnetic reconnection in the relativistic regime. It is instructive to place in advance some very general remarks on the problem of relativistic particle acceleration in electromagnetic fields. The relativistic particle dynamics in time-dependent electric Eand magnetic fields B are non-trivial and are addressed in the numerical simulation. On introduction we restrict to the special case of a stationary field configuration, in which E and B have exactly transversal orientations. The reasons for this choice are straightforward: First, the scalar produce E·B is a Lorentz invariant. Therefore only in configurations with E⊥B it is in principle possible to remove either E or B con- tributions by transforming into a distinguished Lorentz frame. And second, in a very rough approximation the magnetic field topology starting from the initial Harris equi- librium evolves during the initial stages of reconnection within the (x,y)-plane. Then the inductive electric field c−1∂tEz=−(∂xBy−∂yBx) points exactly perpendicular to
the magnetic field.
Hence, we consider an inertial frame K in whichE·B= 0 is valid. Then two alternatives remain: For E<B we transform into the so-called de Hoffmann-Teller-frame. This is a Lorentz frame K’ moving with velocity
βββHT = uHT
c =
E×B
B2
with respect to K. The relativistic field transformations are derived to E = γ(E+βββ×B)− γ 2 γ+ 1βββ(βββ·E) B = γ(B+βββ×E)− γ 2 γ+ 1βββ(βββ·B)
from the Lorentz transformations of the field strength tensor Fµν = ΛµσFστΛτν with Fµν =∂µAν−∂νAµ and Aµ={Φ,A}. Applied to the transition K→K the fields in the de Hoffmann-Teller-frame ensue to
E = 0 B = 0
E⊥ = 0 B⊥ = γ−HT1 B
As a consequence thereof, the electric field vanishes in the de Hoffmann-Teller-frame K’ and the particle motion is reduced to a simple gyration in the perpendicular (with respect to u/c) magnetic field B⊥. For an observer in K the particle trajectory is composed by the superposition of a gyration (with charge-dependent orientation) and a charge- independent drift motion u perpendicular to theE and B fields.
Alternatively, for E>B we transform into a different frame K’ moving with
β ββ= u
c =
E×B
in which
E = 0 B = 0
E⊥ = γ−1E B⊥ = 0
holds. In this regime particle acceleration by the electric field dominates over the gyro motion. In the frame K’ only a finite electric field contribution remains. In the lab frame K the trajectory is described as the superposition of a continuous driftu and a contin- uous acceleration parallel to E. In all configurations withE⊥B the gyro radius scales as rL∝γ. For the case E>B the particle trajectory can be understood as a gyro mo-
tion, in which the relativistic growth of rL supersedes the magnetic deflection. The case
E⊥B is pervasive in reconnection scenarios.
For completeness we consider the other extreme case EB, in which the electric field acceleration takes place always perpendicular to the gyro motion. In this case the mo- tion is continuously accelerated in z and approaches asymptotically the speed of light. The gyro radius remains constant, i.e. p⊥ is a cyclic variable and p⊥/B is a so-called adiabatic invariant.
Finally, what can we infer from these simple considerations? For stationary recon- nection (i.e. for a reconnection scenario, in which the X-point remains localized to a fixed position (x0,y0,z0)) the ratio of inductive electric field to magnetic field is al-
ways Ez/B01. Consequently, a transformation to the de Hoffmann-Teller frame is
always possible. In the next sections we analyze the particle acceleration in relativistic collisionless reconnection. For the thin current sheets under consideration, the reconnec- tion zone evolves from stationarity in the early stages to highly dynamic configurations in the late time evolution. Remarkably, during late times the accelerating electric field
|Eacc|=|Eind+Ecom|>B0 supersedes the Harris equilibrium field. This is becauseEacc
can be understood as the superposition of a reconnection inductive componentEindand
an additional field contributionEcom =−v/c×Bfrom the comotion of the reconnection
zone with the dynamically evolving current sheet filaments. In a formal way of argu- mentation the transformation to the de Hoffmann-Teller-frame Eacc=−uHT/c×B0
would imply a ‘superluminal’ particle comotion cuHT. This formal reasoning pro-
ceeds in analogy to the argumentation in the context of particle acceleration during ‘superluminal’ collisionless shocks. Intriguingly, such ‘superluminal’ effects are essential to explain the transition from Poynting-flux-driven flows to kinetically-dominated flows - the most famous paradigm for this transition being theσ-problem of the Crab pulsar wind [Lyubarski & Kirk, 2001; Kirk & Skjaeraasen, 2003].