---
abstract: 'It is shown how initial conditions can be appropriately defined for the integration of Lorentz-Dirac equations of motion. The integration is performed [*forward*]{} in time. The theory is applied to the case of the motion of an electron in an intense laser pulse, relevant to nonlinear Compton scattering.'
address:
- |
    $^1$Instituto de Física, Universidade de São Paulo,\
    C.P.66318, 05315-970, São Paulo, SP, Brazil 
- |
    $^2$Institute for Theoretical Physics, University of California,\
    Santa Barbara, 93106-4030, USA 
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    $^3$Institute for Theoretical Atomic and Molecular Physics,\
    Harvard-Smithsonian Center for Astrophysics,\
    Cambridge, Massachusetts, 02138, USA
author:
- |
    M.S. Hussein$^{1}$, M.P. Pato$^{1,2}$, and J.C. Wells$^{2,3\thanks{Present address: Center for Computional Sciences, Oak Ridge National
    Laboratory, Oak Ridge, Tenessee, 37831-6373, U.S.A.}}$
title: Causal Classical Theory of Radiation Damping
---

-20mm

The advent of a new generation of extremely high power lasers that uses chirped pulse amplification has put into focus the classical description of the dynamics of relativistic electrons. Under the action of high intensity electromagnetic field, a major ingredient of the dynamics is the electron self-interaction which implies in the damping of the movement caused by the interaction of the charge with its own field. The derivation of the damping force has been reviewed recently[@Luhm] revealing its relativistic origin asociated to the asymmetry introduced by the Doppler effect in the forward and backward emission of radiation. The inclusion of this force in the equation of motion leads to the nonlinear covariant Lorentz-Dirac(LD) equation[@Dirac] for a point charge.

Nonlinear effects have been observed in recent experiments with intense-laser relativistic-electron scattering at laser frequencies and field strengths where radiation reaction forces begin to become significant[@Hart; @Bula]. Relativistic nonlinear Thompson scattering has also been observed[@Yuan]. These experiments justify the recent attention the classical electrodynamics theory(CED) has received. A self-consistent classical theoretical treatment of the radiative reaction force would also be useful in simulating future electron accelerators[@HP92]. Further, the kind of study reported upon in this paper may be useful in the quantal treatment[@FLow].

A certain number of problems, conceptual and practical, is known to be associated with LD equation. These difficulties may be traced to the fact that it contains a dependence on the derivative of the acceleration which implies in the necessity of imposing, in order to solve it, an extra condition besides the usual initial conditions on position and velocity of classical mechanics. It has been estabilished by Rohrlich that this condition is given by the asymptotic constraint that at the far future, when the fields vanish, the acceleration should also vanish. The solution which is obtained however, when extrapolated to the instant when the external force is applied violates causality. On the other hand, from the practical point of view, a condition put in the far future is awkward to be implemented specially in a scattering situation.

A recent book[@Yagh] has been devoted to this question of causality violation. A clear explanation to its origin was given and it has been shown how it can be solved in the context of the CED. On the other side, attempts has been done in order to replace Rohrlich condition by an equivalent initial condition[@Agui; @Vill]. The purpose of this letter is to apply these modern advances, to the case of the classical description based in LD equation, of the movement of an electron interacting with a short, strong laser pulse. This problem has been recently discussed in the literature Ref. [@Hart], which provided numerical solutions of the Lorentz-Dirac equation. The integration was performed backwards in time so that the unphysical, exponentially growing homogeneous solutions of LD would damp out, resulting in a numerical stable solution.

We are going to show that Lorentz-Dirac equation of motion can be integrated forward in time with conditions specified at $t=0$. The idea is to construct the series solution of LD equation. The initial acceleration is then provided by replacing in the series the velocity by their initial value at the instant when the external force is applied. It is easy to show that the solution obtained with this procedure, when extrapolated to the distant future, satisfies Rohrlich condition. However, we still have to cope with the existence of the unphysical runaway solutions which although formally eliminated troubles the process of numerical integration. Then, by combining the recursive use of the series solution with implicit methods of numerical integration we show that the process of integration forward in time can be performed.

We write Lorentz-Dirac (LD) equation of motion as $$F_{\mu }^{ext}=a_{\mu }-\epsilon \left( \frac{d^{2}v_{\mu }}{d\tau ^{2}}+v_{\mu }a_{\lambda }a^{\lambda }\right)   \label{2}$$ where $v_{\mu }$,$a_{\mu }$ and $F_{\mu }^{ext}$ are, respectively, the four-vector components of the velocity,acceleration and of the external force given explicitly by

$$v_{\mu }=\gamma \left( 1,{\bf \beta }\right) ,  \label{4}$$

$$v^{\mu }=\gamma \left( 1,-{\bf \beta }\right) ,  \label{4a}$$

$$a_\mu =\frac{dv_\mu }{d\tau }$$

and

$$F_{\mu }^{ext}=\gamma \left( {\bf \beta }\cdot {\bf F}_{ext},{\bf F}_{ext}\right) .  \label{6}$$

In these equations, $\tau $ is the dimensionless proper time $d\tau =\omega
_{0}dt/\gamma $, $\gamma $ is the relativistic factor $\gamma =\frac{1}{\sqrt{1-\beta ^{2}}}$ with ${\bf \beta }=\frac{1}{c}\frac{d{\bf r}}{dt}$ and $\omega _{0}$ is the frequency of the laser pulse with which the electron is interacting. We follow here the same units of Ref. [@Hart]. With these definitions, it can be easily verified that $v_{\mu }a^{\mu }=0$ as it should. The quantity $\varepsilon =\omega _{0}\tau _{0},$ where $\tau
_{0}=e^{2}/m_{e}c^{3}=0.626\times 10^{-23}s$ is the Compton time scale. We are assuming that the external force applies at $\tau =0$ and the second term in the right hand side of the equation represents the damping force arising as the charge starts to radiate[@Rohr].

By contracting the Lorentz-Dirac equation with $a_{\mu }$, it is found that $\left( a_{\mu }a^{\mu }\right) \left( \tau \right) =C\exp (2\tau /\epsilon )$ is a solution of the resulting homogeneous equation (no external force) for times greater than $\tau _{0}$. These are the so-called runaway solutions. They are eliminated by imposing the Dirac-Rohrlich (DR) condition

$$\lim\limits_{\tau \rightarrow \infty }a_\mu =0  \label{8}$$

when

$$\lim\limits_{\tau \rightarrow \infty }F_\mu ^{ext}\left( \tau \right) =0.
\label{10}$$

In contrast to the usual initial value problem encountered in classical physics, where all quantities, position, velocity, are fixed at $t=0$, the above asymptotic condition, Eq. (\[8\]) recast the problem into a boundary value one where the $x,v$ are known at $\tau =0$ and $a$ is forced to be zero at $\tau =\infty $, in accordance with the condition on the force, Eq. (\[10\]). A particular situation is that of Ref. [@Hart], in which the problem was solved in the rest frame of the particle at some future time with three final homogeneous conditions. In both situations however, the numerical integration has to be performed backwards in time. Although this procedure eliminates the unphysical runway solutions, it is uncomfortable, from the point of view of applications, to have a condition given at some final time. It is therefore desirable to have an equivalent condition on the acceleration defined at the initial time.

For an electron moving in an electromagnetic field the external force is given by $$F_{\mu }^{ext}=-(\partial _{\nu }A_{\mu }-\partial _{\mu }A_{\nu })v^{\nu }
\label{12}$$ where the quantity $A_{\mu }$ is the vector potential given in units of $m_{0}c/e$. For a linearly polarized laser pulse, $$A_{\mu }\equiv (\Phi /c,{\bf A}),\;\;\;\;{\bf A}=\hat{x}A_{x}(\phi
),\;\;\;\;\Phi =0  \label{16}$$ which is a function of the invariant phase of the traveling wave, $$\phi =k^{\mu }x_{\mu }(\tau )=x_{0}-z\;,  \label{20}$$ where $k^{\mu }=(1,0,0,1)$ is the dimensionless laser wave number.

Following Ref. [@Hart] we use $\phi $ as the independent variable to recast the Dirac Lorentz equation as $$\begin{aligned}
\frac{d^{2}v_{x}}{d\phi ^{2}} &=&v_{x}\left( \left( \frac{d{\bf v}}{d\phi }\right) ^{2}-\left( \frac{d\gamma }{d\phi }\right) ^{2}\right) +\frac{1}{u}\left( \frac{dv_{x}}{d\phi }\left( \frac{1}{\varepsilon }-\frac{du}{d\phi }\right) +\frac{A}{\varepsilon }G(\phi )sin\phi \right)   \label{28} \\
\frac{d^{2}v_{z}}{d\phi ^{2}} &=&v_{z}\left( \left( \frac{d{\bf v}}{d\phi }\right) ^{2}-\left( \frac{d\gamma }{d\phi }\right) ^{2}\right) +\frac{1}{u}\left( \frac{dv_{z}}{d\phi }\left( \frac{1}{\varepsilon }-\frac{du}{d\phi }\right) +\frac{v_{x}}{u}\frac{A}{\varepsilon }G(\phi )sin\phi \right) 
\label{30}\end{aligned}$$ where the laser pulse electric field $$\frac{dA_{x}(\phi )}{d\phi }=AG(\phi )sin\phi :,  \label{32}$$ has been introduced, with $A$ being the maximum amplitude of the pulse, $G(\phi )=e^{-(\phi /\Delta \phi )^{2}}$ is a unit Gaussian envelope of width $\Delta \phi $. Note that $$\left( \frac{d{\bf v}}{d\phi }\right) ^{2}=\left( \frac{dv_{x}}{d\phi }\right) ^{2}+\left( \frac{dv_{z}}{d\phi }\right) ^{2}\;,  \label{33a}$$

$$\frac{d\gamma }{d\phi }=\frac{v_{x}}{\gamma }\frac{dv_{x}}{d\phi }+\frac{v_{x}}{\gamma }\frac{dv_{z}}{d\phi }  \label{33b}$$

and

$$\frac{du}{d\phi }=\frac{d\gamma }{d\phi }-\frac{dv_{z}}{d\phi }.  \label{33c}$$

Substituting these relations into Eqs. (\[28\] ) and (\[30\]) we find $$\begin{aligned}
\frac{d^{2}v_{x}}{d\phi ^{2}} &=&v_{x}Q+\frac{1}{u}\left( \frac{1}{\varepsilon }\frac{dv_{x}}{d\phi }-\frac{v_{x}}{\gamma }\left( \frac{dv_{x}}{d\phi }\right) ^{2}+\left( 1-\frac{v_{z}}{\gamma }\right) \frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }+\frac{A}{\varepsilon }G(\phi )sin\phi \right)
\label{34} \\
\frac{d^{2}v_{z}}{d\phi ^{2}} &=&v_{z}Q+\frac{1}{u}\left( \frac{1}{\varepsilon }\frac{dv_{z}}{d\phi }-\frac{v_{x}}{\gamma }\frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }+\left( 1-\frac{v_{z}}{\gamma }\right) \left( \frac{dv_{z}}{d\phi }\right) ^{2}+\frac{v_{x}}{u}\frac{A}{\varepsilon }G(\phi
)sin\phi \right)  \label{36}\end{aligned}$$ where

$$Q=\left( \frac{dv_{x}}{d\phi }\right) ^{2}\left( 1-\frac{v_{x}^{2}}{\gamma
^{2}}\right) +\left( \frac{dv_{z}}{d\phi }\right) ^{2}\left( 1-\frac{v_{z}^{2}}{\gamma ^{2}}\right) -\frac{2v_{x}v_{z}}{\gamma ^{2}}\frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }  \label{40}$$

and $$\begin{aligned}
\gamma &=&\sqrt{1+v_{x}^{2}+v_{z}^{2}}  \label{44} \\
u &=&\gamma -u_{z}\;.  \label{46}\end{aligned}$$

In Ref. [@Hart], by specifying final homogeneous conditions on the acceleration and the velocity and then integrating [*backward*]{} in time, the solution to these equations were obtained at all times. We now want to show that this problem can also be solved by specifying initial conditions on the motion and integrating [*forward*]{} in time.

For the “initial” velocity of our method, we use the final velocity of the backward integration method of [@Hart]. As for the initial acceleration we employ the first terms of the series generated by expanding the equations of motion, in terms of the small quantity $\varepsilon =\omega _{0}\tau _{0}$. To obtain this series we write the two components of the equations of motion (\[34\]) and (\[36\]) as

$$\begin{aligned}
\frac{dv_{x}}{d\phi } &=&-AG(\phi )sin\phi +\varepsilon \left[ u\left( \frac{d^{2}v_{x}}{d\phi ^{2}}-v_{x}Q\right) +\frac{v_{x}}{\gamma }\left( \frac{dv_{x}}{d\phi }\right) ^{2}-\left( 1-\frac{v_{z}}{\gamma }\right) \frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }\right]  \label{50} \\
\frac{dv_{z}}{d\phi } &=&-\frac{v_{x}}{u}AG(\phi )sin\phi +\varepsilon
\left[ u\left( \frac{d^{2}v_{z}}{d\phi ^{2}}-v_{z}Q\right) +\frac{v_{x}}{\gamma }\frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }-\left( 1-\frac{v_{z}}{\gamma }\right) \left( \frac{dv_{z}}{d\phi }\right) ^{2}\right]  \label{52}\end{aligned}$$

From the above equations we derive the zeroth order for the derivatives of the components of the acceleration

$$\begin{aligned}
\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0} &=&-A\frac{d}{d\phi }[G(\phi )sin\phi ]  \label{54} \\
\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0} &=&-A\frac{d}{d\phi }[\frac{v_{x}}{u}G(\phi )sin\phi ]  \label{55}\end{aligned}$$

Substituting these relations back into $\left( \ref{50}\right) $ and $\left( 
\ref{52}\right) $, we obtain the first order approximation for the components of the acceleration $\left( \frac{dv_{x}}{d\phi }\right)
_{1},\left( \frac{dv_{z}}{d\phi }\right) _{1}.$ On the other hand, the zeroth order of the second derivative of the components of the acceleration are given by

$$\begin{aligned}
\left( \frac{d^{3}v_{x}}{d\phi ^{3}}\right) _{0} &=&-A\frac{d^{2}}{d\phi ^{2}}[G(\phi )sin\phi ]  \label{56} \\
\left( \frac{d^{3}v_{z}}{d\phi ^{3}}\right) _{0} &=&-A\frac{d^{2}}{d\phi ^{2}}[\frac{v_{x}}{u}AG(\phi )sin\phi ]  \label{58}\end{aligned}$$

Taking now the derivatives of $\left( \ref{50}\right) $ and $\left( \ref{52}\right) $ we find the rather lengthy relations

$$\begin{aligned}
\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{1} &=&\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}+\varepsilon \{u\left[ \left( \frac{d^{3}v_{x}}{d\phi
^{3}}\right) _{0}-v_{x}Q^{\prime }-\left( \frac{dv_{x}}{d\phi }\right)
_{1}Q\right] +u^{\prime }\left[ \left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right)
_{0}-v_{x}Q\right] +  \nonumber \\
&&+2\frac{v_{x}}{\gamma }\left( \frac{dv_{x}}{d\phi }\right) _{1}\left( 
\frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}+\frac{1}{\gamma ^{2}}\left[ \gamma
\left( \frac{dv_{x}}{d\phi }\right) _{1}-v_{x}\gamma ^{\prime }\right]
\left( \frac{dv_{x}}{d\phi }\right) _{1}^{2}-\left( 1-\frac{v_{z}}{\gamma }\right)   \nonumber \\
&&\left[ \left( \frac{dv_{x}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0}+\left( \frac{dv_{z}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}\right] +\frac{1}{\gamma ^{2}}\left[
\gamma \left( \frac{dv_{z}}{d\phi }\right) _{1}-v_{z}\gamma ^{\prime
}\right] \left( \frac{dv_{x}}{d\phi }\right) _{1}\left( \frac{dv_{z}}{d\phi }\right) _{1}\}  \label{60} \\
\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{1} &=&\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0}+\varepsilon u\left[ \left( \frac{d^{3}v_{z}}{d\phi
^{3}}\right) _{0}-v_{z}Q^{\prime }-\left( \frac{dv_{z}}{d\phi }\right)
_{1}Q\right] +u^{\prime }\left[ \left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right)
_{0}-v_{z}Q\right] +  \nonumber \\
&&\frac{v_{x}}{\gamma }\left[ \left( \frac{dv_{x}}{d\phi }\right) _{1}\left( 
\frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0}+\left( \frac{dv_{z}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}\right] +\frac{1}{\gamma ^{2}}\left[ \gamma \left( \frac{dv_{x}}{d\phi }\right)
_{1}-v_{x}\gamma ^{\prime }\right] \left( \frac{dv_{x}}{d\phi }\right)
_{1}\left( \frac{dv_{z}}{d\phi }\right) _{1}  \nonumber \\
&&-2\left( 1-\frac{v_{z}}{\gamma }\right) \left( \frac{dv_{z}}{d\phi }\right) _{1}\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0}+\frac{1}{\gamma
^{2}}\left[ \gamma \left( \frac{dv_{z}}{d\phi }\right) _{1}-v_{z}\gamma
^{\prime }\right] \left( \frac{dv_{x}}{d\phi }\right) _{1}^{2}  \label{62}\end{aligned}$$

where $Q^{\prime },\gamma ^{\prime }$ and $u^{\prime }$ are the derivatives of these quantities which can be easily obtained. With these expressions inserted in equations $\left( \ref{50}\right) $ and $\left( \ref{52}\right)
,$ we obtain the next order approximation for the acceleration. This process can be repeated to generate higher order corrections as desired.

The forward integration using this above procedure for the initial acceleration was performed using the subroutine Stiffs taken from the Numerical Recipes. The efficiency of the procedure is highly improved when this perturbative expression of the acceleration is recurrently used at each step of the integration process. The results of the numerical calculations are shown in Figs. 1 and 2, respectively, for the transversal and longitudinal components of the momentum. The parameters are $A=1$, $\epsilon
=0.05$ and $\Delta \phi =10$. These results are identical to those obtained in Ref. [@Hart] using the backward integration method.

In conclusion, it has been shown that LD equation can be integrated forward in time with an appropriate acceleration initial condition. This acceleration is provided by the equation of motion itself, treating the radiation damping term as a perturbation. This peculiar behaviour of LD equation stems from the fact that although the self-interaction term turns it into a third-order differential equation, only its particular solution is physically meaningful: all homogeneous solutions have to be excluded. This constraint makes the solution unique. We have discussed the forward numerical integration procedure in the important case of the nonlinear Compton scattering. The essential feature of the calculations is the crucial role played by the recurrent use of the perturbative series expansion of the acceleration in order to achieve stable and accurate results. We may conclude that the final Dirac-Rohrlich constraint can indeed be replaced by an equivalent initial condition as long as the radiation damping term can be treated as a local perturbation in the iterative solution of the equation of motion. This does not mean that nonlinear effects have been neglected. Indeed, by perturbatively iterating the equation of motion, a local expression for the acceleration is generated, in which the radiation affects the motion as much as it is affected by it. The results presented here clearly show that nonlinear LD equations can still be integrated [*forward*]{} in time if the appropriate, albeit unconventional, initial conditions were utilized.

Acknowledgments
===============

This work was partly done at Institute for Theoretical Atomic and Molecular Physics-Harvard(ITAMP), Institute for Theoretical Physics(ITP)-Santa Barbara and IFUSP-São Paulo. Partial support was supplied by the CNPq -Brazil, FAPESP-Brazil, The National Science Foundation under Grant No. PHY94-07194 (ITP). The ITAMP is supported by the National Science Foundation.

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 [**Figure Captions:**]{}

Fig. 1 The transversal component of the momentum vs. proper time, obtained with the forward integration method. See text for details.

Fig. 2 The same for the longitudinal component of the momentum.
