A first numerical investigation of a recent radiation reaction model and comparison to the Landau-Lifschitz model

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A ﬁrst numerical investigation of

a recent radiation reaction model and

comparison to the Landau-Lifschitz model

Christian Bild, Dirk - Andr´e Deckert, Hartmut Ruhl

(Dated: October 24, 2022)

In [1] we presented an explicit and non-perturbative derivation of the classical radiation reaction

force for a cut-oﬀ modelled by a special choice of tubes of ﬁnite radius around the charge trajectories.

In this paper, we provide a further, simpler and so-called reduced radiation reaction model together

with a systematic numerical comparison between both the respective radiation reaction forces and

the Landau-Lifschitz force as a reference. We explicitly construct the numerical ﬂow for the new

forces and present the numerical integrator used in the simulations, a Gauss-Legendre method

adapted for delay equations. For the comparison, we consider the cases of a constant electric ﬁeld,

a constant magnetic ﬁeld, and a plane wave. In all these cases, the deviations between the three

force laws are shown to be small. This excellent agreement is an argument for plausibility of both

new equations but also means that an experimental diﬀerentiation remains hard. Furthermore, we

discuss the eﬀect of the tube radius on the trajectories, which turns out to be small in the regarded

regimes. We conclude with a comparison of the numerical cost of the corresponding integrators and

ﬁnd that the integrator of the reduced radiation reaction to be numerically most and the integrator

of Landau-Lifschitz least eﬃcient.

CONTENTS

I. Introduction 1

A. Delayed-self force equations 2

B. A reduced DSR model for small 3

II. Numerical study 4

A. The diﬀerential ﬂows of the full DSR and

reduced DSR equations 4

B. The numerical integrator 5

C. Remark on the uniqueness of DSR solutions 7

D. The units for the simulations 8

E. Main Result I: Comparison of DSR, reduced

DSR, and LL dynamics 8

F. Main Result II: The -dependence of the

solution to the DSR equations 10

III. Summary and Outlook 11

IV. Acknowledgement 11

A. The Landau-Lifschitz equation 11

1. Tests of the numerical solver 12

2. Numerical eﬃciency 13

References 14

I. INTRODUCTION

In our previous work [1], and as many before us,

we have reviewed Dirac’s original derivation of the self-

reaction force, i.e., the Lorentz-Abraham-Dirac (LAD)

equation for a single charged particle interacting with its

own retarded Li´enard-Wiechert and an additional exter-

nal electromagnetic ﬁeld. To avoid the well-known diver-

gences in case of point charges, Dirac himself [2] imposed

a rigid extended charge model, parameterized by  > 0,

which can be shrunk to a point-like charge by means of

a limiting procedure →0. The energy-momentum con-

servation principle between the kinetic degrees of freedom

of the charge and the electromagnetic ﬁeld leads to

pα(τ2)−pα(τ1) (1)

=qZτ2

τ1

dτ Fαβ(z(τ)) + Fextαβ(z(τ))uβ(τ),

where pα(τ) denotes the four-momentum of the charge at

world-line parameter τ,qthe total charge, and Fαβ

,Fαβ

ext

the ﬁeld tensors of the retarded Li´enard-Wiechert ﬁeld

generated by the charge trajectory τ7→ zα(τ) and an

external one, respectively, while uβ(τ) denotes the four-

velocity. Based on (1), Dirac proposed a corresponding

force equation for the charge. The latter encompasses

the so-called self-force exerted on the charge by means

of its own Li´enard-Wiechert ﬁeld as well as the force due

to the external ﬁeld. The resulting expression of the cor-

responding self-force is, however, rather implicit and by

virtue of Stoke’s theorem involves a surface integral over

the tube V(τ1, τ2) in space-time (R4, ηαβ ), using the sig-

nature η= diagonal(+1,−1,−1,−1), spanned by the ge-

ometric extension of the charge along the curve segment

described by zα(τ) for τ∈[τ1, τ2], i.e.,

(1) = −Z∂V (τ1,τ2)

d3xβTαβ

(x) + Tαβ

ext (x).

Here, Tαβ

and Tαβ

ext denote the respective energy-

momentum tensors, and d3xβthe corresponding three-

dimensional 3d surface measure. In order to arrive at

a more tangible expression, Dirac formally carried out a

Taylor expansion in and, after subtraction of an inﬁ-

nite inertial mass term, arrived at the well-known LAD

arXiv:2210.11903v1 [physics.comp-ph] 21 Oct 2022

equation [2,3]:

∂τpα

(τ)−Fextαβuβ(τ)

3q2daα(τ)

dτ −uα(τ)daβ(τ)

dτ uβ(τ).

Dirac’s derivation is formal for two reasons: First, in the

limit →0, the ﬁeld evaluated on the charge trajec-

tory Fαβ

(z(τ)) is ill-deﬁned, and thus, Stoke’s theorem

cannot be applied. Second, the Taylor series in is not

controlled during the time integration but only at ﬁxed

instances of time. Our motivation in [1] to review Dirac’s

derivation was to avoid this Taylor expansion entirely

and, already for  > 0, arrive at a more tangible expres-

sion. There, we have shown in [1] that, at least for one

special choice of geometry of the extended charge, the

generated dynamics comprising the direct coupling be-

tween the Lorentz and Maxwell equations can be given

in form of an explicit delay-diﬀerential equation, hence-

forth referred to as delayed-self-reaction (DSR) equation:

∂τpα

(τ)−Fextαβuβ(τ)

=−q2

6 [(zγ−zγ(τ−)) uγ]2

uα−4uα(τ−)uβuβ(τ−)

×1−uδuδ(τ−)+(zρ−zρ(τ−)) aρ

−q2

6 (zγ−zγ(τ−)) uγ

4uα(τ−)aτ(τ−)uτ+uζ(τ−)aζ

+4aα(τ−)uϑ(τ−)uϑ−aα

+2q2

3aϕ(τ−)aϕ(τ−)uα(τ−) =: Lα

(τ).

(2)

In the following, we give a short review of these novel

equation of motion, introduce a further in an ad-hoc fash-

ion simpliﬁed version, referred to as the reduced DSR

model, and conduct a numerical study of these dynami-

cal systems while comparing to the well-known Landau-

Lifschitz (LL) model in settings relevant to phenomena of

radiation damping. As show in [3], the LL dynamics re-

sult from singular perturbation theory of the consistently

coupled Maxwell-Lorentz system in suﬃciently regular

external ﬁelds. In certain regimes, it is to be regarded as

a rigorous approximation of the self-force equation even

in the point-charge limit, and therefore, as a meaningful

reference.

A. Delayed-self force equations

For the sake of self-containedness, we shall give a brief

review the DSR dynamics. Figure 1illustrates the special

geometry of the extended charge on the tube employed

to derive the DSR equation for the segment zα(τ) of the

FIG. 1. The ﬁgure illustrates the tube around the charge tra-

jectory t7→ zα(τ) which makes up the charge model employed

to infer the DSR equation (9).

charge trajectory for τ∈[τ1, τ2]. For given parameter  >

0, the geometry of the extended charge at τis given by

the intersection ∂V (τ) of the forward light-cone located

at zα(τ−) with the equal-time hypersurface through

zα(τ), which is four-perpendicular to the velocity uα(τ).

The charge model is then deﬁned by requiring that the

support of the generated charge current density jα

(x)

equals the surface of the 3d set ∂V (τ) and the ﬁeld fulﬁlls

Fαβ

(x) = 0 for x∈V(τ)

Fαβ

LW[z](x) for x∈R4\V(τ),

where Fαβ

LW[z] is the retarded Lin´erd-Wiechert ﬁeld of a

point charge situated on the world-line τ7→ zα(τ). We

emphasize that the charge-current density is not nec-

essarily homogeneously distributed over the surface of

∂V (τ) but given by jα

(x) = ∂αFαβ

(x). Furthermore,

the charge trajectory does not necessarily intersect the

caps of ∂V (τ) in their centers. Both particularities are

owed to the special choice of the tube which, however,

allows to derive the algebraically explicit expression for

the DSR equation (9). Notably, and somehow expected,

for ﬁxed τ, and small the right-hand side of (9) contains

an inertial mass term as well as the LAD force

Lα

(τ) = −q2

2aα+2q2

3( ˙aα+aϕaϕuα) + O().(3)

The inertial mass term diverges as → ∞. Here, it has

to be emphasized that, even for  > 0, one would need to

renormalize the charge’s inertia. Even for smooth charge

models without divergences, coupling of the Maxwell and

Lorentz equations alone implies a change of the inertial

bare mass of the charge [3,4], which needs to be gauged

to the experimentally measured one.

For the DSR equation we proposed the following renor-

malization scheme in [1]. As the geometry of the tube,

recall Figure 1, is not static but dynamic, one may expect

some of the contributions to the inertia mto depend on

the parameter τ. Therefore, we start with the ansatz

pα(τ) = m(τ)uα(τ).

Exploiting uα(τ)aα(τ) = 0 for for world lines, we identify

dτ m(τ) = Lα

(τ)uα(τ).(4)

Setting

Fα

(τ) = Lα

(τ)−Lβ

(τ)uβ(τ)uα(τ),

Fα

ext(τ) = qF αβ(z(τ))uβ(τ)

results in relativistic four-forces, i.e., ones being four-

perpendicular to the four-velocity of the charge. Finally,

we ﬁnd the following system of equations

dτ 



zα(τ)

uα(τ)

m(τ)

=



uα(τ)

m(τ)(Fα

(τ) + Fα

ext(τ))

uα(τ)Lα

(τ)

,(5)

which we refer to as the DSR system of equations.

For  > 0, the DSR system (5) represents a neutral

system of delay diﬀerential equations, meaning that the

highest derivative of a solution component occurs with

and without delay. A general solution theory, and espe-

cially a study of the case →0, is still pending. Nev-

ertheless, candidates for initial conditions are certainly

twice continuously diﬀerentiable segments of the charge

trajectory

[−, 0] →R4, τ 7→ zα(τ)

together with an initial inertia m(0). The static divergent

inertia in (3) can be renormalized by setting the initial

inertial mass equal to

m(0) = m−q2

2,(6)

which at the same time sets the eﬀective initial inertia to

some experimental value m(0). Note however, that m(τ)

may still be both τ- as well as -dependent. Nevertheless,

there is numerical evidence that the corresponding solu-

tions of the DSR system depend only weakly on which

also raises the hope that a rigorous analysis of the limit

→0 can be done. While we think, this is the case, it

must be noted that the right-hand side of (6) becomes

negative for below the classical electron radius and, in

this case, renders the corresponding dynamics unstable;

as already seen in [4] for the coupled Maxwell-Lorentz

system of rigid charges.

B. A reduced DSR model for small 

In this section, the goal is to formulate a simpliﬁed, yet

still meaningful, version of the system (5), namely with

the force (8) below, from which we can expect that the

corresponding solutions sets are close in certain regimes

and -ranges. The main advantage of (8) is the much

simpler structure, which reduces the numerical cost sig-

niﬁcantly.

This simpliﬁed force can be motivated by regarding

only the following, in the limit →0 for ﬁxed τleading,

terms:

Lα

(τ)≈ − q2

2aα+2q2

3(aα(τ)−aα(τ−))

+2q2

3aγaγuα.(7)

Now we permit us to drop all high-order terms and even

remove the part of the force in uα-direction in order to

avoid a τ-dependend inertia coming from (4) which re-

sults in a constant m(τ) = m−q2

2. The lower two com-

ponents of (5) can now be combined into the equation

maα=Fα

ext +2q2

3aγ(τ−)uγuα

+2q2

aα(τ)−aα(τ−)

.

(8)

We will refer to the system of equations comprised by

the ﬁrst row of (5) and equation (8) together with the

above constant choice of m(τ) as the reduced DSR sys-

tem. Note the additional advantage that the right-hand

side of (8) is linear in the acceleration so that the corre-

sponding ﬂow can be given explicitly in contrast to the

one of (7). In (8), we will again assume  > 2q2

3m(19) to

avoid the discussed dynamical instability.

In view of our objectives in formulating the reduced

DSR system, the numerical cost evaluating (8) is now

considerably lower as compared to the full DSR model

(5), i.e., (17) below, or even the LL model, c.f., (A1)

below, as for the latter, the derivative of the ﬁeld strength

tensor, (A12) below, is needed in the multi-particle case.

A similar equation has been suggested in [5]. Equations

(8) still lead to a change of the inertia mas can be seen

in the case of uniform acceleration. Say, Fα

ext was tuned

such that it produces a solution with four-velocity taking

the form uα= (cosh(gτ),sinh(gτ),0,0), we ﬁnd

2q2

3aγ(τ−)uγuα−2q2

aα(τ−)−aα(τ)



=−4q2

3sinhg

22aα

similar to the change in inertia in the case of the DSR

system discussed in [1]. This illustrates also that m(τ)

may not simply be regarded as the total inertia of the

charge.

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ArstnumericalinvestigationofarecentradiationreactionmodelandcomparisontotheLandau-LifschitzmodelChristianBild,Dirk-AndreDeckert,HartmutRuhl(Dated:October24,2022)In[1]wepresentedanexplicitandnon-perturbativederivationoftheclassicalradiationreactionforceforacut-omodelledbyaspecialchoiceoftubesofni...

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A first numerical investigation of a recent radiation reaction model and comparison to the Landau-Lifschitz model

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