The force on a point charge source of the classical electromagnetic field
Michael K. -H. Kiessling
physics.class-ph gr-qc math-ph math.MP

	

	

miki@math.rutgers.edu
Department of Mathematics, Rutgers University,
                110 Frelinghuysen Rd., Piscataway, NJ 08854, USA
	

It is shown that a well-defined expression for the total electromagnetic force ^ on 
a point charge source of the classical electromagnetic field can be extracted from the postulate of total 
momentum conservation whenever the classical electromagnetic field theory satisfies a handful of regularity conditions. 
 Among these is the generic local integrability of the field momentum density over a neighborhood of the point charge.
 This disqualifies the textbook Maxwell–Lorentz
field equations, while the Maxwell–Bopp–Landé–Thomas–Podolsky 
field equations qualify, and presumably so do the Maxwell–Born–Infeld 
field equations. 
 Most importantly, when the usual relativistic relation between the velocity and the momentum of a point charge with 
“bare rest mass” ≠ 0 is postulated, Newton's law = with =^
becomes an integral equation for the point particle's acceleration; the infamous third-order time derivative of 
the position which plagues the Abraham–Lorentz–Dirac equation of motion does not show up. 
 No infinite bare mass renormalization is invoked, and no ad hoc averaging of fields over a neighborhood of the
point charge.
 The approach lays the rigorous microscopic foundations of classical electrodynamics with point charges.

	
 
  Preprint with erratum implemented (colored correction markings) April 21, 2020



Force on a point charge source of the classical electromagnetic field
    Michael K.-H. Kiessling
    July 24, 2019
=====================================================================







§ INTRODUCTION




 The practical success of the Lorentz formula <cit.> 

    ^_(t)
    = 
      -e[(t,(t)) + 1/c(t) (t,(t))]

for the electromagnetic force exerted by a given, smooth 
electric field  (t,) and magnetic induction field (t,) 
on a moving test point electron with charge -e, position  (t), and velocity  (t) is well established;
here, c is the speed of light in vacuum.
 However, this formula is notoriously ill-defined when the point electron is not idealized
as a “test particle” but treated properly as a source of the electromagnetic fields with which it interacts. 
 The Maxwell–Lorentz  equations for these fields, consisting of the two evolution equations

(t,)
   = 
        - c (t,)   ,


(t,)
   = 
        + c(t,)  + 4πe (t)δ_(t)()  ,


together with the two constraint equations 

        ·(t,)  
   = 
        0  ,


        ·(t,)  
   =
        -4 πe δ_(t)()  ,


make it plain that Maxwell–Lorentz (ML) fields with a single point charge source, at (t), must
have some singularity at =(t). 
 In the remainder of this introduction we first summarize the current state of affairs in dealing with this problem,
and then we recall the major deficiencies of this approach that were pointed out by others already.
 The rest of this paper is devoted to setting up a well-defined classical theory of point charge motion.
 
 We will occasionally invoke a manifestly co-variant geometrical four-vector notation, but mostly we will work with the 
space & time splitting well-suited for the formulation of a dynamical initial value problem we seek, 
even though the Lorentz co-variance is then obscured.
 Note that the above formulas are valid in any flat foliation of Minkowski spacetime into Euclidean space points 
∈^3 at time t∈.





                

 §.§ The current state of affairs




 The field singularity associated with a motion t↦(t) having bounded piecewise continuous acceleration
(t) has been known explicitly for a long time.
 Since the system of Maxwell–Lorentz field equations is linear, 
their general distributional solution can be written as the sum of the general Lipschitz continuous source-free 
electromagnetic field solution, for which eq:Lf makes perfect sense, 
plus the retarded Liénard–Wiechert field <cit.>, <cit.>, := & :=, with
(cf. <cit.>)

-.6truecm
(t,)    =
  -e c^2-||^2/|-|^2 
c(,)_!-/(c-(,)·)^3 
.|_ret-1truecm


     -e 
(,)[(c(,)_!-)]/|-|(c-(,)·)^3
.|_ret-1truecm


-1truecm
(t,) 
   = 
        (,)|__ret(t,)
  ,


where (,) = -/|-| is a normalized vector from  to ,
and where “|_ret” means that (,,)= (,,)(t^ret) 
with t^ret(t,) being defined implicitly by c(t-t^ret) = |-(t^ret)|;
here,  is the acceleration vector of the point charge.
 The electromagnetic Liénard–Wiechert fields  and  exhibit both
a ∝ 1/r^2 and a ∝ 1/r singularity, where r denotes |-(t)|; they each
have a directional singularity at the location of the  point charge source, too.

 In an attempt to give some mathematical meaning to the manifestly ill-defined symbolic expressions
“(t,(t))” and “(t,(t))” when (t,) and (t,) are a sum of a regular source-free
field and  the Liénard–Wiechert fields eq:LWsolE and eq:LWsolB, Lorentz and his contemporaries
averaged the fields (t,) and (t,) over a neighborhood of the point charge at (t), but this
does not lead to unambiguous finite vector values for “(t,(t))” and “(t,(t)),” and when the 
neighborhood is shrunk to the point (t) the infinities are back.
 The conclusion at the time (and also more recently in <cit.>) was that the physical electron cannot be assumed to be a point,
but must have an extended charge distribution and perhaps some other structure, all of which to determine became 
a goal of what Wiechert and Lorentz called “electron theory” <cit.>, nowadays referred to as “classical electron theory.” 
 It's an interesting dynamical theory in its own right; for more recent investigations, see the books <cit.> and 
<cit.>, and the papers <cit.> and <cit.>.
 Since we are interested in the theory of point charge motion, we here do not spend much time with classical electron 
theory, except that we note that some insights gained in its pursuit made their way into the prevailing classical theory 
of point electron motion which was put together by Dirac and Landau & Lifshitz.

 In 1938 Dirac <cit.> invented negative infinite bare mass renormalization to avoid the infinities 
which occur when the averaging surface about (t) is shrunk to (t). 
 Following Fermi's contribution to classical electron theory <cit.>, Dirac averaged the fields over a sphere of 
radius r centered at the electron in its instantaneous rest-frame.
 Also, he worked with linear combinations of the retarded and advanced representations of the fields.
 With m_ denoting the electron's “observable rest mass,”
Dirac assigned an averaging radius-dependent bare mass  (r) to the point electron, defined by

    m_ = lim_r↓ 0( (r) +e^2/2c^21/r);

evidently, (r)↓ -∞ as r↓ 0. 
  As is well-known, Dirac's mass-renormalization computations became the template for the
modern renormalization group approach to quantum electrodynamics and, more generally, quantum field theory.
 However, if electrons are true points without structure, then Dirac's construction is 
logically incomprehensible: if a point electron has a bare mass, then it cannot depend on the
radius r of a sphere over which one averages the Maxwell–Lorentz fields.

 Postponing such logical concerns until later in his life, Dirac obtained the Abraham–Lorentz–Dirac equation, which
in the four-vector notation explained in <cit.> reads

m_
^2/τ^2= 
    -e/c^()·/τ
    + 2e^2/3c^3 
(+1/c^2/τ⊗/τ)·^3/τ^3  

where 
the term in the first line at r.h.s.AbrLorDiracEQ
is an “externally generated” test-particle-type Lorentz Minkowski-force, and
the term in the second line at r.h.s.AbrLorDiracEQ
is von Laue's radiation-reaction Minkowski-force.

 While AbrLorDiracEQ is free of infinities if ^() is smooth, 
the third proper time derivative in the von Laue Minkowski-force 
means that AbrLorDiracEQ is a third-order ODE for the position of the particle as a function of (proper) time.
 The pertinent initial value problem therefore requires vector initial data for position, velocity, and acceleration. 
 Yet a classical initial value problem of point particle motion may only involve initial data for position and velocity.

 Landau & Lifshitz <cit.> handled this ⃛ problem in the
following perturbative manner. 
 They argued that von Laue's  ⃛ Minkowski-force term must be a small perturbation of ^
whenever test-particle theory works well. 
 In such situations one may compute ⃛ perturbatively by taking the proper time derivative of the test-particle 
equation of motion, i.e.

    ^3/τ^3≈ -e/m_c/τ(^()·/τ).

 R.h.s.dddq depends only on , , .
 If we substitute it for ^3/τ^3 at r.h.s.AbrLorDiracEQ, equation AbrLorDiracEQ becomes 






m_
^2/τ^2= 
 -e/c^()·/τ- 2e^3/3m_c^4 
(+1/c^2/τ⊗/τ)·/τ
(^()·/τ),


an implicit second-order ODE for the position of the electron, compatible with the available initial data.
 Equation LLeq is called the Eliezer–Ford–O'Connell equation in <cit.>.
 The equation presented by Landau–Lifshitz <cit.> differs from LLeq 
by an additional approximation: noting that
/τ(^()·/τ)
= 
(/τ^())·/τ
+
^()·^2/τ^2, they
substitute -e/ c^()·/τ
for ^2/τ^2 in the last term.


 As recently as in <cit.> the  Eliezer–Ford–O'Connell equation LLeq, resp. its Landau–Lifshitz 
approximation, was still presented as the state of affairs
in the classical theory of point charge motion in flat spacetime.
 These equations owe their longevity to their reputation as practically effective equations of motion for the 
computation of (first-order) radiation-reaction-corrected test-particle dynamics of point charges in smooth “external” fields.
 When point charges are replaced by extended charged particles the Landau–Lifshitz
equation can be derived rigorously from the Abraham–Lorentz model with nonzero bare mass, 
using center-manifold theory <cit.>, and also in a “vanishing-particle limit” <cit.>.
 Its solutions are expected to agree reasonably well with empirical electron motion in the 
“classical regime” of weak and slowly varying “external” fields,
and demands for higher precision can be met with improved effective equations, obtained either by adding higher-order classical
radiation-reaction correction terms, or by invoking QED.
 However, LLeq has major shortcomings even as an effective equation of motion for true point charges!





                

 §.§ Critique





 To convey a first feeling for the limitations of the Eliezer–Ford–O'Connell equation LLeq
and its Landau–Lifshitz approximation, we recall the well-known fact that the last term in LLeq vanishes 
for electron motion along a constant applied electric field, and so does its Landau–Lifshitz approximation; cf. <cit.>.
 Thus in this textbook situation LLeq fails to take the energy-momentum loss due to radiation 
by the electron into account, i.e. its solution is identical to the familiar test-particle motion. 
 To radiation-reaction-correct these requires a non-vanishing higher-order term.

 A much more serious limitation of the Eliezer–Ford–O'Connell and 
Landau–Lifshitz equations was discovered recently <cit.>,
by considering the many-body version. 
 In this case each point charge satisfies its own equation LLeq,
indexed by a subscript _k (say) at e_k, m_k, _k, and at
^_k, where ^_k is now the Faraday tensor of the 
Maxwell–Lorentz field given by a sum of the Liénard–Wiechert fields eq:LWsolE and eq:LWsolB
of all the other particles but the k-th, plus the source-free field. 
 Since nobody knows the past histories of all the particles which enter the 
Liénard–Wiechert formulas eq:LWsolE and eq:LWsolB, one has to stipulate some past motions. 
 But whatever one stipulates, as shown in <cit.>, typically a singularity in the Maxwell–Lorentz
fields will propagate along the initial forward light-cone of each and every point charge, so that this system of 
equations of motion coupled with the Maxwell–Lorentz field equations is typically well-defined only until a point charge 
meets the forward initial light cone of another point charge.
 This is much too short a time span to be relevant to, e.g., plasma physics. 
 This problem cannot be overcome perturbatively by adding a higher-order radiation-reaction correction
term at r.h.s.LLeq. 
 Moreover, it's not just the radiation-reaction term in LLeq which causes trouble — the expression of the
Lorentz force of one particle on another is typically not well-defined on the initial forward light-cones.
 Lorentz electrodynamics for N>1 point charges is in serious trouble!
 
 The above discussion leaves no room for reasonable doubts that ingenious extraction of effective equations of point charge motion 
from the mathematically ill-defined, merely symbolic “Lorentz electrodynamics of point charges,”
is not a winning strategy to arrive at a mathematically well-defined and physically accurate relativistic theory of
point charge motion in the classical realm.
 In the remainder of this paper we explain how to formulate such a theory in a manner which 
 preserves the spirit of Lorentz electrodynamics as much as possible.

 To keep matters as simple as possible we first consider an electrodynamical system featuring only a single point charge.
 The N-body situation will be discussed in section <ref>.
 The motion in a constant applied electric field is revisited in section <ref>.



 



                

§ BASIC DEFINITION OF THE ELECTROMAGNETIC FORCE






 Mechanically the point charge is a point particle with a mechanical momentum 

    (t)
    = (t)/√(1 -1/c^2|(t)|^2),

with (t):=(t) its velocity and ≠ 0 its bare rest mass. 
  
  
  By  Newton's 2^nd law the rate of change with time of the particle momentum equals the force acting on it, 

    (t)
    = (t).

 The force  depends on the non-kinematical qualities of the point particle, in this case its electric charge which 
couples the point particle to the electromagnetic field.

 Electrodynamically the moving point charge is a source / sink for a classical electromagnetic field with 
field momentum (vector-)density ^(t,). 
 Suppose the fields decay sufficiently rapidly as ||→∞ so that ^(t,) is integrable 
w.r.t. ^3s “at spatial infinity.” 
 Suppose also that the field singularity caused by the point charge is mild
enough so that ^(t,)  is locally integrable over any neighborhood of (t), so that
 
    ^ (t) 
    = ∫_^3^ (t,) ^3s

is a well-defined total field momentum vector. 
 Finally, suppose that the motion of the charge is sufficiently regular so that ^ (t) is differentiable
with respect to time.

 Now, following Poincaré (cf. <cit.>), we postulate that in the absence of non-electromagnetic forces 
only those motions are permissible which satisfy the balance law

    (t) 
    = 
    -  ^ (t);

i.e. any momentum gain by the particle is compensated through a corresponding momentum loss by the field, and
vice versa.
 It then follows from eq:dMOMbalance in concert with eq:NewtonSECONDlaw that 
the electrodynamical force on a point charge source of the classical electromagnetic field with a single source is to be defined by

    ^(t) 
    := 
    - ∫_^3^ (t,) ^3s.



 Postulating the balance law eq:dMOMbalance is equivalent to postulating conservation of total momentum,

    (t)
    = ,

for a total momentum defined as the sum of particle and field momenta, i.e.

    (t)
    : = (t) + ^ (t).



We pause for a moment to comment on eq:EMf in the context of Lorentz electrodynamics with point charges.





                

 §.§ Connection with Lorentz electrodynamics





 If we assume the electromagnetic fields satisfy the Maxwell–Lorentz field equations with a point charge source
eq:MLdotB & eq:MLdotE, and eq:MLdivB & eq:MLdivE, in which case
^ (t,) = 1/4π c, then r.h.s.eq:FIELDp is generally ill-defined
(generally ∞ in magnitude), and then r.h.s.eq:EMf has no well-defined meaning either. 
 However, pretending that r.h.s.eq:FIELDp was well-defined, and that so was
r.h.s.eq:EMf, and furthermore pretending that all the ensuing (advanced) multi-variable calculus and analysis
steps were justified, r.h.s.eq:EMf would turn precisely into the expression of the Lorentz force eq:Lf.
 For later convenience we recall those steps: 

(a) pull the time derivative into the integral;

(b) apply the Leibniz rule to get 
    () =() + ;


(c) now use eq:MLdotB to express the partial time derivative 

    of  in terms of , and eq:MLdotE to express the partial 

    time derivatives of  in terms of  and δ_;

(d) use an advanced vector calculus identity in concert 

    with eq:MLdivB & eq:MLdivE to write the so manipulated r.h.s.eq:EMf 

    as a volume integral over the sum of the divergence 

     of Maxwell's stress tensor,  plus the Lorentz force 

    vector density -e(+1/c)δ_;

(e) use Gauss' theorem to conclude that the contribu-

    tion from the stress tensor vanishes;

(f) carry out the volume integral of the force vector 

    density  and thus obtain eq:Lf.

Remarkably, not a single one of these six steps is generally justified within the symbolic system of equations known as
“Lorentz electrodynamics with point charges.”
 Nevertheless this “pseudo derivation” of  eq:Lf from  eq:EMf does suggest that by replacing eq:Lf with
eq:EMf one may accomplish what was intended by Lorentz and his contemporaries.

 As a first encouraging observation we register that there is a large set of field initial data satisfying the constraint equations 
eq:MLdivB & eq:MLdivE for which the formula eq:EMf of the electromagnetic force is initially 
well-defined — even for the Maxwell–Lorentz field theory! 
 In the special case of an electrostatic field of a point charge at rest, the electromagnetic force eq:EMf is 
actually well-defined for all times and consistently equals , which it should in this case; cf. section 3.4.
 This already demonstrates the superiority of the formula eq:EMf over 
Lorentz' eq:Lf, which is ill-defined even in this simplest non-test charge situation.

 Unfortunately, replacing Lorentz' formula eq:Lf with eq:EMf does not convert Lorentz electrodynamics
with point charges into a well-defined theory:
there is an even larger set of Maxwell–Lorentz field initial data satisfying the constraint equations 
eq:MLdivB & eq:MLdivE for which eq:EMf is not well-defined already at the initial time. 
 Also, even with favorable special Maxwell–Lorentz field initial data the electromagnetic initial force typically
cannot be continued into the future; i.e. typically the expression eq:EMf is not
well-defined for continuous stretches of time, the static special case being an exception. 
 Moreover, the initial energy density of an ML field with a point charge source is never locally integrable, i.e. 
for such data the field energy in any finite volume containing the point charge is infinite even if their field momentum 
eq:FIELDp exists.
 





	

§ THE ROLE OF THE ELECTROMAGNETIC VACUUM LAW






 In the following we will discuss eq:EMf for some electromagnetic field theories 
which (are expected to) yield a typically differentiable eq:FIELDp.
 Incidentally, eq:EMf was also the starting point of Abraham <cit.> and Lorentz <cit.> 
for computing the electromagnetic force on a charged particle in their classical theories with extended electron models;
cf. <cit.>.
 While Abraham and Lorentz and their peers chose to replace point electrons by extended structures but otherwise 
continued to work with the Maxwell–Lorentz field equations, we instead continue to work with point electrons but replace 
the Maxwell–Lorentz fields with solutions to the pre-metric Maxwell equations <cit.>  which satisfy different 
electromagnetic vacuum laws — which furnish locally integrable energy-momentum densities of fields with point charge sources.
 (“Pre-metric” means, the spacetime metric plays no role; see <cit.>. It enters through the vacuum law.)
 Since any classical electromagnetic field theory will be about some distinguished subset of solutions of 
the pre-metric Maxwell field equations, the issue is indeed to identify the physically correct classical electromagnetic 
vacuum law!
 We refrain from trying to make such a definitive identification but instead consider two well-known proposals: the nonlinear 
system proposed by Born & Infeld <cit.> (see also <cit.>), and the linear higher-order derivative system of
 Bopp <cit.>, Landé & Thomas <cit.>, and Podolsky <cit.> (see also <cit.>).





           

 §.§ The pre-metric Maxwell field equations




 The pre-metric Maxwell field equations are a four-dimensional (complex) analog of the familiar three-dimensional 
· = 0=.
 Explicitly, the continuity equation 

    ρ (t,) + ·(t,)
    =
    	0

for the charge density ρ and the current vector-density  is a four-dimensional analog of · =0. 
 It implies that ρ and  can be expressed as linear combination of first-order space and time derivatives of
two complex three-dimensional fields, +i and -i; viz. 
4πρ = · (+i) and 4π = c  (-i) -  ( +i).  
 The “4π” factor occurs for historical reasons, and the speed “c” at this point is just a conversion factor.
 Sorted into real and imaginary parts these are precisely the pre-metric Maxwell equations, which we write as one pair of 
homogeneous equations for  and , 

(t,) + c (t,) 
   = 
  , 

        ·(t,)  
   = 
        0  ,

and one pair of inhomogeneous equations for  and ,

-(t,) + c(t,)  
   =  
 4π(t,)  ,


        ·(t,)  
   =
        4 πρ(t,) . 


 Note that the constraint equations eq:MdivB & eq:MdivD only impose on the initial data (0,) and (0,) 
which need to be supplied when viewing (as we will do) eq:MdotB & eq:MdotD as initial value problems for  and ,
respectively. 
 To see this for eq:MdivB, take the divergence of eq:MdotB; for
eq:MdivD, take the divergence of eq:MdotD and the time derivative of eq:MdivD, 
and recall eq:MrhojLAW.

 The pre-metric Maxwell equations are familiar from Maxwell's theory of electromagnetic fields in 
material media, though here they are used for fields sourced by point charges in an otherwise empty space.





           

  §.§.§ Their general solution for point charge sources



 The pre-metric Maxwell equations are easily solved if
the charge density ρ(t,) = -e δ_(t)() 
and the current vector-density (t,)=  -eδ_(t)()(t),
provided that t↦(t) is continuously differentiable so that the continuity equation eq:MrhojLAW
is automatically satisfied in the sense of distributions, and provided that |(t)|<c.

 The pre-metric Maxwell equations in themselves can be viewed as two independent systems of linear first-order PDE 
with constant coefficients, a homogeneous system for the field pair (,), and an inhomogeneous system for the field 
pair (,).
 Their general distributional solutions are readily written down, in the inhomogeneous case
conditioned on the motions of the point charges being given.
 For later convenience we collect the general solutions here; it suffices to do this for when 
there is only a single point charge. 

  The homogeneous system is solved by a linear combination of 
first-order derivatives of a vector potential field (t,)∈^3 and a scalar potential field A(t,)∈, viz.

    (t,) = (t,),


    (t,) = -∇ A(t,) - (t,).

 Of course, this representation is found in every textbook on classical electrodynamics.
 
 Similarly we can handle the inhomogeneous equations. 
 Assuming the map t↦(t) to be continuously differentiable, with a bounded Lipschitz continuous derivative (t)
satisfying the speed limit ||<c, 
the general solution to the system eq:MdotD,  eq:MdivD then is
 
    (t,) =  (t,)  + (t,),


    (t,) =  (t,)  +  Z(t,) + (t,).

 Here,  and  are the Liénard–Wiechert fields eq:LWsolE and eq:LWsolB,
and the vector potential field (t,)∈^3 and a scalar potential field Z(t,)∈ generate
the general solution to the associated homogeneous system. 
 Note the sign difference between the homogeneous (,) system and the homogeneous system
associated with the (,) field pair.

 We remark that +i =  +  ( +i) and that -i =  + (Z+iA) +  (+i). 




           

  §.§.§ Gauge invariance



 As is well-known, the r.h.s.s of eq:BrepA, eq:ErepA are invariant under the gauge transformation

(t,)    ↦(t,) + Υ(t,) ,

A(t,)     ↦A(t,) - Υ(t,) .

 Similarly,  the r.h.s.s of eq:DrepZ, eq:HrepZ are invariant under the gauge transformation

(t,)    ↦(t,) + ℧(t,) , 
 
Z(t,)     ↦Z(t,) - ℧(t,) .


 The gauge transformations can be merged in complex notation: 
the addition of a four-dimensional “pseudo gradient” of a complex scalar ℧ +iΥ, i.e.
(-,)(℧+iΥ), to the complex four-dim. vector field (Z+iA, +i) does not change 
+i and -i.

 




           

 §.§ Electromagnetic vacuum laws




 The Maxwell–Lorentz field theory is concerned exclusively with those solutions of the
pre-metric Maxwell field equations whose imaginary and real parts (referring to the 
fields +i and -i) are related by

        (t,)  
   = 
        (t,)    ,


        (t,)  
   =
        (t,)   ,


which crosslinks the homogeneous with the inhomogeneous pair of equations.
 Equations eq:MlawBisH & eq:MlawEisD are known as Maxwell's 
law of the electromagnetic vacuum (“law of the pure ether” in Maxwell's words).
 As explained in the introduction, Maxwell's law of the electromagnetic vacuum selects solutions of the pre-metric
Maxwell field equations which are too singular to allow a well-defined coupling with the classical (relativistic or not)
theory of point particle motion. 
 But there are more suitable electromagnetic vacuum laws which express the real parts 
of +i and -i in terms of the imaginary parts. 

 As shown by Mie <cit.>, <cit.>, in a Lorentz co-variant electrodynamics the vacuum law follows
from a Lorentz-scalar Lagrangian (density) . 
 The notion of Lorentz invariance makes it obvious that the spacetime metric enters at this point. 
 In the orthodox version  depends only on the Lorentz invariants ||^2-||^2 and (·)^2, but
Lagrangians which in addition depend on the Lorentz invariant (·)^2 - |-1/c|^2
have also been considered in the literature (see below).
 The fields  and  are in either case obtained by functional differentiation from the action =∫^3st, 
viz. = δ_ and = - δ_. 
 If  depends only on the invariants  ||^2-||^2 and (·)^2, this is equivalent to conventional
partial differentiation of the Lagrangian density, viz. = ∂_ and = - ∂_.


 We next list the field Lagrangians and the implied electromagnetic vacuum laws for the ML, the MBI, and the MBLTP
field equations,  in historical order.


           

  §.§.§ Schwarzschild's field Lagrangian and Maxwell's vacuum law




 Schwarzschild's <cit.> Lagrangian, given by

 4π_
= 
12 (||^2-||^2),

yields  Maxwell's “law of the pure ether,” eq:MlawBisH & eq:MlawEisD, obeyed by
the Maxwell–Lorentz fields.
 We already reviewed the Maxwell–Lorentz field equations in the introduction. 

 Even though eq:SchwaL is not an admissible field Lagrangian for a classical electrodynamics 
with point charges, the success of the Maxwell–Lorentz field equations in the realm of weak-field phenomena 
(i.e. far away from the hypothetical point charge sources) and the low-frequency / long wavelength regime
(i.e. visible light, infra-red, radio waves and such) suggests that 
every admissible Lagrangian must reduce to it in the weak-field and low-frequency / long wavelength regime.


           

  §.§.§ The Born–Infeld field Lagrangian and vacuum law




 The Born–Infeld field Lagrangian <cit.>, given by

 4π_
= 
b^2 - √(b^4 - b^2(  ||^2-||^2) - (·)^2 )

yields the Born–Infeld (BI) law of the electromagnetic vacuum, 

           = 
- 1/b^2(·) /√(1-1/b^2(||^2-||^2) - 1/b^4(·)^2)   , 

           =
+ 1/b^2(·) /√(1-1/b^2(||^2-||^2) - 1/b^4(·)^2)  ,


expressing the pair (,) in terms of the pair (,).
 The parameter b is Born's field strength constant.
 In the limit b→∞ the BI law converges to Maxwell's law. 

 Since mathematically the pre-metric Maxwell field equations are quite
naturally interpreted as a pair of evolutionary equations eq:MdotB & eq:MdotD
for the fields  and , with initial data which are
constrained by eq:MdivB & eq:MdivD, it is desirable to rather express the field pair 
(,) in terms of the pair (,). 
 Happily eq:BIlawHofEB, eq:BIlawDofEB can be converted into

   =- 1/b^2×(×)/√(1+1/b^2(||^2+||^2) +1/b^4|×|^2) ,

 

   =- 1/b^2×(×)/√(1+1/b^2(||^2+||^2) +1/b^4|×|^2) .


  Complemented with eq:BIlawHofBD & eq:BIlawEofBD the pre-metric Maxwell field equations eq:MdotB & eq:MdotD
turn into the Maxwell–Born–Infeld (MBI) evolution equations for the fields  and , their initial data being constrained by
eq:MdivB & eq:MdivD. 

 In the absence of any sources the initial value problem for the MBI field equations is globally well-posed for 
classical initial data with sufficiently small energy <cit.>. 
 However, it is also known that certain smooth plain-wave data can lead to a singularity after a finite time, see
<cit.>.
 Speck in his thesis showed that this can be extended to finite energy data which coincide with such plane wave data on 
a sufficiently large bounded domain in space. 
 It is not known whether such blow-up in finite time happens for all finite-energy data in an open neighborhood 
of these finite-energy “local plane-wave” type data.

 For the MBI field equations with fixed point charge sources it has been shown <cit.> that a unique 
finite energy electrostatic solution exists which is real analytic except at the locations of the point charges; this
holds for any finite number N of point charges with arbitrary signs, magnitudes, and placements.
 However, it is not yet known whether the nonlinear BI law leads to an at least locally well-posed 
initial value problem for a physically interesting class of Maxwell–Born–Infeld fields with point charge sources.
 



           

  §.§.§ Bopp's field Lagrangian and vacuum law




 In the 1940s  Bopp, Landé & Thomas, and Podolsky argued that a more accessible linear vacuum law is available
if one is willing to admit higher-order derivative electromagnetic field equations.
 Bopp <cit.> obtained the equations from a Lagrangian given by

 4π_
= 
    12(||^2-||^2) 
     + 12 1ϰ^2 [(·)^2 - |-1c|^2]

which yields the Bopp–Landé–Thomas–Podolsky (BLTP) electromagnetic vacuum law


        (t,)  
   = 
       (1  + ϰ^-2□ ) (t,)   ,


        (t,) 
   =
        (1  + ϰ^-2□ ) (t,)   ;


here, □≡ c^-2∂_t^2 -Δ  is the classical wave operator.
 The parameter ϰ is “Bopp's reciprocal length” <cit.>; see <cit.> for empirical constraints on ϰ.
 The singular limit ϰ→∞ of the BLTP law yields Maxwell's law. 

 The pre-metric Maxwell field equations eq:MdotB & eq:MdotD and eq:MdivB & eq:MdivD,
when supplemented by the BLTP law of the vacuum eq:BLTPlawBandH & eq:BLTPlawEandD, become the
Maxwell–Bopp–Landé–Thomas–Podolsky (MBLTP) field equations. 
 Different from the Maxwell–Lorentz and Maxwell–Born–Infeld field equations, they are higher-order derivative field 
equations, requiring initial data not only for  and , but in addition also for  and ∂_t =:.

 We pause for another moment and comment on the asymmetrical role played by the pair of equations 
eq:BLTPlawBandH & eq:BLTPlawEandD, despite their symmetric appearance.
 When judged in their own right, 
eq:BLTPlawEandD is a second-order evolution equation for the electric field , given , and 
eq:BLTPlawBandH is a second-order evolution equation for , given . 
 However, since eq:BLTPlawBandH and eq:BLTPlawEandD are coupled with 
the pre-metric Maxwell evolution equations eq:MdotB & eq:MdotD for the
fields  and  (constrained by eq:MdivB & eq:MdivD), appearances are misleading
in the case of eq:BLTPlawBandH.
 A well-defined initial value problem for the fields is obtained only if eq:MdotB & eq:MdotD 
and eq:BLTPlawEandD are treated as genuine evolution equations for the fields , , and , while 
eq:BLTPlawBandH is not treated as an evolution equation for  — against all appearances.

 Indeed, given the field initial data (0,), also (0,) is fixed initially, so eq:MdotB 
yields (0,).
 And given the field initial data (0,), also (0,) is  fixed initially, so 
the time derivative of eq:MdotB yields (0,). 
 And with the initial data (0,) given, also Δ(0,) is fixed, and then 
r.h.s.eq:BLTPlawBandH is completely determined initially.  
  Thus eq:BLTPlawBandH defines  initially in terms of  and its second partial derivatives; 
note that this also implies that ·=0. 
 And then, with  so defined initially, and the particle's initial position and velocity given,
eq:MdotD now yields (0,). 
 Lastly, with the initial field data (, )(0,), and (0,) given, 
(0,) is initially determined by eq:BLTPlawEandD.

 This scheme now propagates in time, i.e. eq:BLTPlawBandH remains the defining equation for  
in terms of  and its second partial derivatives, while eq:MdotB,  eq:MdotD, and eq:BLTPlawEandD
are genuine evolution equations for , , and . 
 In <cit.> it is shown that MBLTP field initial data 
(, , , )(0,) launch a unique global distributional solution of the MBLTP field equations,
conditioned on the motions being given.

 There is a small variation on this theme, which takes advantage of the convenience of having the general distributional
solution of the pre-metric Maxwell field equations for the pair (,) available with eq:DrepZ & eq:HrepZ.
 Thus, prescribing the motion t↦(t) for t≤ 0 conveniently, though twice
continuously differentiable with subluminal velocity (t), and choosing smooth and spatially rapidly decaying fields
Z(0,), (0,), and ()(0,), such that ·=0, equations eq:DrepZ & eq:HrepZ 
fix (0,) and (0,). 
 Prescribing also (, )(0,) fixes (0,) and (0,) initially, so
eq:MdotB and its time derivative yield ()(0,) and ()(0,). 
 Thus, eq:BLTPlawBandH, while still not an evolution equation for , is now an elliptic vector Helmholtz equation for (0,),
which has a unique solution that vanishes at spatial infinity, thus determining the initial (0,) completely.

 We next collect the pertinent formulas for the field momentum vector-densities, then show that there are 
field initial data, satisfying the Maxwell constraint equations, for which eq:EMf is initially well-defined.
 Lastly we address the electrodynamical admissibility of the vacuum laws. 




	

 §.§ The electromagnetic field energy-momentum density




 A Lorentz invariant field Lagrangian also determines the field energy-momentum density.
 For the Maxwell–Lorentz (ML) field theory, the field energy density ^ and 
field momentum vector-density ^ are of course well known and 
given by

    4π^
    = 12(||^2 +||^2)  ,


    4π c ^
    =  .

  Recall that = in Maxwell–Lorentz field theory.

 For the Maxwell–Born–Infeld (MBI) field theory, the field energy density ^ and 
field momentum vector-density ^ are given by

    4π^
    = √(b^4+b^2(||^2+||^2) + |×|^2) -b^2  ,


    4π c ^
    =  .


 For the Maxwell–Bopp–Landé–Thomas–Podolsky (MBLTP) field theory, the field energy density ^ and 
field momentum vector-density ^ are given by

4π^
= 
     ·+  ·- 12(||^2 +||^2) 
     -  12ϰ^2  [(∇·)^2 + |∇- 1/c|^2 ] ,


4πc ^
= 
     + - 
     -  1ϰ^2  (∇·)(∇- 1/c).




	

 §.§ Field data yielding an electromagnetic force initially




  Consider first the ML and MBI field theories. 
 Both operate with the same formula for the field momentum density, eq:PiML respectively eq:PiMBI. 
 Initial data for the fields  and  compatible with the constraint equations eq:MdivB & eq:MdivD 
for which eq:FIELDp is initially well-defined are easily obtained as follows.

 Set (0,) = e1/|-(0)| -  Z(0,) -()(0,) and (0,) = (0,),
with Z(0,) and ()(0,) and (0,) smooth and rapidly decaying at spatial infinity together with
their derivatives. 
 Then ()(0,) is integrable over ^3, i.e. ^ (0) exists.
 
 As to the derivative of ^ (t) at t=0, consider first the ML field theoy. 
 With the above choice of initial data, also (0,) is smooth and so is (0,).
 Given in addition the assumed decay at spatial ∞, 
steps (a), (b), and (c) from section II.A are now justified to manipulate eq:EMf and yield


    ^(0) = ∫_^3[ +  +4π1c](0,) ^3s,


which is well-defined.
 Note though that (ρ)(0,) is not well-defined, so that one cannot apply steps (d) and (e)
of section II.A and arrive at the Lorentz formula for ^(0).

 Consider next the MBI field theory.
 If we formally carry out steps (a), (b), and the analog of (c) of section II.A, we get


    ^(0) = ∫_^3[ +  +4π1c](0,) ^3s,


which may or may not be well-defined, depending on (0,).
 Clearly the  term is the same as in the ML setup, pairing a δ distribution with a smooth test function.
 The  term is also well-defined, and integrable, because the BI law eq:BIlawEofBD guarantees 
(i) that || is uniformly bounded whenever || is, and (ii) that for our initial data (0,) is differentiable 
everywhere except at =(0), having uniformly bounded partial derivatives which (iii) decay not slower than 1/r^2 
at spatial infinity.
 However, unless (0,) vanishes at =(0), 
eq:BIlawHofBD yields that (0,) has a 1/r^2 singularity at =(0), 
and it is easy to check that its curl 
will then generally blow up like 1/r^3, which is too strongly.
 If (0,) vanishes like |-(0)| or faster at =(0), then
the  term is well-defined, and integrable.
 Note that eq:Lf is still not well-defined even
with such idealized initial data because (0,) is not well-definable at =(0).

 Since typically the ()(0,) term is not absolutely integrable over a 
neighborhood of =(0), one typically is not allowed to carry out the manipulations by means 
of which one arrives at eq:EMfMBIinitA.
 Nevertheless, since eq:PiMBI has at most 1/r^2 singularities, 
 eq:EMf may still be well-defined for MBI fields with a regularly moving point charge.
 This is yet to be sorted out, though.

 Lastly, as to the MBLTP field theory, it is not necessary to separately discuss the possibility of an initially well-defined 
force because below we will report that we actually were able to obtain a generically well-defined force which persists into 
the future. 





	

 §.§ Electrodynamic admissibility of the vacuum laws




 The examples discussed in the previous subsection demonstrate that the definition eq:EMf can yield a 
well-defined expression for the electromagnetic force on a point charge source of certain classical electromagnetic fields,
whereas the Lorentz formula eq:Lf fails to be well-defined.
 This is encouraging but does not suffice to accomplish our goal, which is to show that a generically well-defined, in fact 
well-posed and physically interesting classical electrodynamics with point charges is possible.
 Our next step thus is to inquire into field evolutions for which  eq:EMf remains well-defined over time.


	

  §.§.§ The ML vacuum law is electrodynamically inadmissible



 For the sake of completeness, we note that the electrostatic special case
leads to a consistent proper solution of Lorentz electrodynamics in which the Maxwell–Lorentz field equations are coupled
with eq:EMf instead of eq:Lf.
 Indeed, assuming that (t)=(0) and (t)= for all time, field initial data 
(0,)= e1/|-(0)| and (0,) = propagate in time unchanged, viz. 
(t,)= (0,) and (t,) = for all time. 
 But then ()(t,) = for all time, and so  by eq:EMf also ^(t)= for all time, 
consistent with (t)= and (t)=(0) for all time. 

 Unfortunately this very special situation does not have an open dynamical neighborhood in Lorentz electrodynamics even when
eq:Lf is replaced by eq:EMf.
 Suppose we perturb the static field data (0,)= e1/|-(0)| and (0,) = 
by replacing the vanishing magnetic initial induction by a smooth and compactly supported (0,) = (0,) 
such that  (0,)(0,) is integrable over ^3. 
 Even leaving the particle initial data as before, by eq:EMfMLinit the point charge will now feel a nonzero
initial force and begin to move. 
 But then by the Maxwell–Lorentz field equations the magnetic induction will not
evolve into a smooth (t,); a 1/r^2 singularity is formed after an arbitrarily short time span after the 
initial instant, so  is only integrable at t=0, and the force  eq:EMf does not remain
well-defined. 
 This disqualifies the Maxwell–Lorentz field equations.


	

  §.§.§ The BI vacuum law may be electrodynamically admissible



 Everything we wrote about the electrostatic special case in Lorentz electrodynamics, with eq:Lf replaced by
eq:EMf, carries over to Born–Infeld electrodynamics. 
 However, this time there may be an open neighborhood of initial data for which 
the ensuing MBI field evolution with a large open set of assumed motions of their point source leads to a well-defined
eq:EMf as time goes on.
  To be sure, no such result has been proven rigorously yet, but 
the author is optimistic that the BI law is admissible, i.e. that an open set of solutions to the MBI field equations 
with point charge sources has integrable field momentum densities, and that the total field momentum is differentiable 
in time — for not too long time intervals. 


	

  §.§.§ The BLTP vacuum law is electrodynamically admissible



 In <cit.> we establish the electrodynamic admissibility of the BLTP law, which here we summarize for
the simplified case of fields with a single point charge source. 
 We begin with a definition.

 Electrodynamically admissible initial data are of the following form: (, , , )(0,)
=∑_j=0^1(_j, _j, _j, _j)(0,). 
 Here, (_0, _0, _0, _0)(0,) is the t=0 evaluation of a C^0,1 finite-energy source-free 
MBLTP solution which is globally bounded by (ℬ_0, 𝒟_0, ℰ_0, ℰ̇_0), 
having global Lipschitz constants L__0 and L__0.
 Moreover, (_1, _1, _1, _1)(0,) is the “co-moving electromagnetic field”  at t=0 of a 
fictitious point charge whose world line coincides with the tangent world line of the actual point charge at t=0.




 Replacing the “co-moving electromagnetic field,” of a fictitious point charge whose world line 
coincides with the tangent world line of the actual point charge at t=0, by the retarded Liénard–Wiechert-type 
fields of a fictitious point charge whose subluminal C^1,1 world line merely is tangent to the world line of the 
actual point charge at t=0, does not result in more general initial data, then, because the difference 
of two such fields at time t=0 is regular enough and satisfies the source-free MBLTP field equations at t=0. 



 The proof in  <cit.> of the electrodynamical admissibility of the BLTP law of the electromagnetic vacuum 
consists in showing that the electrodynamically admissible initial data stipulated above launch field evolutions for 
which the force eq:EMf is well-defined for an open set of physically acceptable motions. 
 This proof is greatly facilitated by the fact that the electromagnetic force in the BLTP vacuum can be computed 
explicitly; the details are given in <cit.>.
 For the convenience of the reader the result of this computation is summarized next; 
it has also been announced in a conference proceedings, see <cit.>.




	

§ THE ELECTROMAGNETIC FORCE IN THE BLTP VACUUM





 By the linearity of the MBLTP field equations the solution launched by the above stated type of initial data
decomposes into the pertinent sum of a vacuum field plus a Liénard–Wiechert(-type) field.
 The vacuum field need not be represented explicitly for it suffices to know that it has the required regularity.
 The field solutions _1(t,) and _1(t,) for t> 0 are for a.e. |-(t)|>0
given by the Liénard–Wiechert fields eq:LWsolE and eq:LWsolB, 
-.6truecm

_1(t,)    = (t,)  ,


-1truecm
_1(t,) 
   =  (t,)
  .


 The MBLTP field solutions _1(t,) and _1(t,) for t≥ 0 are given by (cf. <cit.>)



_1(t,) =   
   
 -e^ϰ^212(^,)-^/c1-(^,)·/c
|_ret
  + 
e^ ϰ^2 ∫_-∞^t^ret(t,)
J_2(ϰ√(c^2(t-t')^2-|-(t')|^2))c^2(t-t')^2-|-(t')|^2^n 
c (-^(t') - ^(t')(t-t'))t', 
 

_1(t,) =  
    -e^ ϰ^2 12 
^(^,)/c1-(^,)·/c
|_ret
  + 
 e^ ϰ^2 ∫_-∞^t^ret(t,)
J_2(ϰ√(c^2(t-t')^2-|-(t')|^2 ))c^2(t-t')^2-|-(t')|^2^n 
^(t')(-^(t') 
)t' ;


note that the time integrations from -∞ to 0 here do not involve some unknown past motion but only the auxiliary straight-line 
motion which encodes the Lorentz-boosted electrostatic MBLTP field of the point charge, to which expressions eq:EjsolMBLTP
and eq:BjsolMBLTP reduce at t=0.

 With the help of these solution formulas the electromagnetic force of the MBLTP field on its point charge source
can be computed as follows. 
 Since each electromagnetic field component is the sum of a vacuum field and a sourced field, the bilinear ^
decomposes into a sum of three types of terms, the vacuum-vacuum terms, the source-source terms, and the mixed vacuum-source terms.
 The vacuum-vacuum contribution to r.h.s.eq:EMf vanishes because the total momentum of a vacuum field is conserved;
 the vacuum-source contribution to r.h.s.eq:EMf yields the force on the point source due to the vacuum field;
 lastly, the source-source contribution to r.h.s.eq:EMf is a “self”-field force in BLTP electrodynamics. 
 Thus, eq:EMf is given by

    ^(t) 
    =
    ^[,](t)
    +
    ^[,;](t)
 
where

    ^[,](t) =
    - e [ _0(t,(t))+ 1c(t)×_0(t,(t))  ]
 
is the Lorentz force eq:Lf evaluated with a vacuum field (i.e. a “test particle contribution” 
to the total force), and



^[,;](t)
  =  - /t ∫_B_ct(_0)
( ^_(t,) - 
^_(0,-_0-_0t)) d^3s  


with ^_
given by eq:PiMBLTP with (_1,_1,_1,_1) in place of (,,,).

 We re-emphasize that there is no such thing as the self-field force in electrodynamics; only the total force, i.e. 
the sum at r.h.s.eq:totalF, has an absolute meaning (in the chosen Lorentz frame).
 In particular, our “self”-field force does generally not agree with the expression studied in <cit.> and <cit.>, 
which depends on the complete actual past motion of the point particle and cannot be used to study its initial
value problem in which only the particle's initial position and velocity are prescribed (given the initial fields).


 The “self”-field force can be evaluated using retarded spherical coordinates (r,ϑ,φ) to carry out the
^3s integrations over the ball B_ct(_0), after which one can differentiate w.r.t. t.
 This yields



^[,;](t)
   =  e^2/4π [ .
 - 𝐙_ξ^[2](t,t) + 𝐙_ξ^∘^[2](t,t)   

           -  ∑_0≤k≤1 c^2-k(2-k)
∫_0^t 
[𝐙_ξ^[k](t,t^r)
-
𝐙_ξ^∘^[k](t, t^r)]
(t- t^)^1-k t^ 

           -   ∑_0≤k≤2 c^2-k
∫_0^t 
[𝐙_ξ^[k](t,t^r)
-
𝐙_ξ^∘^[k](t, t^r)]
(t- t^r)^2-k t^r  . ],

where ξ(t) ≡ (,,)(t) and 
ξ^∘(t) ≡ (_0+_0t,_0,0), where 
𝐙_ξ^[2](t,t) :=lim_t^r→ t𝐙_ξ^[k](t,t^r),
and where

𝐙_ξ^[k](t,t^r) = 
 ∫_0^2π ∫_0^π
(1-1c^(t^r)cosϑ)
 π_ξ^[k](t,(t^r) + c(t-t^r)) 
sinϑϑφ ,
 
with  
=(sinϑcosφ, sinϑsinφ , cosϑ)
a normal vector to the retarded sphere of radius r=c(t-t^r).
 Also, setting

K_ξ(t',t,)     :=
J_1(ϰ√(c^2(t-t')^2-|-(t')|^2 ))√(c^2(t-t')^2-|-(t')|^2)^n,

𝐊_ξ(t',t,)     := 
J_2(ϰ√(c^2(t-t')^2-|-(t')|^2 ))c^2(t-t')^2-|-(t')|^2^n 
 (-(t')- (t')(t-t')), 

the π_ξ^[k](t,) with k∈{0,1,2} read, for ≠,


 π_ξ^[0](t,) =
    - ϰ^4 1/4[

((,) -1/c)(1/c(,) )/
      (1-1/c ·(,))^2 
           ]_ret
    + ϰ^41/2[

 (,)
-1/c/ 1-1/c ·(,) 
             ]_ret 

∫_-∞^t^ret_ξ(t,)
(t')𝐊_ξ(t',t,)t' 

     - ϰ^41/2[  1/c(,) /
      1-1/c ·(,) ]_ret 
∫_-∞^t^ret_ξ(t,)
 c𝐊_ξ(t',t,)t' 

     - ϰ^4 ∫_-∞^t^ret_ξ(t,) 
 c𝐊_ξ(t',t,)t' ∫_-∞^t^ret_ξ(t,) 
(t')𝐊_ξ(t',t,)t' 

     - ϰ^4 c∫_-∞^t^ret_ξ(t,)  K_ξ(t',t,)t' 
 ∫_-∞^t^ret_ξ(t,)  K_ξ(t',t,) (t')t' ,



 π_ξ^[1](t,) = 
    - ϰ^2 
[


(,)((,)[ ((,)-1/c) ])·1/c/
       c^2 (1-1/c ·(,))^4 
     +
(,)  ((,)-1/c) /
      2c^2 (1-1/c ·(,))^3 
]_ret
    - ϰ^2[

(,) ((,)-1/c) /
      c^2 (1-1/c ·(,))^3 
]_ret 

∫_-∞^t^ret_ξ(t,)
(t')𝐊_ξ(t',t,)t' 

     + ϰ^2[(,)[(,)((,)_!-1/c) /
      c^2(1-1/c ·(,))^3 
]]_ret  
 ∫_-∞^t^ret_ξ(t,) 
 c𝐊_ξ(t',t,)t' 

     +ϰ^3 
 [1/1-1/c ·(,) ]_ret 
 ∫_-∞^t^ret_ξ(t,)
 K_ξ(t',t,)[(t^ret_ξ(t,)))+(t')]t' ,
 


π_ξ^[2](t,s) =     - ϰ^2
[1/(1-1/c ·(,))^2 1/c
 - [1-1c^2||^2]
 ((,)-1/c) (1/c(,)) /
      (1-1/c ·(,))^4 
]_ret 

      +ϰ^2 [[1-1c^2||^2](,)
 (,)_!-1/c /
      (1-1/c ·(,))^3 
]_ret ∫_-∞^t^ret_ξ(t,) 
 c𝐊_ξ(t',t,)t' 

    -  ϰ^2[[1-1c^2||^2]

 (,)_!-1/c /
      (1-1/c ·(,))^3 
]_ret 

∫_-∞^t^ret_ξ(t,) 
(t')𝐊_ξ(t',t,)t' ,

and |_ret means that (t̃), (t̃), (t̃) 
are evaluated at t̃ = t^ret_ξ(t,).
 Note that r.h.s.eq:selfFexpl vanishes at t=0.


 Several of the spherical angular integrations can be carried out explicitly in terms of
well-known functions; see <cit.>. 
 Here we are content with the remark that for C^1,1 motions t↦(t) (which for t≤ 0 
coincide with the auxiliary straight line motion t↦(0) + (0)t employed to define the field initial data,
and should not be confused with any actual motion for t<0),
one can easily show that all the terms in ^[,;](t) are well-defined. 
 Since eq:vacuumF is also well-defined for the regular vacuum field, the electrodynamical
admissibility of the BLTP vacuum law follows.





	

§ BLTP ELECTRODYNAMICS HAS NO ⃛-PROBLEM











 Having established that the BLTP law is electrodynamically admissible, in the sense that the solutions
of the MBLTP field equations for prescribed C^1,1 motions t↦(t) 
yield a field momentum vector density which is integrable over space, and its integral differentiable in time,
we now explain that the so obtained electromagnetic force eq:totalF, with 
^[,](t) given by eq:vacuumF and
^[,;](t) by eq:selfFexplff, 
when substituted at r.h.s.eq:NewtonSECONDlaw, yields a well-posed initial value problem for point charge motion, 
and thus a well-posed classical BLTP electrodynamics with a point charge. 
 It is understood here that the relativistic velocity-momentum relation eq:MOMinTERMSofVELO is assumed,
 with ≠ 0.
 This well-posedness result is a special case of a result
proved in <cit.> for an arbitrary finite number of point charges.
 We here explain the main idea of the proof for fields having a single point charge source. 

 A key feature of the total electrodynamical force in a BLTP vacuum, 
^[,] + ^[,;], as indicated by our notation,
is the dependence on only ,,; a third- (and higher-)order time derivative of (t) does not show up. 
 Therefore, already purely formally, our initial value problem for the point charge motion is of second order, as desired.

 Next, inserting eq:MOMinTERMSofVELO at l.h.s.eq:NewtonSECONDlaw and carrying out the differentiation 
we obtain a familiar expression which can be rewritten as a regular matrix acting on the vector ,
the matrix depending only on  (and ), not on  and not on . 
 Applying the inverse of this matrix at both sides of eq:NewtonSECONDlaw (with the force in a BLTP vacuum in place),
our equation of motion becomes

    = 
     W[]·( 
    ^[,]
    +
    ^[,;])

where the velocity  is expressed in terms of the momentum  by inverting eq:MOMinTERMSofVELO, i.e.

    = 1//√(1 +||^2/^2 c^2 );  ≠ 0,

and  where  

    W[]
    := 
    1/1/√(1 +||^2/^2 c^2 )[ - ⊗/^2c^2 + ||^2].

 Together with the usual definitions (t):=(t) and (t):=(t) this is a complicated, nonlinear 
and implicit, second-order differential-integral equation of motion for the position of the point charge.

 We approach the problem stepwise. 
 Temporarily ignoring the relationships (t)=(t) and (t)=(t), and instead treating the
maps t↦(t) and t↦(t) and t↦(t) as a-priori unrelated for t>0 (of course,
for t≤ 0 the maps are determined by the stipulated auxiliary motion), we note that then
the acceleration t↦(t) enters the BLTP “self” force term 
^[,;] only in a linear fashion. 
 Therefore, when treating t↦(t) and t↦(t) as given, eq:Volterra becomes 
a linear integral equation for the acceleration t↦(t). 
 Better yet, inspection shows that it is a linear Volterra integral equation for which we 
prove the following key result, see <cit.>.

 Given C^0,1 maps t↦(t) and t↦(t), with Lip()=v, Lip()=a, and
|(t)|≤ v<c, which for t≤ 0 coincide with the stipulated unaccelerated auxiliary motion,
the Volterra equation as a fixed point map has a unique C^0 solution 
t↦(t) = α[( · ),( · )](t). 
 Moreover, the solution depends Lip­schitz continuously on the maps
t↦(t) and t↦(t). 

 The careful proof in <cit.> fills many pages.

 Essentially as a corollary of the above Proposition we obtain the well-posedness result for the joint initial value problem 
of the MBLTP field and its point charge source.
 Indeed, now substituting α[( · ),( · )](t) for (t) in 
^[,;], Newton's equation of motion, supplemented with the 
relativistic velocity-momentum relation ELPinverse and the definition (t)=(t) of the velocity,
can be formally integrated to become a fixed point map for
a curve in the phase space of the point charge, viz. for some T>0 and t∈[0,T),


(t)  = 
(0)     + ∫_0^t 1/^   (t')/√(1+ |(t')|^2/^2c^2)t',


(t)= 
(0)     + ∫_0^t  ^[,](t') t' 
     + ∫_0^t ^[,;α[( · ),( · )]](t') t',


where  at r.h.s.eq:iteraP is given in terms of  through ELPinverse.
 The following theorem is a special case of the N-body result  proved in <cit.>.

 Given (0) and (0) and the stipulated MBLTP field initial data, there is a T>0 such that 
the fixed point problem eq:iteraQ & eq:iteraP has a unique C^1,1 solution t↦(,)(t)
for t∈(0,T) extending continuously to [0,T).
 Moreover, if in a finite time the particle does not reach the speed of light or infinite acceleration, then T=∞.
 In any event, total energy-momentum conservation holds.


 When ^[,] is replaced by an external, smooth and short-ranged force field,
the dynamics is global, i.e. T=∞. 
 This was shown in <cit.>.


 For the BLTP electrodynamics it is easy to generalize the single-particle formulation to the N-body formulation.
 By the linearity of the MBLTP field equations we can associate each particle with its own Liénard–Wiechert(-type) 
 MBLTP field, and thus a “self”-field force, same expressions as before except that the index _1 is replaced by _j, 
say, also to be attached to position, velocity, momentum, and acceleration vectors, and to the charge and mass parameters.
 The total MBLTP field is the sum of a vacuum field of the kind considered above plus all the Liénard–Wiechert(-type) 
 fields.
 The total force now gets an extra contribution in form of the Lorentz force eq:Lf on particle _j exerted 
by the Liénard–Wiechert-type fields of all the particles but _j.
 This requires extra regularity estimates for all _n and _n, n∈{1,...,N}, 
which are established in <cit.>.
 In addition to the avoidance of the possible finite-time blow-up scenarios for the single particle dynamics, 
global well-posedness now also requires that in finite time  no two particles reach the same location.
 While this can conceivably happen, it is not known whether it will happen generically or not.



  Above we noted already that our “self”-field force r.h.s.eq:selfFexpl vanishes at the initial instant t=0.
  Thus in a BLTP vacuum the inertia of the particle is initially entirely due to its bare mass! 


 To summarize, the work of <cit.> on BLTP electrodynamics demonstrates that it is feasible to set up a
classical electrodynamics with point charge sources as a well-posed joint initial value problem for the fields and the
particles, which is of second order in the particle positions.
 No Landau–Lifshitz type approximation has been invoked because the infamous ⃛-problem does not show up;
no negative infinite bare mass renormalization <cit.>, no additional comparison axioms <cit.> (cf. also
<cit.>), and no “separating off of singularities” <cit.> are invoked. 

 Whether BLTP electrodynamics is already a physically acceptable classical theory is a different question.
 Particularly embarrassing is the fact that the triumph of avoiding unphysical third- (and higher-)order time derivatives
of the point charge's position (t) in its force law is paid for by a high prize, namely by introducing (presumably) unphysical 
higher-order time derivatives in the electromagnetic field equations! 
 As a consequence, four MBLTP fields, namely ,,,, require initial data.
 However, according to phenomenological electromagnetism, once the fields 
(0,) and (0,) are determined / prescribed initially (constrained by (<ref>) and (<ref>)),[In 
   the physics literature on classical Lorentz electrodynamics, one usually finds (0,) and (0,) prescribed, but recall that 
   = (and =) in Lorentz electrodynamics.] 
one does not have any freedom left to also choose _0(0,) and _0(0,), yet
the MBLTP field equations do require such a choice.
 (Incidentally, neither the founding fathers of the BLTP theory, nor Feynman <cit.>, nor recent authors
<cit.>, <cit.>, seem to have been worried about these “loose ends” of this field theory.)
 Be that as it may, it is generally agreed upon, and mathematically realized in the structures of
the Maxwell–Lorentz field equations and of the Maxwell–Born–Infeld field equations,
that once the fields (0,) and (0,) are prescribed, the initial field data are fixed. 
 Therefore, to implement this rule also into the MBLTP field theory one needs a prescription which expresses 
the data (0,) and (0,) in terms of (0,) and (0,). 

 In <cit.> we show that this can be accomplished by
postulating that the fields _0(0,) and _0(0,) maximize the field energy functional initially, given 
_0(0,), _0(0,), and given the Liénard–Wiechert(-type) fields associated with the data (0), (0).
 Thus the initial field energy is made as little negative as possible.
 One can make this constraint Lorentz invariant 
by stipulating, e.g., that the field energy maximization refers to the Lorentz frame in which the total particle momentum vanishes.
 In the Lorentz frame in which the initial value problem is formulated the relevant constraint is obtained through
a Lorentz boost.





	

§ THE ELECTROMAGNETIC FORCE:  N POINT CHARGES






 In the case of BLTP electrodynamics it is straightforward to generalize the formula for the electromagnetic
force on a point charge from when there is only a single charge to when in total N point charges are present.
 This is possible because  the linear pre-metric Maxwell equations are then complemented with 
the linear BLTP vacuum law as closure relation.
 When nonlinear vacuum laws are used, for instance the Born–Infeld law, all these linear algebra-based conclusions
are not available. 
 In the following we first give a general distributional definition of the electromagnetic force on a point 
charge source when N point charges are present.
 Then we extract from this definition the N-body analog of the
earlier given basic definition eq:EMf of the electromagnetic force when only a single charge is present. 



	

 §.§ Distributional definition of the electromagnetic force




 In the electromagnetic field-theory part of electrodynamics one already implements the fact that 
the system of N moving charged point particles is associated with a distribution-valued four-vector field on spacetime,
namely (cρ,)(t,), the inhomogeneity term in the linear pre-metric Maxwell field equations.
 It is only natural to also write the mechanical quantities associated with the point particles as 
distribution-valued fields on spacetime and thereby treat particles and fields on an equal footing.

 The momentum vector-density

    ^
    (t,)
    = ∑_n _n(t) δ__n(t)()

and the symmetric stress tensor field

    T^ (t,)
    = ∑_n  
    1/m_n^_n(t)⊗_n(t)/√(1 + |_n(t)|^2/m_n^2c^2) δ__n(t)()

jointly satisfy the local law of particle momentum balance 

    ^ (t,) + ∇· T^ (t,)
    = ∑_n^_n(t)δ__n(t)();

here, ^_n(t) is precisely the force term in eq:NewtonSECONDlaw, now understood for the n-th particle.

 Similarly, for admissible vacuum laws, the electromagnetic field momentum vector-density ^
and the symmetric stress tensor of the fields[To avoid awkward minus signs elsewhere, we 
      define T^ with the opposite sign compared to the 
      convention introduced by Maxwell in what nowadays is called the “Maxwell stress tensor.”]
T^ 
jointly satisfy the local law of field momentum balance 

    ^ (t,) + ∇· T^ (t,)
    = ∑_n_n(t)δ__n(t)().

 The source (sink) terms, _n(t),  for the electromagnetic field momentum density & field stress
are defined by eq:fieldMOMbalanceLAW.
 For admissible vacuum laws eq:fieldMOMbalanceLAW is indeed well-defined in the sense of distributions 
for fields with point charge sources at the _n(t) which move with subluminal velocities _n(t).
 The vector _n(t) is the electromagnetic field momentum gained (lost) per unit of time at 
location _n^(t) of  the source (sink).
 Therefore _n(t) has the physical dimension of a “force.”

 Now we postulate that the total momentum vector-density of the interacting system of electromagnetic field 
and its charged particle sources is given by the sum of the respective field and particle expressions defined earlier,

    (t,):=^(t,) + ^(t,)

is the total momentum vector-density.
 Similarly, the symmetric total stress tensor is postulated to be the sum

    T(t,):=T^(t,)+T^(t,).

 Our postulates eq:totalMOMENTUMdef & eq:totalSTRESSdef in concert with the two balance laws 
eq:chargeMOMbalanceLAW & eq:fieldMOMbalanceLAW imply the local balance law for the total momentum
vector-density,

    (t,) + ∇· T (t,)
    = ∑_n(_n(t) + ^_n(t)) δ__n(t)().


 Now postulating that the total momentum density and the stresses jointly satisfy the
local conservation law for the total momentum vector-density, 

    (t,) + ∇· T (t,)
    = ,

then by comparing eq:MOMconservation with eq:MOMtotalBALANCE, and invoking relativistic locality
(spacelike separated events do not affect each other), one deduces the identities

    ∀ n & ∀ t ≥ 0  (a.e.): ^_n(t) ≡ - _n(t).

 This is the general distributional definition of the electromagnetic force on a point charge source of the classical
electromagnetic field when N point charges are present.
 It may be seen as a zero-gravity implementation of what Einstein–Infeld–Hoffmann surmised <cit.>.

 In the next subsection we will extract the N-body analog of eq:EMf, the electromagnetic force when only a single 
charge is present. 

 We close this subsection by recalling that the energy density of the charged particle distribution 

    ^ (t,)
    = ∑_n m_n^c^2 √(1 + |_n(t)|^2m_n^2c^2) δ__n(t)()

and its momentum vector-density eq:TokCHARGES jointly satisfy the local law of particle energy balance

    ^ (t,) + c^2 ∇·^ (t,) 
    = ∑_n^_n(t)·_n(t)δ__n(t)().

 Similarly,  the field energy density ^(t,)
and the field momentum density ^ (t,) jointly satisfy the local law of field energy balance

    ^ (t,) + c^2 ∇·^ (t,) 
    = ∑_n_n(t)·_n(t)δ__n(t)().

 When the total energy density is postulated to be additive, i.e.

    (t,):= ^(t,) + ^(t,),

then the postulated local law of total momentum conservation eq:MOMconservation now entails 
the local conservation law for the total energy,

    (t,) + c^2 ∇·(t,) 
    = 
    0.


 Similarly one can obtain the local conservation law for total angular-momentum.



	

 §.§ Integral formula for the electromagnetic force




 The distributional definition eq:EMfN of the electromagnetic force, with _n defined by eq:fieldMOMbalanceLAW, 
gives rise to the following N-body analog of the one-body formula eq:EMf. 
 Let _j denote the Voronoi cell of the j-th point charge at the initial time, and ∂_j its boundary. 
 Let T>0 denote the instant of time until which all N point charges remain in their initial Voronoi cells.
 Then integrating  eq:fieldMOMbalanceLAW over _j yields, for t∈(0,T) and ∀ j∈{1,...,N}, 

    ^_j(t)
    = 
    - ∫__j^ (t,) ^3s 
    - ∫_∂_j (T^·ν_j) (t,) ^2s,

and the initial force is defined as its limit when t↓ 0.
 If N=1, so that j=1, we have _1 = ^3, the surface integral vanishes, and
in this case eq:EMfNint reduces to eq:EMf.

 If T<∞ then to go beyond T one can reset the clock to a new “initial” time, say T-, 
and replace the initial Voronoi cells with those at time t=T-, and repeat.
 
 As shown explicitly for BLTP electrodynamics, so also for BI electrodynamics one should be able to show that with fixed
initial data the r.h.s.eq:EMfNint depends on the vector functions t'↦_n(t'), t'↦_n(t'), 
and  t'↦_n(t') for t'∈(0,t), with lim_t'↓ 0_n(t') = _n(0)
and lim_t'↓ 0_n(t') = _n(0) given particle initial data, i.e.
no higher-order time derivative of the position beyond the second one should show up.
 Then with t'↦_n(t') and t'↦_n(t') considered given, the system of relativistic Newton equations of motion
eq:NewtonSECONDlaw with eq:EMfNint at its right-hand side
becomes a system of generally nonlinear integral equations for the maps t'↦_n(t') as functionals of 
the _n( . ) and _n( . ) (or _n( . )). 
 Whenever this system has a unique solution which depends Lipschitz continuously on the _n( . ) and _n( . ) (or _n( . )),
the electrodynamical initial value problem is locally (in time) well-posed. 




	

§ MOTION ALONG A CONSTANT ELECTRIC FIELD





 The problem of determining the classical dynamics of a single point charge which moves along a static,
spatially homogeneous electric field (approximately achieved by the field between the plates 
of a charged capacitor) has already been mentioned in the introductory section, where we recalled that 
the Eliezer–Ford–O'Connell equation of motion, and also its Landau–Lifshitz approximation, 
fail to account for the radiation-reaction on the motion and merely reproduce the test particle motion.
 We now demonstrate that the initial-value problem for BLTP electrodynamics formulated in this paper 
does take the radiation-reaction on the motion into account.

 For simplicity we restrict the discussion to the case where the particle is initially at rest and
surrounded by its own electrostatic field and by the electrostatic field of the capacitor. 
 Note that this completely fixes the initial data for the field and for the particle.
 No “past hypothesis,” about how these initial data were established, is needed.
 
 The textbook idealization of a single point charge placed in a 
truly uniform capacitor field is obtained as a limiting case from our setup, as follows.
 Consider a system of 2N+1 charges, with j=1 for the point electron whose dynamics we are interested in,
and N positive and N negative singly charged particles distributed uniformly over the respectively charged two
capacitor plates. 
 We  take the formal N→∞ limit in which the capacitor plates become
infinitely charged, but also infinitely extended and separated, leaving a homogeneous, 
static vacuum field ^hom behind in which the point charge _1 is situated. 
 While the total electrical field energy diverges in this limit, the field momentum is initially zero and 
remains well-defined later on; also the particle momentum vanishes initially for our data.
 The electromagnetic force on particle _1 is derived from total momentum balance — note that the total momentum 
of the single-particle-plus-field system is not by itself conserved because this is only a subsystem of a formally 
“infinitely-many-particles-plus-field” system.
 Even though all the other charges have been “moved to spatial infinity,” they still exert an influence on the remaining point
charge _1 through their field ^hom.
 The balance equation for the momentum of the single-particle-plus-field subsystem thus reads 

    ( _1(t) + ^ (t))
     = 
    -e ^hom ,

which yields the total electromagnetic force on particle _1,

    ^_1(t)
    = 
    - ∫_B_ct(_0)^ (t,) ^3s 
    -e ^hom ,

for t≥ 0.
 Technically we obtain eq:EMfCAPACITOR from eq:EMfNint for j=1.
 The point charge's Voronoi cell _1→^3 in the limit N→∞, and eq:EMfNint for j=1 gives eq:EMfCAPACITOR.
 We remark that the ^hom term in  eq:EMfCAPACITOR comes from the boundary integral in eq:EMfNint;
it is easy to see that ^hom does not contribute to ∫_^3^ (t,) ^3s.
 Moreover, in eq:EMfCAPACITOR we have replaced the domain of integration ^3 by the ball B_ct(_0) because 
^ (t,) ≡ outside of this ball.
 The “self”-field force -∫_B_ct(_0)^ (t,) ^3s is in this example identical 
with r.h.s.eq:selfF, for ^ (0,) ≡.

 To evaluate -∫_B_ct(_0)^ (t,) ^3s 
we invoke equations eq:selfFexplff with 𝐙_ξ^∘^[2](t,t^r) ≡.
 Due to the highly symmetrical setup these expressions simplify drastically, although one still cannot carry out each and 
every integration in terms of elementary functions — but this is not necessary for our purposes here.

 We now consider first the early time regime.
 Since, as noted above, the “self”-field force r.h.s.eq:selfFexpl vanishes at the initial instant t=0, 
the initial force is identical to the force -e ^hom on a charged test particle.
 Since the “self”-field force varies continuously with t≥ 0, it will remain small for a certain amount of time 
(which we are not going to determine here). 
 During this early dynamical phase the motion of the particle is therefore well approximated by the test particle
dynamics, with the inertia of the particle given by its bare mass. 
 
 We now show that although the radiation-reaction force vanishes initially, it typically does not remain zero, unlike the
Eliezer–Ford–O'Connell radiation-reaction force and its Landau–Lifshitz approximation.
 By “typically” we mean that for almost all ϰ parameter values the radiation-reaction force does not vanish
(we do not rule out that there might be special, isolated values of ϰ for which this might happen). 
 For this purpose, assume to the contrary that the radiation-reaction force r.h.s.eq:selfFexpl would vanish for an open interval 
of ϰ values.
 Since r.h.s.eq:selfFexplff reveals that the radiation-reaction force is an analytic function of ϰ, it 
follows that it then has to vanish for all ϰ. 
 This in turn means that each and every Maclaurin coefficient of its power series expansion in ϰ has to vanish.
 But The lowest-order term, which is ∝ϰ^2 and can be evaluated exactly in closed form (see below),
does not vanish, but the term ∝ϰ^3 does not,
 hence the radiation-reaction force cannot typically vanish. 
 
 To compute the O(ϰ^2) contribution, divide the expressions for π^[k]_ξ by ϰ^2 and
take the limit ϰ→ 0. 
 The only two terms which survive in the limit are those in the first line of r.h.s.pi1 and r.h.s.pi2,
respectively. 
 Carrying out the pertinent integrations in Zdef, and noting that the result only depends on t^r, not on t,
so that the third line at r.h.s.eq:selfFexplff vanishes at O(ϰ^2), we obtain
 


 ^_1(t) = 
    
- 1/2 e^2 ϰ^2  (t)/(t) 
[ 2 c/|(t)| - c^2/|(t)|^2 ln1+1/c(t)/1-1/c(t) ]

     - 3/2  e^2 ϰ^2   ∫_0^t 
(t^)1/|(t^)|^2 [ 
2 -131c|(t^)|  -  |(t^)|^2/c^2/1-|(t^)|^2/c^2  - 
 (c/|(t^)| -1/6 )
ln1+1/c|(t^)|/1-1/c|(t^)|
] .  c t^  + O(ϰ^3) 


=     -1/2 e^2 ϰ^2 (t)/(t) 
[1-c/|(t)|(1+[1-c/|(t)|]1/2ln1+1/c(t)/1-1/c(t) 
) ]
 + O(ϰ^3)


 The term in the first line at r.h.s.eq:SELFforceEXPLICITexpand is the contribution from the first line
at r.h.s.pi2, the term in the second line at r.h.s.eq:SELFforceEXPLICITexpand is the contribution from the first line
at r.h.s.pi1. 
 Since for straight-line motion (t) and (t) are parallel, and (t)=(t), one can carry out the time integration in 
the second line at r.h.s.eq:SELFforceEXPLICITexpand in terms of elementary functions of , and a few algebraic manipulations 
then give eq:SELFforceEXPLICITexpandMERGE.
 To O(ϰ^2) this is the exact expression for our BLTP radiation-reaction force 
in this problem where the particle starts from rest.
 It vanishes. 
only as ||↓ 0 and as ||↑ c, and otherwise points against . 
A similar computation (to be published) reveals that the term ∝ϰ^3 does not vanish.
This demonstrates that the BLTP electrodynamical initial value problem accounts for the
radiation-reaction on the motion of a point charge along a uniform electric field.

In the vicinity of the initial time when |(t)|/c≪ 1 we 
can expand and obtain, to leading order in ||/c:



^_1(t)
=
- 16 e^2 ϰ^2c (t)
 [1+ O(|(t)|^2c^2 ) ]
+ O(ϰ^3).

Note that eq:SELFforceEXPLICITexpandEXPAND is a radiation-friction force
of the familiar “Newtonian friction” type (i.e. proportional to -).

In Fig. 1 we show v in units of c as a function of t in units of c / e^2ϰ^2
both for the test particle motion and for the BLTP motion with
radiation-friction force given by eq:SELFforceEXPLICITexpandMERGE.
 The applied electric field strength is 0.1 in units of eϰ^2,  
which is a strong field for this problem.
The radiation-friction effect is clearly visible.




 We note that for very weak applied field strength the radiation-reaction is captured by the “Newtonian-friction” 
approximation eq:SELFforceEXPLICITexpandEXPAND, and the point charge's velocity will saturate at 
_∞= -6c/eϰ^2^; see Fig. 2 for an applied field strength
of 0.01 in units of eϰ^2.




The sharp borderline between the “weak field” and “strong field” regimes determines a critical field strength for this
problem. 
 For field strengths just slightly above the critical value ≈ 0.0519 the velocity temporarily reaches a quasi-plateau, 
before it makes the final transition to approach the speed of light; see Fig. 3 for an applied field strength
of 0.052 in units of eϰ^2.




 Although the ∝ϰ^3 term eq:SELFforceEXPLICITexpandMERGE is not an accurate formula for the radiation-reaction 
force in the physically interesting regime of very large ϰ values, it is entirely adequate for
demonstrating that the radiation-reaction does not vanish identically in the BLTP version of this standard textbook-type problem.
 Moreover, we have treated the dynamics properly as a physical initial value problem with the same data as in the test
particle formulation, unlike the treatment in <cit.> where
 “the whole path traversed by the particle up to the present time contributes to [the self-force]” (quoted from <cit.>).






	

§ SUMMARY AND OUTLOOK





 In this paper we have shown that a well-defined electromagnetic force on a point charge source of the classical electromagnetic 
field can be extracted from momentum balance among charges and field, whenever the electromagnetic vacuum law which supplies the 
closure relation for the pre-metric Maxwell field equations leads to a finite field momentum vector which is differentiable with 
respect to time.

 For the BLTP law of the vacuum we even reported that we were able to prove,
in collaboration with S. Tahvildar-Zadeh, that the BLTP electromagnetic force as defined in eq:EMf 
furnishes a well-posed joint initial value problem for
fields and point particles which is of second-order in the particle positions; see <cit.> for the details.
 To the best of our knowledge BLTP electrodynamics is the first classical electrodynamical theory of point charges 
and their electromagnetic fields which has been shown to be dynamically well-posed, free of infinite “self” energies etc. 
and ill-defined Lorentz “self” forces, and free of the ⃛-problem.
 Incidentally, neither Bopp, Landé–Thomas, nor Podolsky considered the definition of the force given in this paper,
but tried (in vain) to implement the ill-defined Lorentz force formula into their theory.

 We have illustrated the well-posed BLTP electrodynamical initial value problem by revisiting the standard textbook
problem of a point charge released from rest in a constant applied electrostatic field. 
 Our discussion confirms that the test-particle approximation is valid in the initial 
dynamical phase, with radiation-reaction corrections first in form of a linear , “Newtonian friction”-type  term
(at least in the small ϰ regime), and eventually in a non-linear manner.
 
 A most interesting finding, valid for arbitrary ϰ, 
is that in the initial phase of the dynamics the particle inertia is determined entirely by
its bare rest mass, not by the mass of the (electromagnetically) “dressed” particle. 
 The latter is generally thought to control the inertia in scattering scenarios.
 The predominance of scattering experiments, in particular in high energy physics, has led to the general belief that
only the “dressed particle” mass is observable in experiments, not the bare mass.
 Our findings by contrast suggest that the bare mass may be observable by cleverly setting up an initial value problem
in the laboratory.

 True, BLTP electrodynamics may not be the most realistic classical theory, but it surely is a “proof of concept,”
signaling that analogous results should be feasible also for putatively more realistic models, in particular the BI electrodynamics.
 We remark that a well-defined joint initial value problem for the MBI fields and their point charge sources was formulated
with the help of a Hamilton–Jacobi-type theory in <cit.>, but it is not clear whether that theory is well-posed,
nor is it clear that its dynamics is independent of the invoked foliation of spacetime it needs for its formulation.
 Since the setup given in the present paper is truly Lorentz co-variant and foliation-independent, it
should shed light on the formulation given in <cit.> by comparing the two.
 Incidentally, neither Born & Infeld, nor Schrödinger, nor Dirac, proposed the
electromagnetic force given in this paper but instead tried to implement the ill-defined Lorentz force 
formula into the Born–Infeld electrodynamics; cf. <cit.>.

 Inside the family of well-posed classical models one may hope to find the classical limit of the elusive, 
mathematically well-defined and physically viable, special-relativistic quantum theory of electromagnetism. 

 Having obtained a rigorous control over  the classical electromagnetic radiation-reaction problem, an
important next goal in the realm of classical physics is to get a rigorous hand on the gravitational radiation-reaction problem.
 As a first step, armed with the insights gained from the special-relativistic theory of motion formulated in this paper
we have embarked on an assessment of the Einstein–Infeld–Hoffmann <cit.> legacy; cf. <cit.>. 




§ ACKNOWLEDGMENT

The ideas and insights reported here evolved over many years, during which I have benefited 
especially from discussions with (in alphabetical order): Walter Appel, Holly Carley, Dirk Deckert, Detlef Dürr, 
Vu Hoang, Markus Kunze, Volker Perlick, Maria Radosz, Jared Speck, Herbert Spohn, and Shadi Tahvildar-Zadeh
in particular, for which I am very grateful.
 After the first version of this paper was made public, the author received several comments which were implemented
in the revision. 
 Many thanks go: to Bob Wald for drawing attention to Ref. <cit.>; to Mario Hubert for catching a sign error in the
Born–Infeld section; to Herb Johnson for prompting me to explain the MBLTP initial value problem more clearly; 
to the referee for the helpful comments which improved the presentation, and for prompting me to work out the 
example discussed in section VII.




§ REFERENCES



apsrev

10
LorentzFORCE 
        Lorentz, H.A., 
                La théorie électromagnetique de Maxwell et son 
                        application aux corps mouvants,
        Arch. Néerl. Sci. Exactes Nat. 25, pp. 363–552 (1892).
lienard 
        Liénard, A., 
                Champ électrique et magnétique produit
                        par une charge concentrée en un point et animée
			d'un mouvement quelconque,
        L' Éclairage électrique 16 p.5;
        ibid. p. 53; ibid. p. 106 (1898).
Wiechert 
        Wiechert, E.,   
                Elektrodynamische Elementargesetze, 
        Arch. Néerl. Sci. Exactes Nat. 5, 549-573 (1900). 
JacksonBOOKb 
        Jackson, J.D.,   
                Classical electrodynamics, 
       	J. Wiley & Sons, New York 
	2^nd ed. (1975); 3^rd ed. (1999).
GHW 
        Gralla, S.E.,
        Harte, A.,
        and
        Wald, R.M.,
        A rigorous derivation of electromagnetic self-force,
        Phys. Rev. D 80:024031 (2009).
LorentzENCYCLOP 
        Lorentz, H.A., 
                Weiterbildung der Maxwell'schen Theorie: Elektronentheorie.,
        Encyklopädie d. Mathematischen Wissenschaften V2,
        Art. 14, pp. 145–280 (1904). 
YaghjianBOOK 
        Yaghjian, A.D., 
                “Relativistic dynamics of a charged sphere,”
        Lect. Notes Phys. m11, Springer, Berlin (1992).
Spohn 
       Spohn, H.,
       Dynamics of charged particles and their radiation fields,
       Cambridge UP (2004).  
AppKieAOP 
        Appel, W.,
        and
	Kiessling, M.K.-H.,
        Mass and spin renormalization in Lorentz electrodynamics,
        Annals Phys. 289, 24–83 (2001).
DiracA 
        Dirac, P. A. M., 
                Classical theory of radiating electrons,
        Proc. Roy. Soc. A 167, 148–169 (1938).
Fermi 
	Fermi, E.,	
                Über einen Widerspruch zwischen der elektrodynamischen und der relativi­stischen 
			Theorie der elektromagnetischen Masse,
        Phys. Zeitschr. 23, 340-344 (1922). 
LandauLifshitz 
  Landau, L. D., 
  and 
  Lifshitz, E.,
                The Classical Theory of Fields,
                Pergamon (1951). 
PoissonETal 
   Poisson, E., Pound, A., and Vega, I.,
   The motion of point particles in curved spacetime,
 Living Rev. Rel. 14,7(190) (2011). 
BurtonNoble 
	Burton, D.-A., 
        and
        Noble, A.,     
                Aspects of electromagnetic radiation in strong fields,
	        Contemp. Phys. 55, 110–121 (2014). 
APMDb 
        Ares de Parga, G.,
        Mares, R.,
        and
        Dominguez, S.
        An unphysical result for the Landau–Lifshitz equation of motion for a charged particle,
        Rev. Mex. Fis. 52, 139–142 (2006). 
DeckertHartenstein 
	Deckert, D.-A.,
        and
        Hartenstein, V.,
                On the initial value formulation of  classical electrodynamics
	        J. Phys. A: Math. Theor. 49,  445202 (19pp.) (2016). 
MillerBOOK 
        Miller, A. I., 
                Albert Einstein's special theory of relativity,
        Springer, New York (1998).
AbrahamD 
        Abraham, M., 
		Die Grundhypothesen der Elektronentheorie,
	Phys. Z. 5, 576-579 (1904). 
HehlObukov 
  Hehl, F. W., 
  and
  Obukov, Y. N.,
  “Foundations of Classical Electrodynamics,”
  Birkhäuser, Basel (2003).
BornInfeldBb 
	Born, M., and Infeld, L.,
	Foundation of the new field theory,
	Proc. Roy. Soc. London A 144, 425–451 (1934).
BiBiONE 
	Białynicki-Birula, I.,	
		Nonlinear electrodynamics: Variations on a theme by Born and Infeld, 
	pp. 31-48 in “Quantum theory of particles and fields,” 
        eds. B. Jancewicz and J. Lukierski, World Scientific, Singapore (1983).
BoppA 
        Bopp, F.,
        Eine lineare Theorie des Elektrons,
        Annalen Phys. 430, 345–384 (1940).
BoppB 
        Bopp, F.,
        Lineare Theorie des Elektrons. II, Annalen Phys. 434, 573–608 (1943).
Lande 
	Landé, A., 
        Finite Self-Energies in Radiation Theory. I, 
        Phys. Rev. 60, 121–126 (1941).
LandeThomas 
	Landé, A., and Thomas, L.H.,
        Finite Self-Energies in Radiation Theory. II,
        Phys. Rev. 60, 514–523 (1941).
Podolsky 
        Podolsky, B.,
        A generalized electrodynamics. Part I: Non-quantum,
        Phys. Rev. 62, 68–71 (1942).
PodolskySchwed 
          Podolsky, B.,
          and
          Schwed, P.,
          A review of generalized electrodynamics,
            Rev. Mod. Phys. 20, 40–50 (1948).
MieFELDTHEORIEa 
	Mie, G.,
	Grundlagen einer Theorie der Materie, I,
	Annalen Phys. 37, 511-534 (1912).
MieFELDTHEORIEb 
	Mie, G.,
	Grundlagen einer Theorie der Materie, II,
	Annalen Phys. 39, 1-40 (1912). 
Schwarzschild 
	Schwarzschild, K.
		Zur Elektrodynamik.I,
	Nachr. Ges. Wiss. Göttingen, 126–131 (1903).


SpeckMBI 
	Speck, J.R.,
		The nonlinear stability of the trivial solution to the Maxwell-Born-Infeld system,
                J. Math. Phys. 53, 083703(83pp) (2012).
SerreA 
Serre, D., 
	Les ondes planes en éctromagnététisme non-linéaire, 
	Physica D 31, 227-251 (1988).

Brenier 
        Brenier, Y.,
        	Hydrodynamic structure of the augmented Born–Infeld equations,
	Arch. Rat. Mech. Anal. 172, 65-91 (2004). 
KieMBIinCMP  
	Kiessling, M.K.-H.,  
		On the quasi-linear elliptic PDE -∇·(∇u/√(1-|∇u|^2)) = 4π∑_k a_k δ_s_k
                        in physics and geometry,
	Commun. Math. Phys. 314, 509–523 (2012); Correction ibid. 364, 825–833 (2018). 
CKP 
        Carley, H. K.,
        Kiessling, M. K.-H.,
        and
        Perlick, V.,
        On the Schrödinger spectrum of a hydrogen atom with BLTP interactions between electron and proton,
        Int. J. Mod. Phys. A 34, 1950146 (2019).
KTZonBLTP 
	Kiessling, M.K.-H.,
        and
        Tahvildar-Zadeh, A. S.,
        Bopp-Landé-Thomas-Podolsky electrodynamics as initial value problem,
        in preparation (2020).
KTZonEIHa 
	Kiessling, M.K.-H.,
        and
        Tahvildar-Zadeh, A. S.,
        The Einstein–Infeld–Hoffmann legacy in mathematical relativity. I: The classical motion of charged point particles,
        16pp. in Proc. of the 15th Marcel Grossmann Meeting on General Relativity, Rome, Italy (2018); 
        Int. J. Mod. Phys. D 28 1930017 (2019).
GratusETal 
	Gratus, J., Perlick, V., and Tucker, R.W.,
        On the self-force in Bopp–Podolsky electrodynamics,
        J. Phys. A: Math. Theor. 48, 435401 (28pp.) (2015). 
Zayats 
        Zayats, A.E.,
        Self-interaction in the Bopp-Podolsky electrodynamics:
          Can the observable mass of a charged particle depend on its acceleration?,
        Annals Phys. (NY)342, 11–20 (2014). 
VuMaria 
	Hoang, V., and Radosz, M.,
        On the self-force in higher-order electrodynamics,
        Preprint (41pp.) arXiv:1902.06386 (2019).
QuinnWald
   Quinn, T.C.,
   and
   Wald, R.M.,
   An axiomatic approach to electromagnetic and gravitational radiation reaction of particles in curved spacetime,
   Phys. Rev. D 56, 3381–3394 (1997).
Kijowski
        Kijowski, J.,
          Electrodynamics of moving particles, Gen. Rel. Grav. 26, 167-192 (1994).
DetweilerWhiting
        Detweiler, S., 
        and 
        Whiting, B. F., 
        Self-force via a Green’s function decomposition,
        Phys. Rev. D 67, 024025 (2003).
FeynmanBOPP 
       Feynman, R. P., 
               A relativistic cut-off for classical electrodynamics,
       Phys. Rev. 74, 939–946 (1948).
EIH 
	Einstein, A.,
	Infeld, L.,
	and
	Hoffmann, B., 
	           The gravitational equations and the problem of motion,
	Annals Math. 39, 65-100 (1938).
KieMBIinJSPa  
	Kiessling, M.K.-H.,
		Electromagnetic field theory without divergence problems. 
		        1. The Born legacy,
	J. Stat. Phys. 116, 1057-1122  (2004).
KieMBIinREGENSBURG  
	Kiessling, M.K.-H.,
        On the Motion of Point Defects in Relativistic Fields, pp.299–335
        in: Finster F., Müller O., Nardmann M., Tolksdorf J., Zeidler E. (eds) 
        “Quantum Field Theory and Gravity.” Springer, Basel (2012).


