OOn the Radiation Reaction Force
T. Matolcsi Abstract
The usual radiation self-force of a point charge is obtained in a math-ematically exact way and it is pointed out to that this does not call forththat the spacetime motion of a point charge obeys the Lorentz–Abraham–Dirac equation.
Introduction
The equation of spacetime motion of a point charge under the action of anexternal force is an old problem of electromagnetism. The original work ofAbraham (treated e.g. in [2], Ch. 16.3) and then Dirac’s formulation (treatede.g. in [3], Ch III.3) suffers from a hardly acceptable mass renormalizationaccording to which the finite mass of a particle is the difference of the positiveinfinite electric mass and the negative infinite mechanical mass. Other, quickdeductions based on Larmor’s formula and some other ‘natural’ requirements(see [2], Ch 16.2) are not convincing either. Moreover, the LAD equation admitsphysically untenable run-away solutions. There are some attmepts to excludesuch non-physical solutions by imposing extra conditions. (e.g. [11], [12]). Thereare a number of articles (e.g.[7], [5], [6], [9], [4]) considering various continuumcharge distributions instead of a point charge and then taking or not the limitto a point. Though casting more and more new light on the problem and itspossible solution, neither way seems satsifactory.A recent article [8] – though its result rests on an erroneous basis, see [20] –points out clearly to Dirac’s unjustified applications of Gauss–Stokes theorem,Taylor expansion and the limit procedure when an extended charge is shrunkto a point.In some papers (e.g. [10], [13],[9]) special applications of Distribution Theorycan be found when treating point charges.
In the present paper, by a systematic use of Distribution Theory, it is shownthat not a mathematically incorrect derivation of the radiation reaction forcebut a physical misapprehension is the source of why the LAD equation does notwork well .The coordinate-free formulation of spacetime expounded in [1] is used whichmakes formulae shorter and more easily comprehensible; find enclosed a briefsummary of the fundamental notions and some special notations.Spacetime points and spacetime vectors (which are often confused in coordi-nates) are distinguished:– M, mathematically an affine space, is the set of spacetime points, itselements are denoted by normal letters, x , y etc.,– M , mathematically a vector space, is the set of spacetime vectors, itselements are denoted by boldface letters, x , y etc., Department of Applied Analysis and Computational Mathematics, Eötvös Loránd Uni-versity, Budapest, Hungary a r X i v : . [ phy s i c s . c l a ss - ph ] F e b the set of time periods is T ,– the exact treatment requires the tensorial quotients of M by T , here wedo not refer to them explicitely,– the Lorentz product of the vectors x and y is denoted by x · y (which is x k y k in coordinates); x is timelike if x · x < − , , ,
1) of the Lorentz form),– an absolute velocity is a futurelike vector u for which u · u = − u , S u := { x ∈ M | u · x = 0 } , the set of u -spacelikevectors, is a three dimensional Euclidean vector space,– the action of linear and bilinear maps is denoted by a dot, too; e.g theaction of a linear map L on a vector x is L · x (which is L ik x k in coordinates),– the adjoint of a linear map L is the linear map L ∗ defined by ( L ∗ · x ) · y = x · ( L · y ) for all vectors x , y ; shortly, L ∗ · x = x · L ,– L is a Lorentz transformation if and only if L ∗ = L − ,– the tensor product x ⊗ y of the vectors x and y is a linear map definedby ( x ⊗ y ) · z := x ( y · z ) (in coordinates it is x i y k ); ( x ⊗ y ) ∗ = y ⊗ x ; x ∧ y := x ⊗ y − ( x ⊗ y ) ∗ and x ∨ y := x ⊗ y + ( x ⊗ y ) ∗ are the antisymmetricand symmetric tensor product, respectively,– the formulae of electromagnetism are written in the choice c = 1, (cid:126) = 1and the electric charge is measured by real numbers.– D denotes the differentiation in spacetime ( ∂ k in coordinates). The anti-symmetric derivative of a vector valued function f is denoted by D ∧ f ( ∂ k f i − ∂ i f k in coordinates). The spacetime divergence of f is denoted by D · f ( ∂ k f k in coor-dinates). Similarly, D · T is the spacetime divergence of a tensor valued function T ( ∂ k T ik in coordinates).– D denotes differentiation of other functions, e.g. parameterizations.The word ‘distribution’ will appear in two different senses: 1. having itseveryday meaning, 2. being a mathematical notion. To distinguish betweenthem, I write distribution for the first one and Distribution for the second one.The usual setting of Distribution Theory is based on R n ([16], [17],[18], [19]).It is a quite simple generalization that spacetime and an observer space are takeninstead of R and R . Another simple generalization is that vector and tensorDistributions are included, too. The present article can be understood withouta thorough knowledge of Distributions; besides the elementary notions the onlynon trivial one is pole taming , described in the Appendix.The guiding principle is that only quantities definable in Distribution The-ory can make sense. Distributions are denoted by calligraphic letters. Locallyintegrable functions and the corresponding Distributions are distinguished innotation; e.g. F is an electromagnetic field function, F is the correspondingelectromagnetic field Distribution. Electrodynamics of point charges
The existence of a material point in spacetime is described by a world linefunction whose variable is the proper time; the range of a world line function2s a world line , a one dimensional submanifold in spacetime.Let r be a given continuously differentiable world line function. For a worldpoint x , s r ( x ) will denote the retarded proper time corresponding to x i.e. theproper time point of r for which x − r ( s r ( x )) is future lightlike or zero; evidently,it is zero if and only if x is on the world line.In other words, s r ( x ) is defined implicitly by the relations( x − r ( s r ( x ))) · ( x − r ( s r ( x ))) = 0 , − u · ( x − r ( s r ( x ))) ≥ u .According to the implicit function theorem, s r is continuously differentiableoutside the world line; differentiating the first equality above, we obtain that2( x − r ( s r ( x )) · (cid:0) − ˙ r ( s r ( x )) ⊗ D s r [ x ] (cid:1) = 0 i.e.D s r [ x ] = x − r ( s r ( x ))˙ r ( s r ( x )) · ( x − r ( s r ( x ))) ( x / ∈ Ran r ) . Though s r is not differentiable on the world line, it is continuous there whichcan be seen as follows.Let s − < s < s + be arbitrary proper time points of the world line. If T → denotes the set of futurelike vectors then (cid:0) r ( s − ) + T → (cid:1) ∩ (cid:0) r ( s + ) − T → (cid:1) is abounded open set containing r ( s ). If x is in this set then r ( s r ( x )) is in it, too;consequently the function x x − r ( s r ( x )) is bounded.The proper time passed between two points of the world line is less or equalthan the inertial time, thus | s r ( x ) − s | ≤ − (cid:0) r ( s r ( x ) − r ( s )) (cid:1) · (cid:0) r ( s r ( x ) − r ( s )) (cid:1) == − (cid:0) r ( s r ( x ) − x ) + ( x − r ( s )) (cid:1) · (cid:0) ( r ( s r ( x ) − x ) + ( x − r ( s )) (cid:1) == 2 (cid:0) x − r ( s r ( x )) (cid:1) · ( x − r ( s )) − ( x − r ( s )) · ( x − r ( s )) . Because of the mentioned boundedness, the right hand side tends to zero as x tends to r ( s ). Let r be a given twice differentiable world line function of a point charge e .Let us introduce the following functions: R r ( x ) := x − r ( s r ( x )) , u r ( x ) := ˙ r ( s r ( x )) , a r ( x ) := ¨ r ( s r ( x )) , (1) L r := R r − u r · R r = − D s r , d r := a r + ( a r · L r ) u r . (2)Then it is known (see [2]) that the electromagnetic field produced by a pointcharge e with a given world line function r is the regular Distribution F [ r ]defined by the locally integrable function F [ r ] := e π u r ∧ L r ( − u r · R r ) + e π d r ∧ L r − u r · R r ; (3)3he first term is the tied field F [ r ] td which is never zero, the second term is the radiated field F [ r ] rd which is zero if and only if the charge is not accelerated.It is emphasized by the notation [ r ] that the formulae above make sense onlyif the world line function is given because the retarded proper time is definedonly in that case . As usual, for an electromagnetic field function F , T := − F · F −
14 Tr F · F (4)is considered the energy-momentum tensor whose negative spacetime divergence − D · T is the spacetime force density.The problem is that T , in general, is not locally integrable, thus T doesnot define a Distribution a priori, so it has no physical meaning. In particular,this occurs for a point charge; that is why the quotation mark appeared in thesection title. According to (3), the ‘energy-momentum tensor’ is T [ r ] = T [ r ] td + T [ r ] rtd + T [ r ] rd (5)where T [ r ] td := − F [ r ] td · F [ r ] td −
14 Tr F [ r ] td · F [ r ] td , T [ r ] rtd := − ( F [ r ] td · F [ r ] rd + F [ r ] rd · F [ r ] td ) −
14 Tr( F [ r ] td · F [ r ] rd + F [ r ] rd · F [ r ] td ) , T [ r ] rd := − F [ r ] rd · F [ r ] rd −
14 Tr F [ r ] rd · F [ r ] rd , for which we have the following equalities: − F td [ r ] · F td [ r ] = e π u r ⊗ L r + L r ⊗ u r − L r ⊗ L r ( u r · R r ) , (6) − Tr( F td [ r ] · F td [ r ]) = e π u r · R r ) , (7) − (cid:0) F td [ r ] · F rd [ r ]+ F rd [ r ] · F td [ r ] (cid:1) = e π d r ⊗ L r + L r ⊗ d r + 2( u r · d r ) L r ⊗ L r ( − u r · R r ) , (8)Tr( F td [ r ] · F rd [ r ]) = 0 , (9) − F rd [ r ] · F rd [ r ] = e π | d r | L r ⊗ L r ( u r · R r ) , (10)Tr( F rd [ r ] · F rd [ r ]) = 0 . (11)Evidently, T [ r ] is differentiable outside the world line; its properties nearthe world line will be examined in the forthcoming sections.4 Radial expansion of the retarded proper time
The retarded proper time is not differentiable on the world line, so it has not aTaylor polynomial around the world line. Nevertheless, it can be expanded in‘radial directions’ in powers of the ‘radial distance’ as follows.We take the parameterization (60) of spacetime around a given world lineRan r ; in this parameterization, at every proper time point s of r , the spacelikevectors Lorentz orthogonal to ˙ r ( s ) are taken into account; then according tothe notations introduced in (62), ρ and n describe distances and directions,respectively, in that Euclidean space. The difference between the parameterproper time and the retarded proper time isΘ( s , q ) := s − s r (cid:0) r ( s ) + L ( s ) q (cid:1) . Θ is a continuous function, non-negative and Θ( s , q ) = 0 if and only if q = 0.Thus, lim q → Θ( s , q ) = 0 (12)holds as well.In the sequel, r is supposed to be three times continuously differentiable, thevariables are omitted for the sake of brevity and Ordo denotes a function whoselimit at zero is zero.First, let us observe that according to (12)Θ = Ordo( ρ ) . Then starting with s r = s − Θ, we have r ( s − Θ) = r ( s ) − ˙ r ( s )Θ + 12 ¨ r ( s )Θ −
16 ... r ( s )Θ + Θ Ordo(Θ) . (13)Subtracting both sides from the parameterization (63), we get (see (1)) R r = ρ n + ˙ r Θ −
12 ¨ r Θ + 16 ... r Θ + Θ Ordo(Θ) . (14)The left hand side is a lightlike vector or zero. The Lorentz square of bothsides, with the equalities˙ r · ˙ r = − , n · n = 1 , ˙ r · n = 0 , ˙ r · ¨ r = 0 , ˙ r · ... r = − ¨ r · ¨ r, results in0 = ρ − (1 + ρ n · ¨ r )Θ + 13 ρ n · ... r Θ −
112 ¨ r · ¨ r Θ + ( ρ Θ + Θ )Ordo(Θ) . In another form,1 = Θ ρ (cid:18) ρ n · ¨ r − ρ n · ... r Θ + 112 | ¨ r | Θ − ( ρ Θ + Θ )Ordo(Θ) (cid:19) , (15)which says that lim ρ → ρ = 1. Since Θ is non-negative, lim ρ → ρ = 1 holds aswell. Then Θ = ρ (1 + A ) where A = Ordo( ρ ) (16)5nd (15) can be rewritten in the form1 = Θ ρ + Θ ρ ( ρ n · ¨ r + ρ Ordo( ρ )) . Putting (16) in the first term on the right hand side, we get2 A + A ρ = − Θ ρ ( n · ¨ r + Ordo( ρ )) . Let ρ tend to zero. Since A = Ordo( ρ ), we getlim ρ → Aρ = − n · ¨ r, in other words, A = ρ ( − n · ¨ r + Ordo( ρ )); thus,Θ = ρ − ρ n · ¨ r + B ) where B = Ordo( ρ ) . (17)Then (15) can be written in the form:1 = Θ ρ (1 + ρ n · ¨ r ) − Θ ρ (cid:18) ρ n · ... r Θ − | ¨ r | Θ + ρ Ordo( ρ ) (cid:19) . (18)Putting (17) in the first term on the right hand side, we get (cid:18) − ρ n · ¨ r − ρB + ρ (cid:0) ( n · ¨ r ) + 2 n · ¨ rB + B (cid:1)(cid:19) (1 + ρ n · ¨ r ) == 1 − ρ ( n · ¨ r ) − ρB + ρ (cid:0) ( n · ¨ r ) + Ordo( ρ ) (cid:1) . Then (18) divided by ρ becomes0 = − ( n · ¨ r ) − Bρ + 14 (cid:0) ( n · ¨ r ) +Ordo( ρ ) (cid:1) − Θ ρ (cid:18) n · ... r Θ ρ − | ¨ r | Θ ρ + Ordo( ρ ) (cid:19) from which lim ρ → Bρ = −
34 ( n · ¨ r ) − n · ... r + 112 | ¨ r | . Then B = ρ (cid:18) −
34 ( n · ¨ r ) − n · ... r + 112 | ¨ r | + Ordo( ρ )) (cid:19) . Then we have the final resultΘ = ρ (cid:18) − ρ n · ¨ r ρ (cid:18) n · ¨ r ) − | ¨ r |
12 + n · ... r (cid:19) + ρ Ordo( ρ ) (cid:19) . (19)For the sake of simplicity, without the danger of confusion, we shall omit theterms Ordo( ρ ), writing approximative equalities:Θ ≈ ρ (cid:18) − ρ n · ¨ r ρ (cid:18) n · ¨ r ) − | ¨ r |
12 + n · ... r (cid:19)(cid:19) , (20)Θ ≈ ρ (1 − ρ ( n · ¨ r )) , Θ ≈ ρ . (21)It is emphasized again that these formulae do not come from a Taylor ex-pansion because of the non differentiability of the retarded proper time on theworld line. 6 Further radial expansions
Using the previous results, we give the radial expansion of the functions occur-ring in the formulae of the electromagnetic field (3).According to (20) and (21), the retarded velocity and acceleration introducedin (1) have radial expansion u r ≈ ˙ r − ¨ r Θ + 12 ... r Θ ≈ ˙ r − ρ ¨ r + ρ (cid:0) ( n · ¨ r ) ˙ r + ... r (cid:1) , (22) a r ≈ ¨ r − ... r Θ ≈ ¨ r − ρ ... r . (23)Further, (14) gives us R r ≈ ρ n + ˙ r Θ −
12 ¨ r Θ + 16 ... r Θ ≈≈ ρ n + ¨ r − ρ ¨ r + ( n · ¨ r ) ˙ r ρ (cid:18)(cid:18) n · ¨ r ) − | ¨ r |
12 + n · ... r (cid:19) ˙ r + ( n · ˙ r )¨ r + ... r (cid:19) ! , ( − u r · R r ) ≈ ρ (cid:18) ρ ( n · ¨ r )2 − ρ (cid:18) ( n · ¨ r ) − | ¨ r | n · ... r (cid:19)(cid:19) , − u r · R r ≈ ρ − ρ n · ¨ r ρ (cid:18) n · ¨ r ) − | ¨ r | n · ... r (cid:19)! , u r · R r ) ≈ ρ − ρ n · ¨ r + ρ (cid:18) n · ¨ r ) − | ¨ r | n · ... r (cid:19)! , − u r · R r ) ≈ ρ − ρ n · ¨ r ρ (cid:18) n · ¨ r ) − | ¨ r | n · ... r (cid:19)! , u r · R r ) ≈ ρ − ρ n · ¨ r + ρ (cid:18) n · ¨ r ) − | ¨ r | + 83 n · ... r (cid:19)! . Thus, for the quantities introduced in (2), L r ≈ n + ˙ r − ρ (cid:18) ¨ r n · ¨ r ) (cid:16) ˙ r + n (cid:17)(cid:19) ++ ρ (cid:18) n · ¨ r ) − | ¨ r | n · ... r (cid:19) ˙ r ++ ρ (cid:18) n · ¨ r ) − | ¨ r | n · ... r (cid:19) n ++ ρ (cid:18) n · ¨ r )¨ r r (cid:19) , (24) d r ≈ ¨ r + ( n · ¨ r ) ˙ r + ρ (cid:18)(cid:18) | ¨ r | − ( n · ¨ r ) − n · ... r (cid:19) ˙ r − ( n · ¨ r )¨ r − ... r (cid:19) (25)7 Radial expansion of the‘energy-momentum tensor’
Let us take (6)-(11) and, for the sake of simplicity, let us rewrite (24) in theform L r ≈ n + ˙ r − ρ A + ρ B . Then – ∨ denotes the symmetric tensor product –( u r ∨ L r ) ≈ u ∨ ( n + u ) − ρ (cid:0) u ∨ A + ¨ r ∨ ( n + u ) (cid:1) ++ ρ (cid:16) ˙ r ∨ B + (( n · ¨ r )¨ r + ... r ) ∨ ( n + ˙ r ) + 2¨ r ∨ A (cid:17) , ( L r ⊗ L r ) ≈ ( n + ˙ r ) ⊗ ( n + ˙ r ) − ρ (cid:0) ( n + ˙ r ) ∨ ¨ r (cid:1) ++ ρ (cid:16) ( n + ˙ r ) ∨ B + A ∨ A (cid:17) . Further, rewriting (25) in the form d r ≈ ¨ r + ( n · ¨ r ) ˙ r + ρ ( β ˙ r − ( n · ¨ r )¨ r − ... r )we have( d r ∨ L r ) ≈ (¨ r + ( n · ¨ r ) ˙ r ) ∨ ( n + ˙ r )++ ρ (cid:16) − (cid:0) ¨ r + ( n · ¨ r ) ˙ r (cid:1) ∨ A + ( β ˙ r − ( n · ¨ r )¨ r − ... r ) ∨ ( n + ˙ r ) (cid:17) , ( u r · d r ) ≈ − ( n · ¨ r + ρβ ) , (26)and ( u r · d r )( L r ⊗ L r ) ≈ − ( n · ¨ r ) (cid:0) ( n + ˙ r ) ⊗ ( n + ˙ r ) (cid:1) ++ ρ (cid:0) ( n · ¨ r )( n + ˙ r ) ∨ A − β ( n + ˙ r ) ⊗ ( n + ˙ r ) (cid:1) ;finally, | d r | ≈ | ¨ r | − ( n · ¨ r ) . All these together give that the ‘energy-momentum tensor’ (5) has the radial8xpansion16 π e T [ r ] = ˙ r ⊗ ˙ r − n ⊗ n + ρ − + n ⊗ ¨ r +¨ r ⊗ n − ( n · ¨ r )(2 ˙ r ⊗ ˙ r − n ⊗ n + ) ρ −− ρ (cid:18) ¨ r ⊗ ¨ r r ⊗ ... r + ... r ⊗ ˙ r (cid:19) + − ρ (cid:18) n ⊗ ... r + ... r ⊗ n )3 + ( n · ¨ r )( n ⊗ ¨ r + ¨ r ⊗ n ) (cid:19) ++ 1 ρ (cid:18) | ¨ r | + 3( n · ¨ r ) + 4 n · ... r (cid:19) ˙ r ⊗ ˙ r ++ 1 ρ (cid:18) | ¨ r | − ( n · ¨ r ) (cid:19) n ⊗ n ++ 1 ρ (cid:18) | ¨ r | n · ... r (cid:19) ( ˙ r ⊗ n + n ⊗ ˙ r )++ 1 ρ (cid:18) | ¨ r | n · ¨ r ) n · ... r (cid:19) + Ordo( ρ ) ρ . (27) The radial expansion (27) shows that the ‘energy-momentum tensor’ T [ r ], forevery proper time value s of the given world line function r , has a pole of fourthorder and a pole of third order in the three dimensional Euclidean space Lorentzorthogonal to ˙ r ( s ). We can tame its poles, obtaining tm T [ r ] as follows. Forthe sake of avoiding misunderstandings, we shall write simply T instead of T [ r ]when integrating functions.For an x in the neighbourhood of the world line in question let ˆ s ( x ) denotethe proper time value for which˙ r (ˆ s ( x )) · (cid:0) x − r (ˆ s ( x ) (cid:1) = 0 , (28)holds, i.e. ˆ s is the first component of the inverse of the parameterization (60). Proposition 7.1.
If the support of the test function φ is disjoint from the rangeof r then (cid:0) tm T [ r ] | φ (cid:1) := Z M T ( x ) φ ( x ) dx ; if the support of the test function φ is in a world roll H I,R I then (cid:0) tm T [ r ] | φ (cid:1) :=:= Z M T ( x ) φ ( x ) − e π r (ˆ s ( x )) ⊗ ˙ r (ˆ s ( x )) + | x − r (ˆ s ( x )) | φ ( r (ˆ s ( x ))) ! dx !! (29) where the double exclamation mark says the integral must be taken in the radialparameterization of the world roll H I,R I and in a given order. roof The integral (29) in the radial parameterization reads Z T ∞ Z Z S (0) (cid:16) T ( r ( s ) + ρ n ) φ ( r ( s ) + ρ n ) −− e π r ( s ) ⊗ ˙ r ( s ) + ρ φ ( r ( s )) (cid:17) (1 + ρ ( n · ¨ r ( s )) d n ! ρ dρ ! d s with n = L ( s ) n .Let us write the radial expansion of the ‘energy-momentum tensor’ in theform A ( n ) ρ + B ( n ) ρ + C ( n ) + Ordo( ρ ) ρ . (30)Then the function (cid:18) A ( n ) ρ + B ( n ) ρ + C ( n ) + Ordo( ρ ) ρ (cid:19) (1 + ρ n · ¨ r ) = (31)= A ( n ) ρ + A ( n )( n · ¨ r ) + B ( n ) ρ + B ( n )( n · ¨ r ) + C ( n ) + Ordo( ρ ) ρ (32)is to be pole tamed for each fixed s . A ( n ) is an even tensor product of n (linear function of an even tensor mulipleof n ), so the first term of (32) can be tamed: the function A ( n ) (cid:0) φ ( r ( s ) + ρ n ) − φ ( r ( s )) (cid:1) ρ is to be integrated in the radial variables (because the derivative of φ drops outfrom the Taylor polynomial).The part containing φ ( r ( s ) + ρ n ) remains as it is, A ( n ) ρ φ ( r ( s ) + ρ n ); (33)the part containing φ ( r ( s )) becomes e π (cid:18) r ( s ) ⊗ ˙ r ( s ) + ρ (cid:19) φ ( r ( s )) (34)which is obtained from the integral14 π Z S (0) (cid:18) ˙ r ( s ) ⊗ ˙ r ( s ) − L ( s ) n ⊗ L ( s ) n + (cid:19) d n by the use of Z S ( ) L ( s ) n ⊗ L ( s ) n d n = 4 π (cid:0) + L ( s ) ˙ r (0) ⊗ L ( s ) ˙ r (0) (cid:1) = 4 π (cid:0) + ˙ r ( s ) ⊗ ˙ r ( s ) (cid:1) .A ( n ) n · ¨ r + B ( n ) is a linear function of an odd tensor product of n , so thesecond term in (32) can be tamed: the function A ( n ) n · ¨ r + B ( n ) ρ φ ( r ( s ) + ρ n ) (35)10s to be integrated in the radial variables (because φ ( r ( s )) drops out from theTaylor polynomial).The third term in (32) defines a regular Distribution, so the function B ( n ) n · ¨ r + C ( n ) + Ordo( ρ ) ρ φ ( r ( s ) + ρ n ) (36)is to be integrated.Summarizing: the sum of (33), (35) and (36) gives the first term in (29) and(34) gives the second term because in the radial parameterization ˆ s ( x ) = s and | x − r (ˆ s ( x )) | = ρ . (cid:4) tm T [ r ] will be called the Distribution of energy-momentum (which is notan energy-momentum distribution!) of the point charge with a given world linefunction r . The Distribution of energy-momentum of a point charge is a mathematical ob-ject. Has it some physical meaning? The negative spacetime divergence of theenergy-momentum tensor of a continuous charge-current density is the forcedensity; that is why we can hope here a similar physical meaning.Recall that λ Ran r is the Lebesgue measure of the world line Ran r (see (53)). Proposition 8.1. − D · tm T [ r ] = 14 π e (cid:0) ... r + ( ˙ r · ... r ) ˙ r (cid:1) λ Ran r = (37)= 14 π e r ∧ ... r ) · ˙ rλ Ran r . (38) Proof
The Distribution tm T [ r ] restricted to an open subset disjoint fromthe world line equals the regular Distribution corresponding to the restrictionof T [ r ]. Since D · T [ r ] = 0 except the world line, we have to take test functions φ whose support is in a world roll H I,R I around the world line. For them( − D · tm T [ r ]) | φ ) = (tm T [ r ] | · D φ ) == Z M T ( x ) · D φ [ x ] − e π r (ˆ s ( x )) ⊗ ˙ r (ˆ s ( x )) + | x − r (ˆ s ( x ) | · D φ [ r (ˆ s ( x ))] ! dx !! == lim R → Z M \ H I,R . . . !!Because of D · T = 0 in M \ H I,R „ the first term in the integrand equalsD · ( T φ ).For examining the second term let us consider ρ and n as functions of thespacetime points x instead of the radial parameters ( s , q ), i.e. let us introduceˆ ρ ( x ) := | x − r (ˆ s ( x )) | , ˆ n ( x ) := x − r (ˆ s ( x )) | x − r (ˆ s ( x )) | ; (39)11oreover, let ˆ u ( x ) := ˙ r (ˆ s ( x )) , ˆ a ( x ) := ¨ r (ˆ s ( x )) . Then (ˆ ρ ˆ n )( x ) = x − r (ˆ s ( x ))and ˆ n · ˆ n = 1 , ˆ u · ˆ n = 0 . (40)By differentiation we get − ˆ u ⊗ Dˆ s = D(ˆ ρ ˆ n ) = ˆ n ⊗ Dˆ ρ + ˆ ρ Dˆ n , (41)from which, by (40) we deduce Dˆ ρ = ˆ n . (42)Further, Dˆ u = ˆ a ⊗ Dˆ s , and 0 = D(ˆ u · ˆ ρ ˆ n ) = ˆ ρ ˆ n · (ˆ a ⊗ Dˆ s ) + ˆ u · ( − ˆ u ⊗ Dˆ s ) , therefore Dˆ s = − ˆ u ρ ˆ n · ˆ a (43)and as a consequence, D · (ˆ ρ ˆ n ) = 4 −
11 + ˆ ρ ˆ n · ˆ a , (44)Further, let Z ( s ) := (4 ˙ r ( s ) ⊗ ˙ r ( s ) + ) · D φ [ r ( s )]6 , ˆ Z ( x ) := Z (ˆ s ( x )) . Then we find that D ˆ Z = ˙ Z (ˆ s ) ⊗ Dˆ s from which by (43) (D ˆ Z ) · ˆ n = 0 follows.Then taking into account (44) and (42),D · ˆ Z ⊗ ˆ ρ ˆ n ˆ ρ = ˆ Z (cid:18)(cid:18) −
11 + ˆ ρ ˆ n · ˆ a (cid:19) ρ − ˆ ρ ˆ n · n ˆ ρ (cid:19) == − ˆ Z (1 + ˆ ρ ˆ n · ˆ a )ˆ ρ = − ˆ Z ˆ ρ + ˆ Z (ˆ n · ˆ a )(1 + ˆ ρ ˆ n · ˆ a )ˆ ρ , in other words, − ˆ Z ˆ ρ = D · ˆ Z ⊗ ˆ n ˆ ρ − ˆ Z (ˆ n · ˆ a )(1 + ˆ ρ ˆ n · ˆ a )ˆ ρ . Thus, we arrive at( − D · tm T [ r ] | φ ) == lim R → Z M \ H I,R D · T φ + e π ˆ Z ⊗ ˆ n ˆ ρ ! − e π ˆ Z ( n · ˆ a )(1 + ˆ ρ ˆ n · ˆ a )ˆ ρ ! . !!12he integral of the second term in the parameterization around the worldline becomes a multiple of Z I R I Z R ρ Z S ( ) Z ( s )( L ( s ) · n ) · ¨ r ( s ) ρ d n dρ d s which is zero because of (54).By Gauss’ theorem, the integral of the first term can be transformed to anintegral on the corresponding world tube in such a way that the tube is directed‘inwards’ i.e. the normal vector is − ˆ n :( − D · tm T [ r ] | φ ) = lim R → Z M \ H I,R D · T φ + e π ˆ Z ⊗ ˆ n ˆ ρ ! !! == − lim R → Z P I,R T φ + e π ˆ Z ⊗ ˆ n ˆ ρ ! · ˆ n dλ P I,R !! == − lim R → Z I R Z S (0) ... ( n , s ) d n d s ! (45)where the integrand is ( T φ ) (cid:0) r ( s ) + R n ( s ) (cid:1) + e π Z ( s ) ⊗ n ( s ) R ! · n ( s )(1 + R n ( s ) · ¨ r ( s )) (46)whith n ( s ) := L ( s ) n .Then Z ⊗ nR · n (1 + R n · ¨ r ) = (4 ˙ r ⊗ ˙ r + ) · D φ [ r ]6 (cid:18) R + n · ¨ rR (cid:19) , and on the base of the radial expansion (27)( T ( r + R n ) · n (1 + R n · ¨ r ) == e π (cid:16) − n R + ˙ r R ++ | ˙ r | n + | ˙ r | ˙ r − ( n · ˙ r ) ˙ r − ... rR ++ Ordo( R ) R (cid:17) . This, together with the expansion φ ( r + R n ) = φ ( r ) + R n · D φ [ r ] + R Ordo( R ) , give that the integral (45) of the Ordo terms in (46), because of the three timesdifferentiability of r , is less than a multiple of R , so their limit is zero when R tends to zero; moreover, the integrals of the terms linear and trilinear in n in(46) are zero. 13he integral of the terms independent of n results in a multiplication by 4 π ,the integral of the terms bilinear in n results in a multiplication by π ( + ˙ r ⊗ ˙ r ).Thus, D φ [ r ] will be multiplied by − π + ˙ r ⊗ ˙ r R + 4 π r ⊗ ˙ r + R = 4 π R ˙ r ⊗ ˙ r and φ ( r ) will be multiplied by π R ¨ r + (cid:0) | ¨ r | ˙ r − ... r (cid:1) . Since4 π R (cid:16) ˙ r ( s )( ˙ r ( s ) · D φ [ r ( s )]) + 4 π R ¨ r ( s ) (cid:17) = 4 π R dd s (cid:0) ˙ r ( s ) φ ( r ( s )) (cid:1) (47)and its integral is zero because the support of φ is in H I,R I , it remains only − e π Z I π
23 ( | ¨ r ( s ) | ˙ r ( s ) − ... r ( s )) φ ( r ( s )) d s which gives the desired result by | ¨ r | = − ˙ r · ... r . (cid:4) The usual formula (38) of the radiation reaction force is obtained in a mathe-matically exact way, without unjustified application of Gauss–Stokes theorem,of Taylor expansion and without a doubtful limit to zero.It is an important fact that the self-force is obtained for a given world linefunction r . Consequently, its physical meaning is the following: If the world line function is r then the self-force is π e ( ˙ r ∧ ... r ) · ˙ r . Accordingly, if the force f is necessary to get a given r without radiation i.e. m ¨ r = f ( r, ˙ r ) would valid without radiation, then besides f the force oppositeto the self-force must be applied for getting the desired world line function r .An actual example is when an elementary particle is revolved in a cyclotronby a homogeneous static magnetic field. In spacetime formulation: there isgiven a constant electromagnetic field F and the world line without radiationwould satisfy m ¨ r = e F ˙ r . Then... r = e m F · F · ˙ r, ˙ r · ... r = e m ( ˙ r · F ) · ( F · ˙ r ) = − e m ( F · ˙ r ) · ( F · ˙ r ) , thus the self-force is 14 π e m (cid:0) F · F · ˙ r − | F · ˙ r | ˙ r (cid:1) ; (48)to keep the particle on a prescribed orbit, the opposite of the above force mustbe applied besides the magnetic field,For better seeing, let us take the standard inertial frame u in which thecyclotron is at rest. Then the u -spacelike component of F , for which F · u = 0,is the magnetic field B , a three dimensional antisymmetric tensor.Using the relative velocity v := ˙ r − u · ˙ r − u with respect to the inertial frame,we have F · ˙ r = F · (cid:18) ˙ r − u · ˙ r − u (cid:19) ( − u · ˙ r ) = B · v p − | v | π e m ( + u ⊗ u ) − u · ˙ r (cid:0) F · F · r − | F · ˙ r | ˙ r (cid:1) = 14 π e m (cid:18) B · B · v − | B · v | − | v | v (cid:19) . Introducing the customary axial vector B corresponding to B we have | B | = | B | , B · v = v × B , B · B · v = ( v × B ) × B . If the relative velocity is orthogonalto B then B · B · v = −| B | v , | B · v | = | B | | v | and the self-force relative tothe inertial frame − π e m | B | v − | v | (49)brakes the motion; the same opposite force must be applied additionally to avoidbraking and to ensure that the particle moves on the prescribed orbit. Besides the unjustified application of Gauss–Stokes theorem, of Taylor expan-sion and a doubtful limit to zero there is another, a fundamental error in usualderivations of the LAD equation which is clearly seen now.
The self-force π e ( ˙ r ∧ ... r ) · ˙ r is valid for a given world line function r ;therefore it is unjustified to put it in a Newtonian-like equation to obtain theLAD equation which would serve to determine r . Thus, the LAD equation isa misconception and its pathological properties are not surprising .The well known fundamental problem ([24]) is that both electrodynamicsand mechanics in their known forms are theories of action :– the Maxwell equations define the electromagnetic field F produced by a given world line function r of a particle,– the Newtonian equation defines the world line function r of a particle ina given force (e.g. the Lorentz force in an extraneous electromagnetic field F ),and at present we have not a well working theory of interaction which woulddefine both the electromagnetic field F and the world line function r together.To see the problem more closely, let us consider a point charge under theaction of a given force f . Then according to the general conception (see e.g.[9]) – formulated in our language – the Newtonian equation is replaced with thebalance equation − D · ( T m [ r ] + T e [ F ]) + f ( r, ˙ r ) λ Ran r = 0 (50)where T m [ r ] is the tensor Distribution of mechanical energy-momentum con-structed somehow of r and T e [ F ] is the tensor Distribution of electromagneticenergy-momentum constructed somehow of F .It seems evident that T m [ r ] = m ˙ r ⊗ ˙ rλ Ran r and then the balance equation m ¨ rλ Ran r = f ( r, ˙ r ) λ Ran r − D · T e [ F ] (51)together with the Maxwell equationsD · F = ˙ rλ Ran r , D ∧ F = 0 (52)would describe that r and F determine each other mutually.It is not sure that there is a conveniently defined T e [ F ] with which (51) and(52) form a consistent system of equations.It is sure, however, that T e [ F ] cannot be replaced with tm T [ r ] which is validfor a given r ; the LAD ‘equation’ is obtained by committing this replacement.15 .3 Equation, equality At last, let the LAD ‘equation’ be looked at from another point of view. Namely,we have to distinguish clearly between an equation and an equality, both denotedby the same symbol =.An equation is a definition : it defines a set (the set of its solutions).An equality is a statement : it states that two sets are the same.Let S ( e, m, f ) be the set of processes of a point charge e with mass m underthe action of a force f .Because of the non-physical run-away solutions, the LAD ‘equation’ doesnot define S ( e, m, f ), thus it is not an equation from a physical point of view.The various attempts to find satisfactory conditions to exclude non-physicalsolutions suggest in this context that, in fact, the LAD ‘equation’ would be anequality, stating that S ( e, m, f ) is a subset of all the solutions.Unfortunately, this is not true, either, as it is shown in the excellent paper[21]: f being the Coulomb force of a point charge, no physical motion in radialdirection is a solution of the LAD ‘equation’. The following example can illustrate the problem of describing the interaction.Let two particles with the same charge and mass rest in the space of a standardinertial frame. They repulse each other by the Coulomb forces which are com-pensated by some constraint to attain the rest. If the constraint is left off at aninstant then the particles start to move. Figure 1 shows what happens (timepasses from the left to the right). ✒❘ ✒❘ ✒❘ ✒❘ ✒❘✠ ✒■ ❘ ✒✠ ❘■ qq Figure 1: Interaction of two point chargesWhile resting, the particles send Coulomb-actions to each other in the pos-itive light cone, represented by sparsely dotted lines. At the instant when theconstraint is left off, the acceleration of each particles is determined by theCoulomb force of the other. Then for some time the acceleration of a parti-cle is determined by the Coulomb force of the other particle and the radiationself-force, and then it sends some action, different from the Coulomb force tothe other particle; this is represented by the densely dotted lines. We have noformula for this acceleration and action. Later the situation becomes more com-plicated: it is not known which action and self-force give rise to an acceleration.16 .5 Final remarks
It is emphasized that the retarded field is meaningful only for a given worldline, therefore it cannot be used for determining the world line.The various attempts to obtain a physically acceptable equation with con-tinuum charge distributions instead of a point charge suffer from the same faultof using the retarded field (and sometimes the advanced one, too).
Appendix
10 Measures
Elementary notions of (vector) measures and integration by them can be foundin [25].The Lebesgue measure of an affine space is the measure given by the usualintegration (customary for R n ).The Lebesgue measure λ H of a submanifold H in spacetime is given by theformula Z H f dλ H := Z Dom p f ( p ( ξ )) p | det( D p [ ξ ] ∗ D p [ ξ ]) | dξ where p is an arbitrary parameterization of the submanifold i.e. a continuouslydifferentiable injective map defined in an affine space, having H as its range and D p [ ξ ], the derivative of p at ξ is injective for all ξ .In particular, for a world line function r , the Lebesgue measure of the cor-responding world line is λ Ran r which gives the integration along the world line.In particular, it is considered a Distribution in spacetime,( λ Ran r | φ ) = Z T φ ( r ( s )) d s (53)for arbitrary test functions φ .
11 Three dimensional integration
Let u be an absolute velocity in spacetime and S the three dimensional Eu-clidean vector space Lorentz orthogonal to u ; S ( ) is the unit sphere aroundzero, its elements are denoted by n .In the usual parameterization given by the angles ϑ, ϕ relative to an or-thonormal bases, the Lebesgue measure of S ( ) is obtained by the symbolicformula d n ∼ sin ϑ dϑ dϕ with which we easily find the equalities Z S ( ) n d n = 0 , Z S ( ) n ⊗ n ⊗ n d n = 0 , (54) Z S ( ) n ⊗ n d n = 4 π + u ⊗ u ) (55)17here is the identity map of spacetime vectors; and so on, the integral of aneven tensor products is a non-zero tensor, the integral of an odd tensor productsis zero.The radial coordinates of q ∈ S are ρ ( q ) := | q | ≥ , n ( q ) := q | q | ∈ S ( ) . Then the radial parameterization of S is( ρ, n ) ρ n . We applied the usual ambiguity that ρ and n denote functions in somerespect and variables in an other respect.The other well known integral formula Z S f ( q ) d q = ∞ Z ρ Z S ( ) f ( ρ n ) d n dρ (56)will be often used, too.
12 Pole taming
There is a well defined method to attach a Distribution to a locally non-integrable function, having a pole singularity in a Euclidean space. Originallythe method was called regularization ([16]). Since the Distribution obtained inthis way is not regular, I call the method pole taming .Only a special case is necessary for us.For the function ρ ( q ) := | q | ( q ∈ S ), ρ m is not locally integrable if m >
0. For m being an even positive integer, a Distribution can be attached toit by pole taming defined as follows.For a test function ψ let q T ( m − ψ ( q ) denote the Taylor polynomial oforder m − ψ − T ( m − ψ is a continuousfunction of order ρ m , therefore ψ − T ( m − ψ ρ m is integrable there. Outside the supportof ψ , however, the term containing the ( m − m is even; in the radial parameterization (11), D ( m − ψ ( )( ρ n , . . . , ρ n ) isa linear function of an odd tensor product of n , thus, integrating in the ordergiven by (56), this term will drop out and an integrable function remains.We repeat for avoiding misundertanding: it is known that, for an integrablefunction, the order of integration by n and ρ can be interchanged in (56). Ifa function is not integrable, it can occur that in one of the orders the integralexists (but in the other order not). Here we take the advantage that the integralexists in the given order. In this way the pole taming of ρ m results in theDistribution defined by (cid:18) tm 1 ρ m (cid:12)(cid:12)(cid:12) ψ (cid:19) := ∞ Z ρ Z S ( ) ψ ( ρ n ) − T ( m − ψ ( ρ n ) ρ m d n dρ ! (57) Áron Szabó proposed this name m = 0 we mean the pole taming formally by taking the correspondingregular Distribution.Later, for the sake of brevity, we also use the formula (cid:18) tm 1 ρ m (cid:12)(cid:12)(cid:12) ψ (cid:19) := Z § ψ ( q ) − T ( m − ψ ( q ) | q | m d q !! (58)where the double exclamation mark calls attention to that the integral must betaken in radial parameterization and in the given order.We have to extend the notion of pole taming as follows. For a k -th tensorpower n ⊗ k := n ⊗ n ⊗ . . . n we put tm n ⊗ k ρ m (cid:12)(cid:12)(cid:12) ψ ! := ∞ Z ρ Z S ( ) n ⊗ k (cid:0) ψ ( ρ n ) − T ( m − ψ ( ρ n (cid:1) ρ m d n dρ ! (59)which makes sense if m − k is odd; therefore– if m is even then k must be even, too,– if m is odd then k must be odd, too.
13 World rolls and world tubesaround a world line
Let r be a twice continuously differentiable world line function. The Lorentzboost (see [1]) L ( s ) := + ( ˙ r (0) + ˙ r ( s )) ⊗ ( ˙ r (0) + ˙ r ( s ))1 − ˙ r (0) · ˙ r ( s ) − r ( s ) ⊗ ˙ r (0)maps ˙ r (0) into ˙ r ( s ).Further, we use the notations u := ˙ r (0) and S for the linear subspaceconsisting of vectors Lorentz orthogonal to u ; then { L ( s ) · q | q ∈ S } is thesubspace consisting of vectors Lorentz orthogonal to ˙ r ( s ). Then p : T × S → M , ( s , q ) r ( s ) + L ( s ) · q (60)is a parameterization of a neighbourhood of the world line.This is a reformulation of a usual setting: ( s , q ) correspond to the so calledFermi coordinates.To be precise, the following details are necessary. p is evidently continuouslydifferentiable and its derivative D p [ s , q ] = (cid:16) ˙ r ( s ) + ˙ L ( s ) · q L ( s ) | S (cid:17) (61)( L ( s ) | S is the restriction of L ( s ) to S ) is injective for all s and for q = ,thus, according to the inverse function theorem and the fact that p ( s , ) = r ( s ),there is an open subset in T × S which contains T × { } where both p and allthe values D p [ · ] are injective. 19et I be a bounded and closed time interval. Then { r ( s ) | s ∈ I } is acompact set and each of its points has a neighbourhood in the range of theparameterization; thus, { r ( s ) | s ∈ I } can be covered by finite many suchneighbourhoods. As a consequence, there is an R I > { r ( s ) | s ∈ I } of the world line is contained in H I,R := { r ( s ) + L ( s ) · q | s ∈ I, | q | < R } for all 0 < R ≤ R I . H I,R is called a world roll of length I and radius R . With the notations ρ ( q ) := | L ( s ) · q | = | q | , n ( s , q ) := L ( s ) · q ρ ( q ) , n ( q ) := q ρ ( q ) , (62)the parameterization of H I,R can be rewritten in the form r ( s ) + ρ ( q ) n ( s , q ) = r ( s ) + L ( s ) · n ( q ) . (63)The cylinder around H I,R , P I,R := { r ( s ) + L ( s ) · q | s ∈ I, | q | = R } is called the corresponding world tube .The tangent space of P I,R at the point r ( s ) + L ( s ) · q is { (cid:0) ˙ r ( s ) + ˙ L ( s ) · q (cid:1) t + L ( s ) · h | t ∈ T , h ∈ S , h · q = 0 } . For q ∈ S we have ( L ( s ) · q ) · ˙ r ( s ) = , ( L ( s ) · q ) · ( L ( s ) · h ) = q · h = 0and ( ˙ L ( s ) · q ) · ( L ( s ) · q ) = ∂∂ s | L ( s ) · q | = 0; thus, (cid:16)(cid:0) ˙ r ( s ) + ˙ L ( s ) · q (cid:1) t + L ( s ) · h (cid:17) · ( L ( s ) · q ) = 0and we find that the outward normal vector of P I,R at the point r ( s )+ ρ ( q ) n ( s , q )is n ( s , q ) . (64)
14 Integration in world rolls
Integrals in a world roll will be computed in the parameterization (60) for whichwe have p | det D p [ s , q ] ∗ D p [ s , q ] | = 1 + ρ ( q ) n ( s , q ) · ¨ r ( s ) . (65)To show it, for the sake of brevity, we introduce the notation z ( s , q ) := ˙ r ( s ) + ˙ L ( s ) · q . Then from (61) we get D p [ s , q ] ∗ D p [ s , q ] = (cid:18) z ( s , q ) (cid:0) L ( s ) | S (cid:1) ∗ (cid:19) · (cid:0) z ( s , q ) L ( s ) | S (cid:1) = (66)= (cid:18) z ( s , q ) · z ( s , q ) z ( s , q ) · L | ( s ) S ( L ( s ) | S ) ∗ · z ( s , q ) S (cid:19) . (67)20mitting the variables for the sake of perspicuity, we can write L | S = L · ( + u ⊗ u ), thus ( L | S ) ∗ = ( + u ⊗ u ) · L ∗ ; accordingly, ( + u ⊗ u ) · L ∗ · z = L ∗ · z + u (( u · L ∗ ) · z which equals z · L + u ( ˙ r · z ) = z · ( L · ( + u ⊗ u )).The block matrix (67) is symmetric, its determinant is z · z − ( z · L | S ) · (( L | S ) ∗ · z ). The second term here equals ( z · L ) · ( z · L ) + 2( z · L ) · u ( ˙ r · u ) − ( ˙ r · u ) = z · z + ( ˙ r · u ) , thus we havedet( D p ∗ D p ) = − ( ˙ r · z ) = − (1 + ρ n · ¨ r ) ;the last equality comes from˙ r ( s ) · z ( s , q ) = − r ( s ) · ˙ L ( s ) · q = − ∂∂ s (cid:0) ˙ r ( s ) · L ( s ) · q (cid:1) − ¨ r ( s ) · L ( s ) · q and the quantity in the parenthesis in the middle term on the right hand sideis zero.Finally, if ρ ( q ) is ‘sufficiently small’ then 1 + ρ ( q ) n ( s , q ) · ¨ r ( s ) is positive,thus it is positive for all the possible ( s , q ) because D p [ s , q ] is injective, thedeterminant cannot be zero.Further, because of n ( s , q ) = L ( s ) n ( q ), using the radial parameterization(11) of S we have Z H I,R . . . ( x ) dx = R Z ρ Z S (0) . . . (cid:0) r ( s ) + ρ L ( s ) n (cid:1) (1 + ρ ¨ r ( s ) · L ( s )) d n dρ. (68)
15 Acknowledgement
I am grateful to T.Gruber for checking all the not too simple formulas and toP.Ván for helping me to compose this article.
References [1] T. Matolcsi:
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