Finite and divergent parts of the self-force of a point charge from its spherically averaged self-field
aa r X i v : . [ phy s i c s . c l a ss - ph ] A ug FINITE AND DIVERGENT PARTS OF THE SELF-FORCE OF APOINT CHARGE FROM ITS SPHERICALLY AVERAGED SELF-FIELD
V. Hnizdo ∗ and G. Vaman † Institute of Atomic Physics, P. O. Box MG-6, Bucharest, Romania
The electromagnetic self-force of a point charge moving arbitrarily on a rectilinear trajec-tory is calculated by averaging its retarded electric self-filed over a sphere of infinitesimalradius centered on the charge’s present position. The finite part of the self-force obtainedis the well-established relativistic radiation reaction, while its divergent part implies thepre-relativistic longitudinal electromagnetic mass of Abraham.
Keywords: self-force, radiation reaction, electromagnetic mass, finite part, divergent part
1. Introduction
Recently, we have obtained the relativistic Lorentz-Abraham-Dirac (LAD) radiation-reaction force [1] of a point charge moving arbitrarily on a rectilinear trajectory by calculatingthe negative of the time rate of change of the momentum of the charge’s retarded self-field [2].Remarkably, the non-manifestly-covariant calculation procedure employed in [2] involved noinertial-mass renormalization or any other explicit removal of infinities. The procedure thusamounted to the extraction of a finite part of the divergent integral for the point charge’sretarded-self-field momentum.In this paper, we revisit the same problem, but calculate the radiation-reaction force usingthe finite part of the average of the charge’s retarded self-field over a sphere of infinitesimalradius centered on its present (current) position. The calculation, unlike that in [2], yields alsothe divergent part of the self-force. It will turn out that the electromagnetic mass impliedby the divergent part is surprisingly the longitudinal electromagnetic mass that Abrahamobtained in his theory of a non-Lorentz-contractable electron before the advent of specialrelativity, but the finite part is again the relativistic LAD radiation reaction for rectilinearmotion.The procedure of calculating the radiation-reaction force of a point charge using a suitableaverage of its self-field has already been employed by several authors [3]–[6]. Teitelboim’s ∗ [email protected] † vaman@ifin.nipne.ro calculation [3] was manifestly covariant, yielding the full relativistic LAD equation of motion.Boyer [4] and Haque [5] used the procedure to obtain the radiation-reaction force of a chargein nonrelativistic uniform circular motion. Mansuripur [6] employed the average of what iseffectively half the difference of the retarded and advanced self-fields (following in this Dirac[1]) over a sphere of vanishingly small radius, centered on the charge’s retarded position inits instantaneous rest frame. The self-force thus obtained is the well-known nonrelativisticLorentz-Abraham radiation-reaction force, which on the Lorentz transformation to the lab-oratory frame becomes the exact LAD radiation reaction. It should be noted that, like thecalculation in [2], Mansuripur’s procedure did not require any renormalization of the charge’sinertial mass.
2. Averaging the retarded electric self-field (Finite part)
We consider a point charge e moving arbitrarily on a rectilinear trajectory w ( t ) along the x -axis. It is convenient to assume that w ( t )ˆ x vanishes at the given instant of time t , so thatthe charge is located at the origin r = 0 of the coordinate system at that time instant. Theradius vector r of observation will thus coincide with the displacement from the position ofthe charge at the given present time t .The x -component of the charge’s retarded electric field can be written as E x ( r , t ) = E x ( r , t ) + E x ( r , t ) , (1)where E x ( r , t ) = − ∂φ ( r , t ) ∂x = − e ∞ X n =0 ( − n n ! ∂ n +1 x w n ( t − r/c ) r , (2) E x ( r , t ) = − ∂A ( r , t ) c ∂t = − ec ∂∂t ∞ X n =0 ( − n n ! ∂ nx w n ( t − r/c ) ˙ w ( t − r/c ) r . (3)Here, φ ( r , t ) and A ( r , t )ˆ x are the pertinent retarded scalar and vector potentials, respectively; ∂ nx denotes the partial differentiation with respect to x of order n , w n ( · ) ≡ [ w ( · )] n and theoverdot indicates time differentiation. The retarded potentials used here were obtained in [2](see Eqs. (8) and (9) there) using the Taylor-series expansions of delta-function expressionsfor the charge and current densities of a moving point charge.To be able to calculate the average of the charge’s retarded electric field over a spherecentered on the present position of the charge, we shall need the following formulas, holdingfor the time t at which w ( t )ˆ x is assumed to vanish: d p dt p w k ( t ) = , p < kk ! ˙ w k ( t ) , p = k , (4) d k +1 dt k +1 w k ( t ) = k ( k + 1)!2 ˙ w k − ( t ) ¨ w ( t ) , (5) d k +2 dt k +2 w k ( t ) = k ( k − k + 2)!8 ˙ w k − ( t ) ¨ w ( t ) + k ( k + 2)!6 ˙ w k − ( t )... w ( t ) . (6)They can be inferred from the differentiation analogue of the multinomial theorem, d m dt m f k ( t ) = X j + ··· + j k = m m ! j ! · · · j k ! f ( j ) ( t ) · · · f ( j k ) ( t )= k X p =0 (cid:18) kk − p (cid:19) f p ( t ) X j + ··· + j k − p = mj ,..., j k − p > m ! j ! · · · j k − p ! f ( j ) ( t ) · · · f ( j k − p ) ( t ) , (7)written in the second line so that its application to f ( t ) = w ( t ) at the time when w ( t )ˆ x = 0is facilitated.Using the fact that w n ( t ) ˙ w ( t ) = ( n +1) − dw n +1 ( t ) /dt and expanding w n ( t − r/c ) in Taylorseries, the fields (2) and (3) can be written as E x ( r , t ) = − e ∞ X n =0 ∞ X i = n ( − n + i n ! i ! c i ( ∂ n +1 x r i − ) d i dt i w n ( t ) , (8) E x ( r , t ) = − ec ∞ X n =0 ∞ X i = n − ( − n + i ( n + 1)! i ! c i ( ∂ nx r i − ) d i +2 dt i +2 w n +1 ( t ) , (9)where the lower limits of the sums over i are adjusted in accordance with formula (4). Thederivatives ∂ n +1 x r i − and ∂ nx r i − can be calculated using Faa di Bruno’s formula [7], d m dx m g ( f ( x )) = X b +2 b + ··· + mb m = mb + ··· + b m = k m ! b ! · · · b m ! g ( k ) ( f ( x )) f (1) ( x )1! ! b · · · f ( m ) ( x ) m ! ! b m , (10)by choosing f ( x ) = x + y + z ( y and z thus being regarded as parameters) and g ( f ) = f ( i − / ; note that only the first two derivatives of thus defined f ( x ) are different from zero: f ′ ( x ) = 2 x and f ′′ ( x ) = 2. This way, we obtain ∂ n +1 ∂x n +1 r i − = n/ X b =0 ( − b +1+ n/ b +1 ( n + 1)!(2 b + 1)!( n/ − b )! (cid:18) − i (cid:19) b +1+ n/ r i − n − b − x b +1 , n even , (11) ∂ n +1 ∂x n +1 r i − = ( n +1) / X b =0 ( − b +( n +1) / b ( n + 1)!(2 b )!(( n + 1) / − b )! (cid:18) − i (cid:19) b +( n +1) / r i − n − b − x b , n odd , (12)where ( − i ) b +1+ n/ , etc. are the Pochhammer symbols ([8], Appendix I). The expressionsfor ∂ nx r i − are obtained by replacing n in (11) and (12) with n −
1, and changing “even” to“odd” and vice versa . These expressions facilitate angular integration of (8) and (9). Since Z d Ω x b = 4 π r b b + 1 (13)and the angular integral of x b +1 vanishes, we obtain for the angular averages of E x ( r , t ) and E x ( r , t ):¯ E x ( r, t ) = 14 π Z d Ω E x ( r , t )= − e ∞ X n =0 ∞ X i =2 n +1 ( − i + n n ( n + 1) (cid:0) i (cid:1) ! (cid:0) − i (cid:1) n +1 (2 n + 3)! i ! (cid:0) i − n − (cid:1) ! c i r i − n − d i dt i w n +1 ( t ) , (14)¯ E x ( r, t ) = 14 π Z d Ω E x ( r , t )= − e ∞ X n =0 ∞ X i =2 n − ( − i + n n (cid:0) i (cid:1) ! (cid:0) − i (cid:1) n (2 n + 1)(2 n + 1)! i ! (cid:0) i − n (cid:1) ! c i +2 r i − n − d i +2 dt i +2 w n +1 ( t ) . (15)In the limit r →
0, only the terms with 1 /r , 1 /r and r can contribute to angular averages(14) and (15). We consider first the finite parts of the limits r → i = 2 n + 3 and i = 2 n + 1 in (14) and (15),respectively, and using formulas (5) and (6):¯ E fin1 x ( t ) = f . p . lim r → ¯ E x ( r, t )= e ∞ X n =0 ( − n n ! c n +3 (2 n + 1)( − n − n +1 h n w n − ( t ) ¨ w ( t ) + 13 ˙ w n ( t )... w ( t ) i , (16)¯ E fin2 x ( t ) = f . p . lim r → ¯ E x ( r, t )= e ∞ X n =0 ( − n n ! c n +3 ( n + 1)(2 n + 3)( − n ) n h n w n − ( t ) ¨ w ( t ) + 13 ˙ w n ( t )... w ( t ) i . (17)The series over n can be summed using ∞ X n =0 ( − n n ! n (2 n + 1)( − n − n +1 β n − = − β (1 + β )(1 − β ) , (18) ∞ X n =0 ( − n n ! (2 n + 1)( − n − n +1 β n = − β (1 − β ) , (19) ∞ X n =0 ( − n n ! n ( n + 1)(2 n + 3)( − n ) n β n − = 2 β (5 + β )(1 − β ) , (20) ∞ X n =0 ( − n n ! ( n + 1)(2 n + 3)( − n ) n β n = 3 + β (1 − β ) , (21)where β = ˙ w ( t ) /c , and we obtain¯ E fin1 x ( t ) = − e β (1 + β )(1 − β ) ¨ w ( t ) c − e β (1 − β ) ... w ( t ) c , (22)¯ E fin2 x ( t ) = e β (5 + β )(1 − β ) ¨ w ( t ) c + e β (1 − β ) ... w ( t ) c . (23)The angular averages of the electric-field components E y and E z , along with those of themagnetic field, can be seen to vanish already on account of symmetry. The retarded-self-fieldforce of a point charge e moving arbitrarily along the x -axis, calculated using the finite partof the self-field’s average over a sphere of infinitesimal radius, centered on the charge’s presentposition, is thus F fin = e [ ¯ E fin1 x ( t ) + ¯ E fin2 x ( t )] ˆ x = 2 e c γ (cid:20) ¨ v ( t ) + 3 c γ v ( t ) ˙ v ( t ) (cid:21) ˆ x , (24)where we now write the velocity ˙ w ( t ) as v ( t ) and γ = (1 − v /c ) − / . It is exactly the samerelativistic radiation-reaction force as that obtained in [2].
3. Averaging the retarded electric self-field (Divergent part)
We now calculate the divergent parts of the fields (2) and (3) averaged over a sphere ofinfinitesimal radius centered on the charge. To do this, we take i = 2 n + 1, 2 n + 2 and 2 n − n in Eqs. (14) and (15), respectively. Since the Pochhammer symbols ( − n ) n +1 and (1 − n ) n vanish, the resulting terms with 1 /r vanish, too. Using formulas (5) and (6), we thus havefor the divergent parts of the limits r → E div1 x ( t ) = − lim r → (cid:18) e ¨ w ( t ) rc (cid:19) ∞ X n =0 ( − n n ( n + 1)(2 n + 1)( n + 1)! (cid:0) − n − (cid:1) n +1 (2 n + 3)! c n ˙ w n ( t ) , (25)¯ E div2 ,x ( t ) = − lim r → (cid:18) e ¨ w ( t ) rc (cid:19) ∞ X n =0 ( − n n ( n + 1)! (cid:0) − n (cid:1) n (2 n )! c n ˙ w n ( t ) . (26)Using now (cid:18) − n − (cid:19) n +1 = ( − n +1 (2 n + 1)!2 n +1 n ! , (cid:18) − n (cid:19) n = ( − n (2 n )!2 n n ! , (27)and summing the series over n , ∞ X n =0 ( n + 1)(2 n + 1)2 n + 3 β n = γ + 1 β (cid:18)
12 ln 1 + β − β − β − β (cid:19) , (28) ∞ X n =0 ( n + 1) β n = γ , (29)where γ = (1 − β ) − / and β = ˙ w ( t ) /c , we obtain for the divergent part of the force of theelectric self-field averaged over a sphere of infinitesimal radius: F div = e [ ¯ E div2 ,x ( t ) + ¯ E div2 ,x ( t )] ˆ x = lim r → e ˙ βrc ! β (cid:18)
12 ln 1 + β − β − β − β (cid:19) ˆ x . (30)Before the advent of special relativity, the so-called longitudinal electromagnetic mass, m em || , was taken to be the proportionality coefficient of the negative of the electromagneticself-force proportional to the acceleration of a charge in rectilinear motion ([9], sec. 27). At adistance r = a from the charge’s location, this coefficient is according to Eq. (30): m em || = e ac β (cid:18) β − β −
12 ln 1 + β − β (cid:19) . (31)Perhaps surprisingly, it happens to equal the mass m em || that Abraham obtained in his modelof the electron as a uniformly charged “rigid” (meaning today non-Lorentz-contractable)spherical shell of radius a ([10], p. 191, Eq. (117); see also [9], p. 39, Eq. (68)).We note here that the electromagnetic mass implied by the momentum of the well-knownHeaviside fields of a uniformly moving charged shell, in its rest frame spherical with radius a ,is given by the simple expression 2 e γ/ (3 ac ) [11], with which m em || of Abraham agrees onlywhen β <<
1. But it is interesting to note also that it can be shown (see Appendix) thatwhen the divergent self-field momentum G of a uniformly moving point charge is evaluatedby approaching the field-momentum density’s singularity on a spherical surface of radius a ,one obtains a rather more complicated expression: G = lim a → e ac β [(1 + 2 β ) β − (1 − β ) γ arctg( βγ )] v . (32)The “electromagnetic-mass” factor multiplying here the velocity v reduces to the value2 e γ/ (3 rc ) only when β <<
4. Concluding remarks
In a recent Comment [12] on the derivation of the LAD equation of motion by Dondera[13], we have advanced a conjecture concerning the Hadamard decomposition of the retarded-self-field momentum of an arbitrarily moving point charge. According to the conjecture, thedivergent electromagnetic mass implied by the self-field momentum of a uniformly movingpoint charge is given by lim ε → e γ/ (3 εc ). The requisite integration of the divergent in-tegral for the retarded-self-field momentum is thus understood there to be done so that thesingularity in the self-field-momentum density is approached on an oblate-spheroidal surfacewhose shape is congruent with that of a Lorentz-contracted sphere.Curiously, Rohrlich has attributed to Abraham an equation of motion ([14], Eqs. (2.7)–(2.8)) that implies the same decomposition of the retarded-self-field momentum as our con-jecture would for a charged rest-frame-spherical shell of radius a . Rohrlich’s attribution iscorrect for the equation’s radiation-reaction force, which Abraham was indeed the first to ob-tain in its full relativistic form ([10], p. 123, Eq. (85)), but not for its electromagnetic-inertiaterm since Abraham’s “rigid” shell of charge does not undergo the Lorentz contraction whenit is moving. Yaghjian has also attributed the same equation of motion ( sans the “bare-mass”term) to Abraham, but he has attributed it to Lorentz as well, saying that it was for a “rel-ativistically rigid” (i.e. Lorentz-contractable, in Lorentz’s parlance “deformable”) sphericalshell of charge ([15], pp. 11-12).Using a spherical average of the retarded electric self-field of a point charge moving arbi-trarily on a rectilinear trajectory, we obtained in this paper both the finite and divergent partsof its electromagnetic self-force. The finite part obtained is the same as the well-establishedrelativistic LAD radiation reaction, but the electromagnetic mass implied by the divergentpart differs at relativistic velocities markedly from that of the electromagnetic momentum ofa uniformly moving rest-frame-spherical shell of charge. This fact indicates that, in contrastto the divergent part of the self-force, the force of radiation reaction is not sensitive to thedetails of the models for the charge used and the methods of calculation employed.Abraham obtained his longitudinal electromagnetic mass from the electromagnetic momen-tum he calculated for a uniformly moving “rigid” spherical shell of charge, the exterior electricfield of which is not the well-known Heaviside electric field of a uniformly moving spheroidal,Lorentz-contracted shell of charge. The fact that our divergent part of the self-force impliesAbraham’s electromagnetic mass is indeed surprising.In closing, we should caution that electromagnetic mass is today a concept of limitedapplicability and usefulness. It is rooted in the electromagnetic programme of the turn ofthe 19th century, in which prominent physicists like Lorentz and Abraham attempted tosupplant classical mechanics by a field theory based on Maxwell’s electromagnetism and theether as the medium sustaining it. But after the advent of special relativity the programmebecame rather quickly outmoded, and its key concepts, the ether and electromagnetic mass,superseded (an illuminating account of this interesting episode in the history of physics isgiven in [16]). Appendix
The well-known Heaviside expressions for the fields of a point charge e moving with aconstant velocity v are given by E ( r , t ) = eγ (1 − β sin θ ) / r r , B ( r , t ) = ( β × E ) , (33)where β = v /c , γ = (1 − β ) − / and θ is the angle between v and r , the latter being thedisplacement from the present position of the charge. The momentum of these fields is G = 14 πc Z d r ( E × B )= 14 πc Z d r (1 − cos θ ) E β , (34)where account is taken of the fact that only the component of the vector product ( E × B )along the direction of β contributes to the integral. The square of the Heaviside electric fieldcan be written as E = γ (1 + β γ cos θ ) e r , (35)and thus the divergent momentum G calculated by approaching the singularity at r = 0 ona spherical surface of radius a is given by G = lim a → e γ πc Z ∞ a drr Z d Ω (1 − cos θ )(1 + β γ cos θ ) β = lim a → e γ ac Z − dξ (1 − ξ )(1 + β γ ξ ) β = lim a → e ac β [(1 + 2 β ) β − (1 − β ) γ arctg( βγ )] v . (36) [1] P. A. M. Dirac: Classical theory of radiating electrons, Proc. R. Soc. A (1938), 148.[2] V. Hnizdo and G. Vaman: Electromagnetic self-force of a point charge from the rate of change ofthe momentum of its retarded self-field,
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