Electromagnetic self-force of a point charge from the rate of change of the momentum of its retarded self-field
aa r X i v : . [ phy s i c s . c l a ss - ph ] J u l ELECTROMAGNETIC SELF-FORCE OF APOINT CHARGE FROM THE RATE OFCHANGE OF THE MOMENTUM OF ITSRETARDED SELF-FIELD
V. HNIZDO ∗ and G. VAMAN † The self-force of a point charge moving on a rectilinear trajectory is obtained,with no need of any explicit removal of infinities, as the negative of the timerate of change of the momentum of its retarded self-field.
Keywords : self-force, Lorentz-Abraham-Dirac equation, delta function
1. Introduction
The problem of the self-force, or radiation reaction, of a non-uniformly movingcharged particle has a long history. The Abraham-Lorentz evaluation of the self-force [1], dating to the early 1900’s, assumed the particle’s charge distribution tobe spherically symmetric, of radius a , and obtained the self-force as an expansionin powers of a (including the power a − ). Dirac [2] was the first to obtain theradiation-reaction force assuming at the outset the particle to be point-like, ina relativistic calculation that used conservation of energy and momentum, and,unlike the Abraham-Lorentz evaluation, both the retarded and advanced self-fieldsof the charge. For a recent survey of the history of this topic see Ref. [3].In this paper, we present a novel calculation of the self-force of a point charge,using only its retarded self-field. As in the account of Jackson of the Abraham-Lorentz evaluation ([4], Sec. 16.3), our starting points are (i) the conservation ofmomentum in the system consisting of a single charged particle and an externalelectromagnetic field, d~p mech dt + d ~Gdt = 0 , (1) ∗ † Institute of Atomic Physics, P. O. Box MG-6, Bucharest, Romania (vaman@ifin.nipne.ro) ~p mech is the purely mechanical, or “bare”, momentum of the particle, and ~G is the momentum of the total electromagnetic field (external plus the particle’sself-field), and (ii) the relation d~p mech dt = ~F ext + ~F s . (2)Here, ~F ext and ~F s are the Lorentz forces on the charged particle due to the ex-ternal field and the particle’s self-field, respectively (in the original calculations ofAbraham and Lorentz, the particle’s momentum was taken to be entirely electro-magnetic, entailing that ~p mech = 0, and so the total force ~F ext + ~F s was assumedto vanish). Equations (1) and (2) can be combined to write − d ~Gdt = − ddt ( ~G − ~G s ) − d ~G s dt = ~F ext + ~F s , (3)where ~G s is the electromagnetic momentum of the self-field only. Equation (3)suggests expressing the self-force ~F s and the external force ~F ext separately as ~F s = − d ~G s dt , (4) ~F ext = − ddt ( ~G − ~G s ) . (5)We shall calculate the self-force ~F s using the ansatz (4), assuming for simplicitythat the particle moves on a rectilinear trajectory. Remarkably, the calculation willturn out to yield the exact space part of the radiation-reaction 4-force of Dirac’sequation [2] (now usually called the Lorentz-Abraham-Dirac (LAD) equation toreflect fully its origins), with no need of renormalization or any other explicitremoval of infinities.
2. Calculation of the self-force
We consider a point particle of charge e , moving on a rectilinear trajectory w ( t ),say along the x -axis. Its charge and current densities are then given by ρ ( ~r, t ) = eδ ( ~r − w ( t )ˆ ~x ) , ~j ( ~r, t ) = ρ ( ~r, t ) ˙ w ( t )ˆ ~x, (6)where overdot denotes differentiation with respect to time, and δ is the Dirac deltafunction. We assume that the charge density can be expanded as the Taylor series: ρ ( ~r, t ) = e ∞ X n =0 ( − n n ! w n ( t ) ∂ nx δ ( ~r ) , (7)2here ∂ nx denotes the partial differentiation with respect to x of order n ; w n ( t ) ≡ ( w ( t )) n . The Taylor-series expansion suggests that we consider only small oscilla-tions about an equilibrium position, but our final results will not depend on anyamplitude of w ( t ).Using Eqs. (6) and (7), the retarded scalar and vector potentials of the movingcharge are calculated to be: φ ( ~r, t ) = Z d~r ′ ρ ( ~r ′ , t − | ~r − ~r ′ | /c ) | ~r − ~r ′ | = e ∞ X n =0 ( − n n ! ∂ nx w n ( t − r/c ) r , (8) ~A ( ~r, t ) = 1 c Z d~r ′ ~j ( ~r ′ , t − | ~r − ~r ′ | /c ) | ~r − ~r ′ | = ec ∞ X n =0 ( − n n ! ∂ nx w n ( t − r/c ) ˙ w ( t − r/c ) r ˆ ~x. (9)Here, we used the facts that ∂ nx ′ δ ( ~r ′ ) = δ ( n ) ( x ′ ) δ ( y ′ ) δ ( z ′ ) and Z ∞−∞ dx ′ f ( ~r − ~r ′ ) δ ( n ) ( x ′ ) = ( − n ∂ nx ′ f ( ~r − ~r ′ ) | x ′ =0 = ∂ nx f ( ~r − ~r ′ ) | x ′ =0 . (10)The y - and z -components of the retarded electric and magnetic self-fields of thecharge, ~E ( ~r, t ) = −∇ φ ( ~r, t ) − c ∂∂t ~A ( ~r, t ) , ~B ( ~r, t ) = ∇ × ~A ( ~r, t ) , (11)are obtained using Eqs. (8) and (9) to be E y,z ( ~r, t ) = e ∞ X n =0 ( − n +1 n ! ∂ y,z ∂ nx w n ( t − r/c ) r , (12) B y,z ( ~r, t ) = ec ∞ X n =0 ( − n n ! ˜ ∂ z,y ∂ nx w n ( t − r/c ) ˙ w ( t − r/c ) r , (13)where ˜ ∂ z = ∂ z and ˜ ∂ y = − ∂ y . 3 .2 Self-field momentum The self-fields create an electromagnetic-self-field momentum ~G s ( t ) = 14 πc Z d~r ~E ( ~r, t ) × ~B ( ~r, t )= 14 πc Z d~r [ E y ( ~r, t ) B z ( ~r, t ) − E z ( ~r, t ) B y ( ~r, t )] ˆ ~x ; (14)the y - and z -components of ~G s ( t ) can be seen to vanish already on account ofsymmetry (the particle is moving only along the x -axis).We evaluate the integral in (14) in the momentum space: ~G s ( t ) = 14 πc π Z d~k h E y ( ~k, t ) B z ( − ~k, t ) − E z ( ~k, t ) B y ( − ~k, t ) i ˆ ~x, (15)where the Fourier transforms are defined by f ( ~k, t ) = Z d~rf ( ~r, t ) e i~k · ~r . (16)Using Eqs. (12) and (16), we express the Fourier transforms E y,z ( ~k, t ) as E y,z ( ~k, t ) = e ∞ X n =0 n ! Z d~r (cid:16) ∂ y,z ∂ nx e i~k · ~r (cid:17) w n ( t − r/c ) r = e ∞ X n =0 i n +1 n ! k y,z k nx Z ∞ dr r w n ( t − r/c ) Z d Ω e ikr cos θ = 4 πe ∞ X n =0 i n +1 n ! k y,z k nx k Z ∞ dr sin( kr ) w n ( t − r/c ) , (17)where, in the 1 st line, we integrated by parts; k x,y,z are the Cartesian componentsof ~k in the momentum space and k = | ~k | . Similar calculation yields, using Eq.(13): B y,z ( − ~k, t ) = 4 πec ∞ X n =0 ( − i ) n +1 n ! ˜ k z,y k nx k Z ∞ dr sin( kr ) w n ( t − r/c ) ˙ w ( t − r/c ) , (18)where ˜ k z = − k z and ˜ k y = k y . Using Eqs. (17) and (18), the self-field momentum415) is now calculated as follows: ~G s ( t ) = e π c ∞ X n =0 ∞ X p =0 ( − p +1 i n + p +2 n ! p ! Z ∞ dk Z d Ω ~k ( k y + k z ) k n + px × Z ∞ dr sin( kr ) w n ( t − r/c ) Z ∞ dr ′ sin( kr ′ ) w p ( t − r ′ /c ) ˙ w ( t − r ′ /c )ˆ ~x = 4 e πc ∞ X κ =0 2 κ X n =0 ( − n + κ n !(2 κ − n )!(2 κ + 1)(2 κ + 3) Z ∞ dk k κ +2 × Z ∞ dr Z ∞ dr ′ sin( kr ) sin( kr ′ ) w n ( t − r/c ) w κ − n ( t − r ′ /c ) ˙ w ( t − r ′ /c )ˆ ~x = 2 e c ∞ X κ =0 2 κ X n =0 ( − n +1 n !(2 κ − n + 1)!(2 κ + 1)(2 κ + 3) c κ +3 × Z ∞ dr w n ( t − r/c ) d κ +3 dt κ +3 w κ − n +1 ( t − r/c )ˆ ~x. (19)Here, in the 2 nd equality, we used the result Z d Ω ~k k y,z k mx = (cid:26) πk m +2 / ( m + 1)( m + 3) , m even0 , m odd , (20)and changed the summation indices using n + p = 2 κ in view of the fact that onlythe terms with n + p even contribute to ~G s ( t ); in the last equality, we used thegeneralized-function identity Z ∞ dk k m sin( kx ) sin( ky ) = π i m h δ ( m ) ( x − y ) − δ ( m ) ( x + y ) i , m even , (21)where only the first term contributed in the radial integration, and then the iden-tity d m f ( t − r/c ) dr m = (cid:18) − c (cid:19) m d m f ( t − r/c ) dt m . (22)Transforming the integration variable r to t ′ = t − r/c , the self-field momentum(19) can now be expressed as ~G s ( t ) = − Z t −∞ dt ′ F s ( t ′ )ˆ ~x, (23) Relation (21) can be obtained from R ∞ dk k m cos( kx ) = πi m δ ( m ) ( x ), m even (cid:0) see Ref.[5], p. 43, Table 1; note that the Fourier transform of f ( x ) is there defined as g ( y ) = R ∞−∞ dx f ( x ) exp( − πiyx ), and sin( kx ) sin( ky ) = cos( k ( x − y )) − cos( k ( x + y )) (cid:1) . F s ( t ′ ) (the subscript anticipates relation (4)) is a force given by F s ( t ) = 2 e ∞ X k =0 (2 k + 2) c − (2 k +3) (2 k + 1)(2 k + 3)! k X n =0 (cid:18) k + 1 n (cid:19) ( − w ( t )) n d k +3 w k − n +1 ( t ) dt k +3 ; (24) (cid:0) k +1 n (cid:1) = (2 k + 1)! / (2 k + 1 − n )! n ! is a binomial coefficient. We now proceed to evaluate the force (24) in closed form. Using the identity df ( t ) dt = − c df ( t − r/c ) dr (cid:12)(cid:12) r =0 , (25)we can write (24) as F s ( t ) = 2 e ∞ X k =0 ( − k +3 (2 k + 2)(2 k + 1)(2 k + 3)! k X n =0 (cid:18) k + 1 n (cid:19) ( − w ( t )) n d k +3 w k − n +1 ( t − r/c ) dr k +3 (cid:12)(cid:12) r =0 . (26)But k X n =0 (cid:18) k + 1 n (cid:19) ( − w ( t )) n w k − n +1 ( t − r/c )= w k +1 ( t − r/c ) k X n =0 (cid:18) k + 1 n (cid:19) (cid:18) − w ( t ) w ( t − r/c ) (cid:19) n = w k +1 ( t − r/c ) "(cid:18) − w ( t ) w ( t − r/c ) (cid:19) k +1 − (cid:18) − w ( t ) w ( t − r/c ) (cid:19) k +1 = ( w ( t − r/c ) − w ( t )) k +1 + w k +1 ( t ) , (27)where we used in the 3 rd line the binomial theorem. Using now (27) in (26), weget F s ( t ) = 2 e ∞ X k =0 ( − k +3 (2 k + 2)(2 k + 1)(2 k + 3)! d k +3 ( w ( t − r/c ) − w ( t )) k +1 dr k +3 (cid:12)(cid:12) r =0 , (28)since d k +3 w k +1 ( t ) /dr k +3 = 0 for all k ≥
0. We can use here for the higher-orderderivatives the formula d m dx m f k ( x ) = X j + ··· + j k = m (cid:18) mj , . . . , j k (cid:19) f ( j ) ( x ) . . . f ( j k ) ( x ) , (29)6hich is the differentiation analogue of the multinomial theorem. This transforms(28) into F s ( t ) =2 e ∞ X k =0 ( − k +3 (2 k + 2)(2 k + 1)(2 k + 3)! X j + ··· + j k +1 =2 k +3 (cid:18) k + 3 j , . . . , j k +1 (cid:19) × d j ( w ( t − r/c ) − w ( t )) dr j (cid:12)(cid:12) r =0 . . . d j k +1 ( w ( t − r/c ) − w ( t )) dr j k +1 (cid:12)(cid:12) r =0 . (30)Utilizing that d j ( w ( t − r/c ) − w ( t )) dr j (cid:12)(cid:12) r =0 = (1 − δ j, ) (cid:18) − c (cid:19) j w ( j ) ( t ) , (31)Eq. (30) can be written more simply as F s ( t ) = 2 e ∞ X k =0 k + 2(2 k + 1) c k +3 X j + · · · + j k +1 = 2 k + 3 j , . . . , j k +1 > w ( j ) ( t ) j ! . . . w ( j k +1 ) ( t ) j k +1 ! . (32)We note that, because of the summation constraints on the orders j i of the deriva-tives w ( j i ) ( t ), only the factors ˙ w k ( t )... w ( t ) and ˙ w k − ( t ) ¨ w ( t ) are allowed in Eq.(32), their numbers being (cid:0) k +11 (cid:1) = 2 k + 1 and (cid:0) k +12 (cid:1) = (2 k + 1) k , respectively.Equation (32) can thus be still more simplified to read F s ( t ) = 2 e ∞ X k =0 k + 2(2 k + 1) c k +3 (cid:20) k + 13! ˙ w k ( t )... w ( t ) + (2 k + 1) k w ( t ) ˙ w k − ( t ) (cid:21) = 2 e ∞ X k =0 k + 1 c k +3 (cid:18)
13 ˙ w k ( t )... w ( t ) + k w k − ( t ) ¨ w ( t ) (cid:19) . (33)Using now the results P ∞ k =0 ( k + 1) x k = 1 / (1 − x ) and P ∞ k =0 ( k + 1)( k + 2) x k +1 =2 x/ (1 − x ) , 0 ≤ x <
1, the series in (33) can be summed, yielding F s ( t ) = 2 e c γ ¨ v ( t ) + 2 e c γ v ( t ) ˙ v ( t ) , γ = (1 − v ( t ) /c ) − / , (34)where we now write the velocity v ( t ) for ˙ w ( t ). This is exactly the space part ofthe relativistic LAD radiation-reaction force [6] ~F LAD = 2 e c γ (cid:20) ¨ ~v + 3 γ c ( ~v · ˙ ~v ) ˙ ~v + γ c ( ~v · ¨ ~v ) ~v + 3 γ c ( ~v · ˙ ~v ) ~v (cid:21) , (35)7hen that is adapted for a rectilinear motion. Returning to Eq. (23), we nowsee that if the limits t → −∞ of ˙ v ( t ) and ¨ v ( t ) vanish, i.e., if the particle moveduniformly in the remote past, then − ~G s ( t ) dt = F s ( t )ˆ ~x. (36)In words, the self-force calculated as the negative of the rate of change of thecharge’s self-field momentum is the LAD radiation-reaction force for the charge’sassumed motion.
3. Conclusions
We calculated the time rate of change of the retarded self-field momentum of apoint charge moving on a rectilinear trajectory of a remote-past constant velocity,and found that its negative equals the charge’s LAD radiation-reaction force. Therequisite integration was performed in the momentum space, where, apart fromthe standard practice of doing the angular integrations first, no renormalization orany other removal of infinities was required (unlike in the traditional treatment,in which the charge has a finite extension a and the self-force contains a termproportional to 1 /a , which diverges in the limit a → REFERENCES [1] H. A. Lorentz:
The Theory of Electrons , 2 nd ed., Stechert, New York 1916.[2] P. A. M. Dirac: Classical theory of radiating electrons, Proc. R. Soc. A ∼ mcdonald/examples/selfforce.pdf.84] J. D. Jackson: Classical Electrodynamics , 3 rd ed., John Wiley, New York1999.[5] M. J. Lighthill: An Introduction to Fourier Analysis and Generalized Func-tions , Cambridge University Press, Cambridge 1958.[6] F. Rohrlich: The dynamics of a charged sphere and the electron,
Am. J.Phys. (1997), 1051.[7] A. M. Steane: Reduced-order Abraham-Lorentz-Dirac equation and the con-sistency of classical electromagnetism, Am. J. Phys.83