The equation of motion for a radiating charged particle without self-interaction term

Volume 145, number 1

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26 March 1990

THE EQUATION OF MOTION FOR A RADIATING CHARGED PARTICLE WITHOUT SELF-INTERACTION TERM L. HERRERA”2

Departamento de Fisica, Facultad de Ciencias, Universidad de las Isla.s Baleares, Pa/ma deMallorca E-07071, Spain Received 19 January 1990; accepted for publication 29 January 1990 Communicated by J.P. Vigier

The motion of a radiating charged particle is studied from the point ofview of relativistic classical mechanics. Thus, the resulting equation of motion emerges from equating the total rate of change of momentum to the external force, without the introduction of a “self-force” term. Doing so, one is forced to abandon either one, or both, of the following restrictions: (a) the external force is non-dissipative, (b) the proper mass ofthe particle is constant. By abandoning (a) we obtain the Mo and Papas equation ofmotion, whereas allowing variations in the proper mass one is led, uniquely, to the Bonnor equation. A new equation of motion is proposed by abandoning both (a) and (b).

The equation of motion of a charged particle has been the subject of lengthy discussions since the pioneering work by Lorentz [1]. However, and in spite of all the work done on this problem, it is still a matter of controversy (the number of articles written on this subject is so large, that, besides papers directly related to this work, we shall only quote some “classical” books and some recent review articles, e.g. refs. [2—6]). The reasons for this situation may be found in the following facts: (a) Lack of experimental data on the radiation reaction effect to test the validity of the different proposed equations. (b) The most widely accepted equation of motion (Lorentz—Dirac) allows strange and undesirable effects such as runaway solutions and pre-acceleration, and does not provide a satisfactory explanation about the origin of radiation energy. To derive the equation of motion ofa charged partide there are, essentially, two different approaches. One of them consists in assuming that the radialion reaction effect is the result of the interaction of

2

On leave from Departamento de Fisica, Facultadde Ciencias, Universidad Central de Venezuela, Caracas, Venezuela. Postal address: Apartado 80793, Caracas 1080 A, Venezuela.

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the particle with its own radiation field (i.e. it is a self-interaction effect). This approach has been followed by the overwhelming majority of researchers, and the differences appearing between different contributions, based on this idea, are basically related to differences in the way of calculating the self-force. The second approach, which as far as we know has been adopted only by Bonnor [6] and by Mo and Papas [7] and which we shall follow here, is based on the idea that the equation of motion (as it should be in the context of classical mechanics) follows from equating the total rate of change of momentum to the external force, without the introduction of a selfforce term. As a consequence of this, one is forced to assume that either (a) the proper mass of the particle is variable (Bonnor), or (b) the external force is dissipative (Mo and Papas), or (c) both (a) and (b) (the equation presented in this work). In what follows we shall consider each of these cases one by one. However before doing so, let us justify our choice. (1) Classical electrodynamics is a linear theory and

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therefore self-interactions are in principle excluded from such a theory. Therefore, to calculate self-forces in the context of the Maxwell theory sounds contradictory. (2) The renormalization procedure which is unavoidable when adopting the first approach (unless distributions are introduced [5,8]) is not required with the second approach. (3) The equations of motion resulting from the second approach present neither runaway solutions nor pre-acceleration, and do not involve more than first derivatives of velocity, (4) The source of the radiation energy is not to be related to a strange “acceleration energy” term (as is the case with the first approach) but is to be found, either in the proper mass of the particle (Bonnor), or in the work done by the external (dissipative) force, or in a combination ofboth factors (eq. (18)). Letthe us now turn toofthe ofbe motion. shall take signature theequation metric to (—2).We Defining the proper time by s cr, then the four-velocity of a point with coordinates xa(t) (a=O, 1. 2, 3) is dxa dr

(1)

satisfying UaUa

=c2,

(2)

and the four-acceleration is defined by Üa

dr satisfying UaUa

=0,

=

(3)

~

dr

~

<0.

(4)

Next, we know from electrodynamics that an accelerated charge e radiates four-momentum at a rate given by [91 2e2 ~U’~UaU’9.

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where m is the proper mass of the particle. Since in general m may be a function of r we can write (6) in the form dm mu’= (7) dx — —

—.

It is important to stress that when calculating the total rate of change of momentum, the term (5) must be included. This point has been sometimes overlooked (see for example eq. (76.1) in ref. [2]). Now by contracting eq. (7) with u0 it is clear that either FaUa 0 (dissipative external force) or dm1 dr0, or both. We shall next consider each one of these cases. (1) dm/dr= 0, FaUa ~ 0 (Mo and Papas). If the proper mass of the particle is constant, then we obtam from (7) 2 (8) FaUa = e 32 C3 which reveals the dissipative character of F~.Next, if we assume that the external force is of electro—

magnetic nature, then it is obvious that we are forced to generalize the Lorentz force as to include dissipative terms. Two comments are in order at this point: (a) The Lorentz force law does not come from Maxwell equations, but is additional to them. Therefore we can always reject the Lorentz force law while retaining Maxwell equations. (b) Ifwe do not introduce any self-force term, and the proper mass of the particle is constant, then the only source of the radiation energy is the work done by the external forces, indicating thereby its dissipative nature. Now, in order to specify the dissipative term in the external force, we shall, following Mo and Papas [71, assume that the total force exerted by an external electromagnetic field on the charge e is given by

(5) Fa=

Therefore, equating the total rate of change of momentum to the external force F’~,and taking into ad-

eF~Y.flup+eiFapulp

(9)

where e 1 is a constant (see eq. (14) below), such that

count the radiated momentum as given by (5), we obtain [6]

~

(10)

C

~mua= dr

(6)

in most physical cases. 15

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Then eq. (7) becomes 2e2 e mz~— __ühut uoe= —F~ufl+elF~ã~. 3 C5 C Contracting this last equation with spectively we obtain

Ua

and

2 e2 eIF’~uaufl= ~ U~Uj~,

(11) Ua

re-

and e =

—

F~upüa.

dm 2 e2 d3 (17) Thus as in the preceding case, the corresponding equation of motion contains only first derivatives of velocity, does not admit either runaway solutions .

..

(12)

—

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(13)

From (12) and (13) it follows that

(F = 0=~.u’~=0) or pre-acceleration (see section 3 in ref. [61) and provides a clear explanation about the origin of radiation energy. The idea that the proper mass of the particle is the source of radiation energy is very attractive and one even might say that it is somehow implicit in the Lorentz—Dirac equation. In fact, it follows from this last equation that (at least) part of the radiation energy comes from the Schott term, which in its turn is related to the internal energy of the particle. Therefore

e 1

~ 3 mc Observe that the ratio 2 e e/C = 23 mc3 =

(14)

—~---~.

—

is a very small quantity. Thus, for an electron we have

[31

2 e2 r0.62x

l023 S

and for macroscopic particles it is smaller still, justifying thereby the inequality (10). Now, from (12)— (14), we may write eq. (11) in the form mi~

F~ÜAupun =

—

~ F~U~ +e

1 FaPI~~. C

it is reasonable to expect that when radiating away internal energy the particle changes its proper mass. On the other hand, it is worth mentioning that in the the way which both the Lorentz—Dirac Schott term andequation, the external forceincontribute to the energy balance, is highly dependent on the type of the external field (and thereof on the type of motion of the particle). Thus, for example, in the case of a uniformly accelerated charge all radiation energy comes from the Schott term whereas in the case of synchrotron radiation (almost) all of the radiation energy comes from the kinetic energy of the particle. Even in the non-radiating approximation (when neglecting the effects of radiation) we know that changes in the kinetic energy of the particle are related only to the electric part of the Lorentz force,

(15)

C

This is the Mo—Papas equation (but note that we do not use relativistic units). It does not admit runaway solutions (FaP=0~.I~~~=0), the radiated energy comes from the work done by the extra term in the “generalized” Lorentz force law (eq. (12)) and as shown in ref. [7]it does not present pre-acceleration. (2) FaUa=0 (Bonnor). If following Bonner we assume that the external force is non-dissipative, then the source for the radiated energy is the proper massonly of the particle. In this case it follows at once from (7) [61 that (16)

the magnetic part being orthogonal to the velocity. Thus, intuitively, one would expect that both the proper mass of the particle and the work done by the external force contribute to the total radiation energy in a way which would depend on the specific kind of the external field acting on the particle. All these comments above suggest considering the following case. (3) FaUa~0, dm/dt0. In this case the corresponding 2equation of emotion can be written as mü°~= ~2 e ~F(4tup C

C

+e 2Fa$üp_ua~,

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(18)

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wherewe have assumed a “generalized” Lorentz force law as in eq. (9), but now the factor e2 does not need to be a constant. Contracting eq. (18) with ~ respectively we obtain 2 + ~ FaflÜpUa, dm = 2e z~i’~ü~ —~

Un

and

(19)

and mãaüa=

(20)

~ C

Combining (19) and (20) we get 2 drn 2e u~u~(l—3me 4/2e3) 2c .

(21)

To proceed further we have to specify the explicit form of the function e2. Since we have been unable to do so from first principles, we shall, just for illustration, present two possible options suggested by very general physical considerations. Option 1. Since we expect the expression within the parentheses in eq. (21) to be positive (for otherwise the mass would increase during radiation), let us take 3k 2e e 2= (22) ~

k being a constant such that 1 ? k>~0. Then eq. (21) becomes 2 1k) dm 2 e t~I~p(

(23)

.

Both the Bonnor and the Mo—Papas equations are easily recovered by choosing k= 0 and k 1 respectively. In the general case however, the specific contribution to the radiation energy from the proper mass and the external force, will depend on the mass of the particle (i.e. it will change with time). We can write eq. (18) in the form

26 March 1990

has to be complemented with eq. (23). Observe that we have still the freedom to “modulate” the conversion of proper mass into radiation energy, through the parameter k. Option 2. are Although the equations of motion presented here not intended to be applied to quantum systems and therefore we should not be worried about the variations of the proper mass of fundamental particles [6], it is possible to introduce a cutoff in the rate of change of the proper mass of the particle by an appropriate choice of e 2. Thus, let us take 2e3 m e 2= —~——tanh C (25) 3cm (rn_mci where m~is some value of the mass which we want to be the minimum allowed value of the particle’s mass (m ~ me). From (21) and (25) it follows that, as the particle radiates, the proper mass decreases, tending to the cut-off value m = mc. At this value, all time derivatives of rn vanish and radiation energy is provided by the external (dissipative) force. As expected, as rn approaches m~,the function e2 tends to beWe constant given eq. (14)with (withthern=rn~). woulde1 like tobyconclude following comments: (1) By an appropriate choice of e2, the two assumptions leading to eqs. (18)— (21) do not contradiet present experimental observations. (2) The resulting equation is obtained on the basis of classical relativistic mechanics and Maxwell equations. (3) Self-interaction effects, which are hardly accounted for by a linear (Maxwell) theory are not considered, avoiding thereby the introduction of a renormalization procedure. (4) The resulting equation allows neither runaway solutions nor pre-acceleration, and provides a clear explanation about the origin ofthe radiation energy. (5) We are aware that the last word on this issue corresponds to experimental data. In the meantime, however, and on the basis of comments (l)—(4) above, we believe that the present approach deserves

rnã~r_~F5fli~U~Ua= ~FaPup+e 2FaPüp,

(24)

which is, formally,identical to eq. (15) above. However, m and e2 are now functions of r and eq. (24)

to be considered further. It is a pleasure to acknowledge financial support from Direccion Nacional de Investigacion CientIfica 17

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y Técnica (DGICYT) (Spain) and Fundacion Polar (Venezuela), and the hospitality from the Physics Department at the Universidad de las Islas Baleares. References [I] H. Lorentz, Theory of electrons (Dover, New York, 1915). [2] L. Landau and E. Lifshitz, Classical theory offields (AddisonWesley, Reading, MA, 1962).

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[3] F. Rohrlich, Classical charged particles (Addison-Wesley, Reading, MA, 1965). [4] C. Teitelboim, D. Villarroel and Ch.G. van Weert, Riv. Nuovo Cimento3 (1980) 1. [5] E.G.P. RoweandG.T. Rowe, Phys. Rep. 149 (1987) 287. [6] W.B. Bonnor, Proc. R. Soc. A 337 (1974) 591. [7] T.C. Mo and C.H. Papas, Phys. Rev. D 4 (1971) 3566. [8]A.Lozada,J.Math.Phys.30 (1989) 1713. [9] A. Shild, J. Math. Anal. AppI. 1(1960)127.