6
PRAMANA © Printed in India Vol. 43, No. 4, _ _ journalof October 1994 physics pp. 273-278 The Lorentz-Dirac equation in curved spaces JOSI~ L LOPEZ-BONILLA, JESIOS MORALES and MARCO A ROSALES* Areas de Fisica, Divisi6n de C. B. I., Universidad Aut6noma Metropolitana-Azcapotzalco, Apdo. Postal 16-306, 02200 M6xico, D. F., Mexico * Depto. de Fisica y Matemfiticas, Escuela de Ciencias, Universidad de las Am6ricas-Puebla, Apdo. Postal 100, Sta. Catarina Mhrtir, 72820 Puebla, Mexico MS received 19 April 1994 Abstract. The Teitelboim-Plebafiski method of average field is used to obtain the Lorentz-Dirac equation with gravitation. Keywords. Li6nard-Wiechert field:curved spaces;gravitational field; Lorentz-Dirac equation. PACS Nos 03-50; 04-90 I. Introduction The Lorentz-Dirac (LD) equation [1] describes the motion of a point charge in Minkowski space taking into account the radiation reaction [2]. Dirac's procedure to derive it consists in a) enclose a part of the world-line of the charge with a 3-cylinder of constant instantaneous distance [1, 3,4], b) establish the energy and momentum balance through such a 3-space, c) take the limit as the cylinder contains only the path of the charge, and d) renormalize the mass of the particle. The result is known as the LD equation in absence of gravitational field. This procedure [I] was extended by DeWitt-Brehme [5] to curved space-times to obtain the corresponding LD equation in a given Riemannian geometry including, hence, the gravity. The calculations in [5] are long and tedious because Dirac's cylinder is used therein and this is not well adapted for retarded fields, and because of the authors need to make covariant Taylor expansions of the Maxwell tensor Tab in a curved space the expression for Li+nard- Wiechert's Faraday tensor F~j is complicated. This, in turn, yields an equally complicated Tab (constructed from Fij ). In view of these facts, the DeWitt-Brehme approach does not seem very appealing. Furthermore, it must be mentioned that these authors made a mistake in their calculations and therefore did not arrive at the right LD equation which was later pointed out and corrected by Hobbs [6] using Dirac's technique. Teitelboim [7] and Plebafiski [8] introduced the.method of average field as an alternative to the process used in [1] and thus obtained, in a simple manner, the LD equation in absence of gravity. For calculations this method is better than that of Dirac because it can be applied with a Bhabha [9]-Synge [3] cylinder (very well adapted to retarded effects) and because it does not need Tab but only Fi~, thus reducing the complexity of the calculations. In the next section we will show that the technique of the average field also works in curved spaces, and that it generates a very simple derivation (compared to those in [5, 6]) of the LD equation including a gravitational field; we have not found any applications of the average field method to Riemannian spaces in the literature. 273

The Lorentz-Dirac equation in curved spaces

Embed Size (px)

Citation preview

PRAMANA © Printed in India Vol. 43, No. 4, __ journal of October 1994

physics pp. 273-278

The Lorentz-Dirac equation in curved spaces

JOSI~ L LOPEZ-BONILLA, JESIOS MORALES and MARCO A ROSALES* Areas de Fisica, Divisi6n de C. B. I., Universidad Aut6noma Metropolitana-Azcapotzalco, Apdo. Postal 16-306, 02200 M6xico, D. F., Mexico * Depto. de Fisica y Matemfiticas, Escuela de Ciencias, Universidad de las Am6ricas-Puebla, Apdo. Postal 100, Sta. Catarina Mhrtir, 72820 Puebla, Mexico

MS received 19 April 1994

Abstract. The Teitelboim-Plebafiski method of average field is used to obtain the Lorentz-Dirac equation with gravitation.

Keywords. Li6nard-Wiechert field: curved spaces; gravitational field; Lorentz-Dirac equation.

PACS Nos 03-50; 04-90

I. Introduction

The Lorentz-Dirac (LD) equation [1] describes the motion of a point charge in Minkowski space taking into account the radiation reaction [2]. Dirac's procedure to derive it consists in a) enclose a part of the world-line of the charge with a 3-cylinder of constant instantaneous distance [1, 3,4], b) establish the energy and momentum balance through such a 3-space, c) take the limit as the cylinder contains only the path of the charge, and d) renormalize the mass of the particle. The result is known as the LD equation in absence of gravitational field. This procedure [I] was extended by DeWitt-Brehme [5] to curved space-times to obtain the corresponding LD equation in a given Riemannian geometry including, hence, the gravity. The calculations in [5] are long and tedious because Dirac's cylinder is used therein and this is not well adapted for retarded fields, and because of the authors need to make covariant Taylor expansions of the Maxwell tensor Tab in a curved space the expression for Li+nard- Wiechert's Faraday tensor F~j is complicated. This, in turn, yields an equally complicated Tab (constructed from Fij ). In view of these facts, the DeWitt-Brehme approach does not seem very appealing. Furthermore, it must be mentioned that these authors made a mistake in their calculations and therefore did not arrive at the right LD equation which was later pointed out and corrected by Hobbs [6] using Dirac's technique.

Teitelboim [7] and Plebafiski [8] introduced the.method of average field as an alternative to the process used in [1] and thus obtained, in a simple manner, the LD equation in absence of gravity. For calculations this method is better than that of Dirac because it can be applied with a Bhabha [9]-Synge [3] cylinder (very well adapted to retarded effects) and because it does not need Tab but only Fi~, thus reducing the complexity of the calculations. In the next section we will show that the technique of the average field also works in curved spaces, and that it generates a very simple derivation (compared to those in [5, 6]) of the LD equation including a gravitational field; we have not found any applications of the average field method to Riemannian spaces in the literature.

273

Jos~ L L6pez-Bonilla et al

2. The Lorentz-Dirac equation

We consider here the Riemannian geometry of the space-time (R4) fixed and produced by physical sources according to some gravitational theory (general relativity, for example), and our aim is to derive the equation of motion of a point charge in such R4 taking into account the Maxwell self-field of the particle. In other words, we accept that the mass and the gravitational and electromagnetic radiations of the charge do not alter the curvature of the space-time. The situation is far too complicated when these effects are included. Although this problem has been studied by Infeld- Wallace [10], Bazafiski [11] and Carmeli 1,13, 14] among others, their technique is only approximate since they use the method employed by Einstein-Infeld-Hoffmann [15] to solve the problem on the motion in general relativity. Nevertheless, their results confirm the expressions of references [5, 6]. The process of [ 10-14] is applicable only to the slow motion 1-15] and nobody has ever investigated the corresponding problem for the fast motion [16, 17].

As it has been mentioned, we will consider that the mass, charge and the radiations of the particle do not produce gravitation: the equation of motion is a consequence of the curvature of R4, the external electromagnetic field (Lorentz force) and its own Maxwell self-field (radiation reaction), but not its recoil by gravitational emission 1,18]. This avoids the approximation techniques and leads to an exact equation of motion, namely, the Lorentz-Dirac (LD) equation in curved spaces [5, 6]. Such an equation is simply obtained with the Teitelboim 1,7]-Plebafiski [8] method which states that the motion of the particle obeys the Lorentz force:

ma b, (s) = 4~zqF b,c, (s) vC" (s) (1)

where q, s, Vc, ac, and m denote the electric charge, proper time, veiocity, acceleration and (non-normalized) mass of the particle, respectively and the effective Faraday

tensor Fbc consists of an external part F~ and another part ffb¢ origirrated from the ¢xt

interaction of the particle with its own electromagnetic field

Fb~(s) = Fbc + Fb~(s). (2) e x t

The subtle point resides in the last term: It represents an average self-field at the position of the particle, and it is defined as

ff bc(s) = lim ( Fb~(X') ~ Lye / (3a)

= lim (Fbc(X') ~ (s - ~l) (3b) ~-',0 \ LW /

where the following remarks must be kept in mind:

a) Fbc(X r) is Li6nard-Wiechert's Faraday tensor produced by the charge at the event Lw

X" off the world-line b) X r denotes the events on a Bhabha [9]-Synge l-3] tube keeping the proper time and the retarded distance constant 1,3, 4, 19]

274 Pramana- J. Phys., Vol. 43, No. 4, October 1994

The Lorentz-Dirac equation in curved spaces

c) ( ~ represents an average to be defined below (see eq. (4)) / \

d) ~ Fbc(X')~ is a function of r, the retarded proper time associated to the event X' \ LW /

on the Bhabha-Synge cylinder / \

e) In the above function r is replaced by s - r/with ~/<< 1 and then (Fbc (X') ~ (s - ~l) \ LW /

is expanded in powers of ~/. Introducing the result into (1), (2), and (3b) we are lead (through mass renormalization) to the equation of motion.

This process was successfully employed by Teitelboim [7] and Plebafiski [8] to derive the LD equation in the plane space-time (i.e., without gravitation). We show that it is also applicable to 4-spaces with a gravitational field, in which case the essential difficulty is to define the average ( ) in the presence of a curvature: we only know (from [7,8]) that ( ) is an integral over a section of the Bhabha-Synge tube and that it takes an average over the corresponding solid angle subtended by the charge, but we ignore the integrand in ( ) . In order to solve this difficulty we use:

i) the analysis of references [20, 21] over the volume element for a surface of constant retarded distance, ii) the parallel propagator [22] that allows the integral of a tensor to also have a tensor character, and iii) the average used by Isaacson [23] (see p. 970 of reference [24]) to obtain an effective energy tensor for gravitational waves:

Hence we have proposed the definition (not found in the literature on electrodynamics):

Fb, c,(X') (z) = ~ gb'gc" FijA1/2d• (4) LW co,~ = constant LW

which, in absence of gravitation, reduces to the average employed in [7, 8]. In (4),

the parallel propagator g~, "transports" the tensor Fij from the point X' (with retarded LW

distance = co = constant and proper time - • = constant) to the corresponding retarded event [3, 4] xr(z); d~; = sin 0d0d~ is the solid angle element in the rest frame of the particle, and A is a bi-scalar [5, 20,21 25] (of great importance in the integration of tensor quantities) constructed in terms of the relevant concept of the "world function" f~ introduced by Ruse and Synge [5, 21, 22, 25-36]. The function f~ is a bi-scalar defined as the square of the geodesic distance between two points and contains all the geometric properties of the space-time; in general, it is very difficult to calculate f~ for a given metric: an approximation has been obtained only for the Schwartzschild [-32] and G6del [34] geometries. Fortunately, in diverse works on gravitation and electrodynamics there is no need to know it explicitly.

In Riemannian spaces it is not easy to perform exact calculations [40]. Therefore, it is necessary to use Taylor theorem as given by Ruse [26] in order to expand the tensor quantities that depend on two events very close to each other: this leads us to assume the uniqueness of the geodesic between any two points in R 4, and to construct with it the important concepts of world function f~ and parallel propagator g,,q. Without f~ there is no way to perform covariant expansions, and without Op,~ it is practically impossible that the integral of a tensor would possess tensor character. In other words, without these quantities it would be impossible to study electro- dynamics in curved spaces.

Pramana - J. Phys., Vol. 43, No. 4, October 1994 275

Josd L L 6 p e z - B o n i l l a et al

From equation (3.52) of [5] we obtain the expression:

with:

A bc = -- -w - 2 f~b i, U cr' v" r " r , Ucr, = A1/ 2 g cr, 1

(5a)

Ab c = ~o9- 3 Db u , v", "Z = Da,b, V a" vb' 2

Abe = -- O9- 2 ~-'~b Ucr' (a" -- o9- 1 Wv") , W = Dj a j" (5b) 3

1U V r' Abe = - - (.D - 2 ~"~b U c r , j, Ur" O j', Abe = -- 03- br';c ' 4 5

f $

Abc = CO- 1 ~b Vc,, V", A b c = Vbr,;~ vr" df l 6 - 8

where the notation of [22] for the covariant derivatives of ~ has been employed, and va¢, is a complicated function of the metric for which no exact expression exists [5] but, fortunately, the following'approximations are known:

1 C , ( R R vb, ),

! A 1/2 = 1 - "~R"'b 'D, ,A' ib ,

12

(5c)

with R o being the Ricci tensor and R = R~. Substitution of relations (5) into eq. (4) and integration over the solid angle of the resulting integrals, which can be found in references [3, 37] lead to

I F b , c , ) ( s - - r l ) " 2 ~q(Ab , Vc, Ac, Vb,)(S) LW "" 3 q q - l (ab'v¢" -- ac' Vb')(S) -- -- --

~[ (Rb , j , V c , - R c , j , V b , ) V J ' ] ( s ) + q f ~ fb ,~ , , , v"d f l+O(~l ) (6a) oo

where Ac = 6a¢/6s is the superacceleration of the particle and

fb'c', ' = Vb',';c" -- Vc',';b" (6b)

Thus, using eqs (1), (2), (3b) and (6) we are able to obtain the famous Lorentz-Dirac equation in curved spaces [5, 6] (a 2 - abab):

c • 8 7 ¢ q 2 , - - 2 ,

mobsa " = 4 ~ q F r c v "4- ~ - [ A . -- a v ) -- e x t

q2(RcjVSV , + Rcr)V c + 47tqv ~ f,j~ dfl (7a) . 1 - o o

276 Pramana - J. Phys., Vol. 43, No. 4, October 1994

The Lorentz-Dirac equation in curved spaces

with the normalization

m°bs = lim ( m + ~ q2rl- ' ) (7b)

The result (7a) coincides with (5.28) of the first article of [6] and with (2.23) of [38]. The expression (5.25) of 15] does not contain the Ricci tensor because, as Hobbs pointed out, their relation (5.11) was erroneously calculated. The integral in (7a) is a consequence of the non-local character of the electromagnetic field generated by the charge, i.e., the emitted radiation leaves a trace as it spreads through the curvature of the space-time. It is clear that this does not happen in special relativity, and in the second article of [6] it is proven that the mentioned integral vanishes in conformally plane spaces [39].

To obtain exact solutions for the LD equation in absence of gravitation is not a simple matter, hence, the difficulties are even bigger in curved spaces and we must invoke approximate situations: particle at low speed and/or weak gravitational field [41, 42]. Under such conditions Carmeli [13] studied the LD equation through the Frenet-Serret formulas [22] but, unfortunately, he used the wrong expression from [5] and so his work was incomplete; nobody has ever tried to correct Carmeli's results. In [41] the incorrect result from [5] was also used to analyze the path of a charged particle moving at a low speed in a weak gravitational field with spherical symmetry, and in [42] the corresponding quantum problem is studied. Sigal [43] made use of eq. (7a) to examine the electromagnetic radiation of a point charge in Robertson-Walker [22] cosmologies.

3. Conclusion

In this paper we have derived (7a) with the average field method, because in eq. (4) the Faraday tensor produced by the charge in motion is averaged titus eliminating some singularities at the position of the particle and allowing that the rest of the singularities cancel out when the mass renormalization is made. Unlike [5, 6] there was no need to know the Maxwell tensor To~ or to calculate its energy flux through any 3-space, which shows the simplicity of the method with respect to the one used by Dirac.

Finally, using (4) it is useful to know that such an average is determined with the radiation coordinates (0, q~, o9, s) introduced by Florides-McCrea-Synge [2 l, 44, 45] which are well adapted to retarded effects. These coordinates are the generalization, for curved spaces, of those studied by Synge [3], Newman-Unti [46, 47] and Graef [48] in special relativity.

References

[1] P A M Dirac, Proc. R. Soc. London A167, 148 (1938) [2] J L Jim6nez, Ciencia (Mexico) 40, 257 (1989) [3] J L Synge, Ann. Mat. Pura Appl. (Italy) 84, 33 (1970) [4] J L L6pez-Bonilla, Rev. Colomb. Fis. 17, 1 (1985) [5] B S DeWitt and R W Brehme, Ann. Phys. 47, 220 (1960) I-6] J M Hobbs, Ann. Phys. 47, I41 and 166 (1968) [7] C Teitelboim, Phys. Rev. D4, 345 (1971)

Pramana - J. Phys,, Vol. 43, No. 4, October 1994 277

Josb L L6pez-Bonilla et al

[8] J Plebafiski, The structure of the field of point charges, (internal report) CINVESTAV- IPN), Mexico (1972)

[9] H J Bhabha, Proc. R. Soc. London A172, 384 (1939) [10] L lnfeld and P R Wallace, Phys. Rev. 57 797 (1940) [11] S Bazafiski, Acta Phys. Pol. 15, 363 (1956) [12] P Havas, Phys. Rev. 108, 1351 (1957) [13] M Carmeli, Phys. Rev. BI38, 1003 (1965) [14] M Carmeli, Ann. Phys. 34, 465 (1965) [15] L lnfeld and J Plebafiski, Motion and relativity (Pergamon Press, 1960) [16] B Bertotti and J Plehafiski, Ann. Phys. 11, 169 (1960) [17] A Kiinel, Ann. Phys. 28, 116 (1964) [18] A Schild, Bull. Acad. Pol. Sci. 9, 103 (1961) [19] G Ares de Parga, J L L6pez-Bonilla, G Ovando and T Matos, Rev. Mex. Fis. 36, 194 (1990) [20] D Villarroel, Phys. Rev. DII, 2733 (1975) [21] J L Lbpez-Bonilla, Acta Mex. Ciencia y Tecnol. (IPN) 1, 23 (1983) [22] J L Synge, Relativity: the general theory (North-Holland, 1976) Chap. II [23] R A Isaacson, Phys. Rev. 166, 1272 (1968) [24] C W Misner, K S Thorne and J A Wheeler, Gravitation (W H Freeman, San Francisco,

1970) [25] B S DeWitt, Phys. Rev. 162, 1239 (1967) [26] H S Ruse, Proc. London Math. Soc. 32, 87 (1931) [27] J L Synge, Proc. London Math. Soc. 32, 241 (1931) [28] J L Synge, Rend. Sere. Nat. Fis. Mllano 271 (1960) [29] J L Synge, Recent developments in general relativity (Pergamon Press, 1962) p. 441 [30] J D Lathrop, Ann. Phys. 79, 580 (1973) [31] J D Lathrop, Ann. Phys. 95, 508 (1975) [32] H A Buchdahl and N P Warner, Gen. Relativ. Gravit. 10, 911 (1979) [33] H A Buchdahl, Austr. J. Phys. 32, 405 (1979) [34] N P Warner and H A Buchdahl, J. Phys. AI3, 509 (1980) [35] R W John, Ann. Phys. 41, 1 (1984) [36] R Schimming and D Matel-Kaminska, Z. Anal. Anw. 9, 3 (1990) [37] Ch G Van Weert, Physica 66, 79 (1973) [38] A O Barut and D Villarroel, J. Phys. 8, 1537 (1975) [39] G Ares de Parga, O Chavoya, J L L6pez-Bonilla and G Ovando, Rev. Mex. Fis. 35, 201

(1989) [40] G N Plass, Rev. Mod. Phys. 33, (1961) [41] C M DeWitt and B S DeWitt, Physics 1, 3 (1964) [42] C M DeWitt and W G Wesley, Gen. Relativ. Gravit. 2, 235 (1971) [43] R Sigal, J. Math. Phys. 12, 2490 (1971) [44] P S Florides, J Mc Crea and J L Synge, Proc. R. Soc. London A292, 1 (1966) [45] P S Florides and J L Synge, Proe. R. Soc. London A323, 1 (1971) [46] E T Newman and T W J Unti, 2. Math. Phys. 4, 1467 (1963) [47] G Ares de Parga, O Chavoya, J L L6pez-Bonilla, E Luna and J Morales, Rev. Mex. Fis.

36, 184 (1990) [48] C Graef, Rev. Mex. Fis. 11, 129 (1962)

278 Pramana - J. Phys., Voi. 43, No. 4, October 1994