Lopez-Bonilla, J L

Morales, J

Ovando, G

IACS Institutional Repository

Indian J. Phys. 74B (2), 167-169 (2000)I J P  B- an international journalA splitting of the Weert superpotentialJ L L6ptz-BonillaThe International Institute of Integral Human Sciences, 15174 de Maisonncuve West, Montreal, Qu£. Canada H3H 1K5J Morales afed G Ovando*Area deFtsica, Divisidn de CBI, Universidad Autdnoma Metropolitans-Azcapotzalco Apdo Postal 16-306. 02200 Mdxico, D Fe-mail address : gatfe@hp9000al.uam.mxR e c e iv e d  7 Ju n e  J999 , a c c e p te d  15  S e p te m b e r  1999Abstract : We show that the concept of Lanczos spintensor (R ev. M o d  P h y s . 34 179 (1962) f 1]) leads in a natural way to the splitting of T ac as proposed by L6pez (P h ys . R ev . D17 2004 (1978) [2]) for the bounded part of the Lilnard-Wiechert electromagnetic field. HKeywords : Lidnard-Wiechert Field; superpotential for the electromagnetic field.PACS Nos. : 03.50.-z; 13.30.-aFirst of all we go back to Refs. [3-9] and recall the notation and quantities employed there. When a point- charge is moving arbitrarily in Minkowski space, it generates the Lidnard-Wiechert electromagnetic field whose Maxwell tensor may be split as [10]T ac  = Ta + Ta ( 1 )in its bounded Ty and radiative portions, which in turnB Rsatisfy the continuity equation outside from the world line of the charge:T a = 0'c .a  U. (2)£?.«= 0. (3)On the other hand, in [3,5,7-9,11,12] it was shown the non-local superpotential ~ ~ K  bac for the radiativeR Rpart,Ta = K iac,b> (4)In Ref. [8], the physical meaning of K ^  was found and itBturns out that this is the intrinsic angular momentum density of the electromagnetic field produced by the point charge.Furthermore, a splitting of the bounded piece was suggested in [2] :Tac = T ac Tac (6)B B Bwhich is quite convenient when studying the angularmomentum of the Lidnard-Wiechert field; however, the actual origin of (6) was not clear.Since the potential K ijr has [6,8] the same algebraic andBdifferential symmetries as that of the Lanczos spintensor [1,15-28], we will prove in this work that this fact brings naturally to the splittingfrom which (2) follows. Similarly, (3) implies the existence of the Weert superpotential [13,14] K iJr, which has the role of a generator for the bounded part:K  arc ~~ Karc  B B + { -as evident from die expressions (5) and (6).(7)To whom correspondence should be addressed. © 2000 IACS168 J L Lopez-Bonilla, J  Morales and G OvandoSplitting o f K tjr :In [4-10,12-14] one finds the expression for the bounded term of the Maxwell tensor associated to the Lidnard- Wiechert case:Th, = q2w-4 ^  gh, + (kbac + kcab)B{khv, +kcvh)-w -* (  1 -2  W)khk, ]. (8)Also Weert [13,14] has shown that (5) implies (8) by means ofKbjr - ^ - w - * [ w - x{A W - 3){vhkj - v t kb)kc-4 (ahkj - a ,k b)kc + gcjkb -  g,hkj ] . (9)The symmetries of the Weert superpotential have been studied for example in Refs. [6,8]; in fact, it is easy to verify by using (9) that they areK b  = jbo % />; = 0,B bK ^  + K ^  + K h =0, 0, (10)B B B D *which are remarkable because ( 10) arc fulfilled also by the Lanczos spintensor Kt/r [1,15-28] in general relativity.At this stage, we proceed to give the steps to generate a splitting in Tac : we write (9) in the form (7) but carefullyB _ „imposing the condition that K nr and K ac satisfy theB Bsymmetries of Lanczos (10) too. And indeed, it is notdifficult to see that there is only one way to rewrite (9) fulfilling such requirement:2ffhjr =■%-»*’ -  tU *T l-  4 (ahkj -  Vjkh)k(4* ftej^b * ] * (II)When comparing this eq. with (7), the following superpotentials come o u t:Kbjc = q2w^[{yv-lWvb - a h)kj -(w -'W v,( 1 2 )2^bje ~ *4 3^w l (v jk b — vbkj)kc + kcjkb ~Scb^j j>(13)satisfying the properties (10). The splitting (6) proposed by Ldpez [2] appears if eqs. (7,12,13) are substituted into (3), withTB= K , t ac ~ Ka c.r (14)and one recovers the expressions (which are not necessary to display here) of the mentioned author.In this manner, we have made clear that (6) is motivated by requiring that the Weert superpotential be the sum of two tensorial objects with the symmetries of the Lanczos spintensor.References[1 ] C Lanczos R ev. M od. P hys. 34 379 (1962)12] C A L6pez P hys. R ev. D17 2004 (1978)[3] N Aquino, O Chavoya, J Ldpez-Bonilla and D Navarrete Nuovo C m . B108 1081 (1993)[4J V Gaftoi, J L6pez-Bonilla, J Morales and M Rosales J. M ath  P h ys  35 3482 (1994)[5J J L6pez-Bonilla, J Morales and M Rosales P ra m a n a  J. P h ys  42 89 (1994)^6] N Aquino, H N Ntiiiez- Y6pez, J Ldpez-Bonilla and A L Salas Brito J  P hys. A28 L375 (1995)f7] J L6pez-Bonilla, G Ovando and J Rivera In d ia n  J . P u re  A p p l M ath . 28 1355 (1997)[8J i  L6pez-Bonilla, G Ovapdo and J Rivera N u o v o  C im  B112 14133 (1907)19] J L6pcz-Bonilla, H N Nunez-Y6pez and A L Salas-BntoJ. P hys. A30 3663 (1997)[10] C Teitelboim P hys. R ev. D1 1572 (1970)[11] J L6pez-Bonilla and G Ovando G en. R el. G ra v  31 1931 (1999)[12] V Gaftoi, J Ldpez-Bonilla and G Ovando N u o v o  C im  B114 423 (1999)113] Ch G van Weert P h ys R ev  D9 339 (1974)[14] V Gaftoi, J L6pcz~Boni!la and G Ovando In t. J. T heor. P h ys  38 939(1999)[15] F Bainpi and G Caviglia G en. R ela t. G rav. 15 375 (1983)[16] G Ares dc Parga, O Chavoya and J Ldpez-Bonilla J. M a th  P hys. 30 1294 (1989)[17] J L6pez-Bonilla, J Morales, D Navarrete and M Rosales C lass  Q u a n tu m  G rav. 10 2153 (1993)[18] P Dolan and C W Kim P ro c . R o y . S o c . (L o n d o n )  A447 557 (1994)[19] V Gaftoi, G Ovando, J L6pcz-Bonilla, J Morales and J J Pena, J. M o sc o w  P h ys. Soc. 6 267 (1996)[20] G Bergqvist J. M a th . P h ys. 38 3142 (1997)[21] S B  Edgar and A Hdglund P ro c . R o y . S o c . (L o n d o n )  A453 835(1997)[22] V Gaftoi, J Ldpez-Bonilia, G Ovando and J Rivera B u l l  A lla h a b a d  M ath . Soc. 12-13 85 (1997-98)[23] i  L6pez-Bomlla and S Rivera In d ia n  J. M a th . 40 159 (1998)A splitting of the Weert superpotential 169p4j v Gaftoi, J Ldpez-Bonilla and G OvandoNuovo dm. B113 1489(1998)[25] R L Agacy Gen. Rel. Grav. 31 219 (1999) r2(>] J Ldpez-Bonilla. J Morales and G Ovando Gen. Rel. Grav. 31 413 (1999)[27] V Gaftoi, J Morales, G Ovando and J J PeSa Nuovo Cim. B113 1297 (1998)[28] J Ldpez-Bonilla, G Ovando and J J Pena Found. Phys. Lett. 12 401 (1999)

A Splitting of the Weert Superpotential

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A Splitting of the Weert Superpotential

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