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General Relativiiy and Gravitation, Vol. 20, No. 1, 1988 On the Permanence of the Null Character of Maxwell Fields Bartolomé Coll' and Joan. Josep Ferrando !2* Received April 7, 1987 A critical review of known results about the permanente conditions for the null character of the solutions to the (vacuum) Maxwell equations, is presented. Concomitants of the electromagnetic field and the metric lensor are constructed, which give the principal directions of the field in covariant form. The known permanence conditions are generalized in order to include al! the (local) null Selds; the above concomitants allow these conditions to be explicitly formulated ín terms of the electromagnetic field. 1. INTRODUCTION One o! the general problems, still open, concerning the Maxwell equations is that of the conditions under which the permanence of ¡he null elec- tromagnetic field may be insured: the (vacuum) Maxwell equations do not guarantee that an electromagnetic ficid which is null at an instant will remain null near it. The only known result about such permanence conditions is the Mariot-Lichnerowicz theorem,* which states that among the Maxwell fields admirting an integrable principal direction, the mall fields are permanent. Tn this paper we solve mainly two problems: (1) to obtain covarianlly the principal directions of an electromagnetic field; this will allow us, in particular, to write the differential equations that a Maxwell field satisfies Y Département de Mécanique Relativiste, Université Paris VI, Tour 66. 4, Place Jussicu. 75252 Paris Cedex 05. France. 2 Permanent address: Departament de Fisica Teórica, Facultat de Fisiques. Burjassot, Valéncia, Spain. 3 Supported in part by “Conselleria de Cultura, Educació ¡ Ciéncia de la Generalitat Valenciana.” See the discussion oí Section 2. s1 0003-7705 850109005 1506,00 O 1 198% Plena Publisimns Corperavon 52 Coll and Ferrando when it admits an integrable principal direction: the above theorem will then become a permanence statement for the null initial data of this dif ferential system. (1i) To obtain a generalization of the Mariot-Lichnerowicz result ín order to include the null fields having no integrable principal directions. The literature on this subject being rather confusing, we devoted Sec- tion 2 to explaining the nature of the permanence problem for the null Maxwell fields, the role played by the Mariot-Lichnerowicz theorem and the possible generalizations of it that one expects to obtain. In Section 3 we solve ina covariant way the algebraic problem of finding the principal directions of an arbitrary electromagnetic field. To do this, we introduce a pair of algebraic concomitants of the electromagnetic field and the metric which project arbitrary time-like directions onto the principal directions. This section is self-consistent, and ¡ts interest exceeds the use of it thatwe make here, Finally, in Section 4 we generalize the Mariot-Lichnerowicz theorem in order to include al! the null Maxwell fields, and use the results of the above section to formulate explicitly the differential system in the electromagnetic field variables with respect to which the null fields are permanent. Under our permanence conditions, the property of admitting an integrable principal direction becomes also permanent. The results without proof of this paper were communicated to the Spanish relativistic meeting E.R.E. 86 [1]. 2. SURVEY OF THE PROBLEM (a) A field Fis said to be a Maxwell field íT it is a solution of the vacuum Maxwell equations? : 8F=0, $*F=0 (0) From the evolution point of view, these equations split, with respect to a timelike direction u, into an evolution system L(wóF=0, l(u)ó*F=0 (E) and a consiraint system u)ÓF=0, — Hu)ó*F=0 (0) $$ and = are. respectively. the divergence and dual operators. $ 1 (w) is the spatial projector with respect to u, and (1) is the e projection (interior product). 54 Coli and Ferrando Taking the time derivative of the last equation and using the first, we obtain h-rote+e-roth=0 (1) e-rote—h-rote=0 But (M4) is in involution, and (£) is well-posed (in the Cauchy sense), so that to every pair (ep, hp) verifying (C] on an instant corresponds, in the neighborhood of it, to a unique Maxwell field. Let the instant be given by (=p and consider the following fields at £o eo(x)=(x, y, 0), — hy(x)=(0,0, 1) where x= [x, y, 2) (Cartesian coordinates) and r?=x?+ y", It is easy to see that these fields verily the systems (WN) and (C], and so they generate in the neighborhood of /, a Maxwell field. Nevertheless, as may be seen by substitution, our data are not solutions of (1) at rg and, consequently, the corresponding Maxwell field, null at £y, is regular in its neighborhood, as stated in Proposition 1. It may be noted that the verification of the first-order system (1) by the initial data (€o, Ap) does not imply the null character of the field out of fo: taking the time derivative of (1) and using (Ej, one obtains a second- order system at 1 which, in general, will not be verified by the solutions to the system (Cu (Nju ((1)). The prolongation of this procedure to any finite order of derivation will always give the same negative result. (c) - Proposition 1 leads naturally to the following: Problem. To find the conditions under which the null Maxwell fields are permanent. The only known answer lo this problem is the Mariot-Lichnerowicz theorem. Remember that the principal directions of an electromagnetic field are given by the null eigen-directioris and that an electromagnetic field is said to be of integrable type 11 it has an integrable principal direction, say £:f a dí =0. One has then: Theorem (Mariot-Lichnerowicz). lí a Maxwell field of integrable type is null on an instant, it is null in the neighborhood. This statement, and an easy proof of it based on the eigen-direction equations, is due to Lichnerowicz [3]. Before him, Mariot [4] obtained it in an adapted Cartan's moving frame, but his proof, based partially in a previous erroneous result [2], is incomplete, and his statement, for the On the Permanence of the Null Character of Maxwell Fields 35 same reason, contains some inaccuracies. In spite of these errors, Mariot seems to have been the first to ask for the permanence of the null fields; he discovered also another fundamental property of them [5]. (d) According to its definition, an electromagnetic ficid F is of integrable type if there exists a function 6 such that Kd0NF A d0=0, (40) »F A d0=0 Q) Thus, it is for the Maxwell fields verifying (2) that the Mariot- Lichnerowicz theorem insures the permanence of the null character of the fields. But, while the Maxwell system (M) isa differential system in F, system (2) is a mixed system, explicitly algebraic in F and implicitly dif férential, depending on a function whose null gradient itself depends on E The analysis of such a system is not easy to make. lt would be desirable to substitute system (2) by an equivalent one, in which the dependence on F of de is explicit. In the following section we introduce the essential algebraic elements for this task. The Mariot-Lichnerowicz theorem does not address the permanence of Maxwell fields of a nonintegrable type. Because of the great variety and physical interest of such fields, it would be convenient to have a permanence theorem for them. We obtain it in the last section. (e) Here we indicate three aspects of the permanence problem we think important. The first concerns the roninvolutive character of the system (E) U (N]: it is nor possible to find initial conditions which ensure, in general, the permanence of the null character of Maxwell fields. In other words: there do not exist common relations to all null Maxwell fields (and, eventually, to some other regular ones) such that their verification on one instant ensure, for a null field at that instant, the permanence of its null character in the neighbor ol the instant. Nevertheless, let us note that this does not deny the possibility of finding initial conditions ensuring the permanence of some, particular, null fields. Thus, for example, the initial conditions given, in MinkowskT's inertial frames, by (Ve¿=0, Vh¿=0), although without physical interest, show that “ie Maxwell fields which are null and constant at one instant remain null and constant everywhere.” The conditions which select al/ the permanent null Maxwell fields are to be imposed in all domains in question (as is already the case for the Mariot-Lichnerowicz conditions). In other words, the system of the Maxwell equations has to be necessarily completed, in all given domains, with a convenient differential system in such a way that the common solutions to both systems be either everywhere null or everywhere regular in the domain. Á permanence theorern for all the null Maxwell fields is thus also a permanence theorem (everywhere regularity) for some class of On the Permanence of the Null Character al Maxwell Fields 57 rotation, and are null fields ¡ff y=0. Let us remember also that the characteristic polynomials of F and *F are of the form A A (4) thus, denoting by £, their principal directions, we have ¡E JE= tal, M£,)+F= FBL, (5) Let us introduce Po(21)=(¿4+0) PU) =(420(4?+ 8) PE(1)= (+8) PF) = (AH BI + a) (6) from the Cayley-Hamilton theorem it follows that P,,(F) (resp, P£(+F)) is an eigentensor of F (resp., of+F) with eigenvalue +a (resp. + B) and commutes with F (resp., with *F). Starting from the known identities P-F=(2-P)g FxP=*FxF= —aPg mM it can be shown that the above eigentensors are of the form P.(F)=ag, PAF)= a F e Pr (*F)= BF, > Pi*F)=pP 'F Where 4 is the tensor defined by AF PF+ TA gg (9) and 'F denotes its transpose. Definition. We call principal concomitants of a 2-form F their eigen- tensors F and '$. (b)- Let us analyze some properties of these concomitants: The expression (9) shows that 4 never reduces to a 2-form and that 1t becomes symmetric iff F = 7, that is to say, if F is null. In such a case, we know that Tx F=Fx T=0; we obtain now the analog of this relation for the regular case, When Fis nonnull, there is, at last, one nonzero cigenvalue, say a; then, as P*(F) are eigentensors of F commuting with it, from the -first two relations (8) it follows that FAF=FxF =0F (10) 38 Coli and Ferrando and, taking into account the second identity in (7) Pr FF = BF (1) From (10), (11) and their transposed relations, it follows that Im 4 and Tm “F are the principal directions of F. Therelore, we may write PE (12) It is then easy to see that: Proposition 2, The principal concomitants of a 2-form F are, up to a scalar factor, the only eigentensors of F and +F that commute with them. An endomorphism $ is said to be a generator of a pair of directions, say (€, ), TImS=(£, ) and Im 'S= (£_ ). The above result may be stated as follows: Proposition 2. An endomorphism Y is the generator of the principal directions of a 2-form F iff it coincides, up to a scalar factor, with a prin- cipal concomitant of F. Now, from (12), it is clear that Ker F and Ker “FF have no timelike directions; we have thus the following rule to obtain the principal direc- tions: Theorem 1. The principal directions f, of a 2-form F are given by LEFIO, £ ="F(x) (13) where x is an arbitrary timelike direction and 4 is the principal con- comitant of F given by (9). This theorem is the “covariant solution” to the problem of finding the principal directions of an electromagnetic field, . (c) The Rainich algebraic relations are the necessary and sufficient conditions for a symmetric tensor T to be the energy tensor of an elec- tromagnetic field; we give here the corresponding relations for F. From (12) it follows that Fx "F ='F x.F =0; conversely, if a tensor F verifies these relations, Im 4 and Im 'F are null directions and so F may be writ- ten as (12). Consequently, every 2-form F having these null directions as principal directions admits 4% as principal concomitants. But, the trace- 1 F(1)= lx) “F: in local cuordinates /.* =P %x%. 60 Coll and Ferrando (16) (resp. by (17)) generate the 2-plane x (resp., 7*). Taking into account (15), we have Proposition 5. The invariant 2-planes, x and x*, of a regular 2-form F are given, respectively, by asi) A,), m=iina,?) where x is any timelike vector such that 7(x)% yx, and the biparametric concomitants oí F, F,,, and F,,*, 4, pel, are given by Fa, UT +18) +0 F— BF), $, =UT— 18) + (BF +0xF) Note that 7 (resp., 11) is an eigenspace of F, with eigenvalue zero, when £ is spacelike (resp. timelike); that is to say, when «=0 (resp. B=0). When F is singular, both invariant 2-planes are null and contain the principal direction of F; they are defined by 1=(p/p a F=0) and n*= íg/q a =+F=0). Let x be an arbitrary. time-like vector; one has xgxmun” and then (xjF 40, (x)*F40. Also, from FaF=Fa+*F=0, one has ¡(Fez and ¡(x) +Fex?. The principal concomitant 4 reduces now to T, so that we have Proposition 6. The invariant 2-planes, x and a+, of a null 2-form F are given by ne (NT +a Amer), mts (MT +) A per) where x is an arbitrary timelike vector. (e) «For the null 2-forms, F and '% reduce to their energy tensor; in the opposite end, for the completely regular 2-forms (240, $0), F and “£F coincide with the Frobenius covariants of the associated matrix. But, as algebraic regular functions on the space of all the 2-forms, our principal concomitants F and 'F seem not to have been considered up to now. Let us note that this method of “covariant resolution” of the problem of eigen-directions of a 2-form, may be extended to arbitrary tensors. lis extension, in particular, to symmetric tensors has been recently made [8]. 4. PERMANENCE OF NULL FIELDS (a) The principal concomitants of a 2-form F allow one to formulate, in terms of F, the diflerential equations imposed on their principal direc- tions by any particular problem. On the Permanence of the Null Character of Maxwell Fields él Theorem Í states that, if xis an arbitrary timelike 1-form, £ =F (x) is a principal direction of F. Thus, the integrability of £, £ a df =0, may be written F(x) a dF(x)=0. Considered as an endomorphism on the space of the 1-forms, + is a A'-valued vector field; when exterior algebra operators act on F, it must be understood they act on the “t-form part” of it." Thus, the above expression for the integrability of £ being valid for all x, itis easy to show that: Proposition 7. An electromagnetic field Fis of integrable type if, and only if, its principal coricomitant PF verifies FidFr=0 (18) Taking into account this result, the Mariot-Lichnerowicz theorem may be stated as a property of a diflcrential system for F: Theorem (Mariot-Lichnerowicz). For the electromagnetic fields F verifying the diflerential system dE=dF=F AdF=0. (19) the null fields are permanent. (b) A real null tetrad (£,% Ap), with (-a=-pP=-g=1 as nonzero scalar products, defines an almost-product structure given by the two 2-planes x(£, +) and ríf, 7); Jet P, P?= g be its structure tensor. Á simple algebraic calculation shows that: _Lemma 1. Let £ be a principal direction for a symmetric tensor 7” verifying the algebraic Raínich conditions. With respect to the real null tetrad [£, », fi, 7), Y may be written T=kLO(+iP+tÓ pt (20) where P is the structure tensor of the tetrad 1?+5*=4*%y and ” denotes symmetrization. From (20), with P=2x SÓ £—g and remembering that ¡(1) "Vf=0, we have UTA VE)= 10€ + [2 e) +ri fo + si) HO YE and thus: 1 That is lo say. on the first index of their local coordinate components. On the Permanence ef the Null Character oí Maxwell Fields 63 When Fis a null field, (23) is satisfied if T is conserved. When F is regular it may be shown that (23) is equivalent to tr 42=0, tr being the contraction of the first covariant index with the first contravariant one. !? We have thus Proposition 10. The necessary and sufficient condition for a Maxwell field F to have a geodesic principal direction is r=0 where 3 is the dillerential concomitant of F given by (23). Theorem 2 then may be stated in the following form: Theorem 2”. ' For the electromagnetic fields satisfying the differential system dE=dF=u[F A(FP)YF]=0 (4) the null character is permanent. Let us note that, as has already been indicated in Section 2(e), this result is also a permanence statement for regularity: if F verifies (24) in a domain and is regular on an instant, then F is regular in the domain. Theorem 2' is the wanted generalization' of the Mariot-Lichnerowicz theorem. (ec) Let w=wx*(v A db) be the rotation of v and denote by D=i(0)V the directional derivative; we have the identities [», D] =0, [42, D]Jo=*Vox do + dex Vo (25) and, for every 2-form A and every 2-tensor K HAXK+'KxA4)=tr K-x4— («dx K+Kxx04) (26) Applying (25) and (26) to ww, we have Dwe==Díva do)=x*[(De a de+un Dedo) ==*d(0 a Do) + (vu) «[ "Vox do + dex Vo] =wd(e a Do) + H0)[ —«edex “Vo — Vox «dv +t1 Vo- *do) =x*d(v A De) +a(3 a de) x "Vo— (Do) xdo + 60 - «(0 A do) 2% has local components of the form Ap”. The antisymmetry in xf follows from the exterior produc! form: meanwhile, the symmetry in Zee is a consequence of the identity +A =0, obtained from (12) 64 Coll and Ferrando that is Dw= i(w) "Vv+ó0-w+ (o, Vo) (27) where X (o, Vo)=*[d(v a Dv) + Dv A do) When X (o, Vo) depends on w and vanishes with it, (27) becomes a propagation system for w; thus, for geodesic fields (v A Du=0) where XA =¿w, we have: Proposition 11. Let v be a geodesic field in a domain. 1f v is not tangent to an instant and is integrable on it, it is integrable in the domain. From this proposition and Theorem 2, we have: Theorem 3. H an electromaguetic field F is a solution of (24) in a domain and is of integrable type on an instant, then it is of integrable type in the domain. This theorem reduces to ¡mirial conditions a part of the conditions required in alí the domain by the Mariot-Lichnerowicz theorem (that part that must be imposed to a geodesic principal direction in order to be integrable). It is thus a nonempty refinement of their statement. In fact, the Mariot-Lichnerowicz theorem ensures the permanence. of the null fields among the ones of integrable type, whereas Theorem 3 ensures the permanence of the null fields of integrable type among the null fields, REFERENCES 1. Coll, B. and Ferrando, J. J. (1986). in Actas de los E.R.£. (Publicacions de la Universital de Valéncia, Valencia, Spain). 2. Marioi, L (1954). C.R.A.S. París, 239, 1189-1190, . Lichnerowicz, 'A. (1960). Ann. Mat. Pura Ed App. 50, 1-96. Mariot, E. (1955). C.R.A.S. Paris, 241, 175-176. , Mariot, L. (1954). C.R.A.S. Paris, 238, 2055-2056. . Robinson, 1. (1961). Jour. Math. Phys, 2, 290-291. . Raynich, G. Y. (1925). Trans, Am. Maih, Soc. 27, 106-136, . Bona, €., Call, B.. and Morales, J. A. (1986). In- Actas de los E.R.E. (Publicacions de la Universitat de Valencia, Valencia, Spain). 9. Debever, R. (1976). Bull. Acad. R. Belgi. Class Sci, 62, 662-677. país a es