STABILITY. Department of Theoretical Physics, Russian Peoples' Friendship University, Moscow. Miklukho Maklay St., 6, Moscow , Russia

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1 PACS Jb DROPLET-LIKE SOLUTIONS TO THE NONLINEAR SCALAR FIELD EQUATIONS IN THE ROBERTSON-WALKER SPACE-TIME AND THEIR STABILITY Yu. P. Rybakov?, B. Saha y, G. N. Shikin?? Department of Theoretical Physics, Russian Peoples' Friendship University, Moscow Miklukho Maklay St., 6, Moscow , Russia y Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, Russia Exact droplet-like solutions to the nonlinear scalar eld euations have been obtained in the Robertson-Walker space-time and their linearized stability has been proved.

2 1 Introduction When the general theory of relativity (GTR) and uantum theory of eld were developed, an interest to study the role of gravitational interaction in elementary particle physics arose. On this context, to obtain and study the particle-like solutions to the consistent systems of wave and gravitational elds present a major interest. To obtain and study the properties of regular localized solutions to the nonlinear classical eld euations (soliton- or particle-like solutions) is connected with the hope to develop a divergence-free theory of elementary particle, which in its turn would describe the complex spatial structure of particle, observed experimentally. In doing so one should keep in mind that the nonlinear generalization of eld theory is necessary irrespective of the uestion of divergence as the consideration of interaction between the elds inevitably leads to the advent of nonlinear terms in the eld euations. Conseuently, nonlinearity should be considered not only as one of the ways to eliminate diculties of theory, but also the reection of objective properties of eld. As it is noticed by N. N. Bogoluibov and D. B. Shirkov [1], the complete description of elementary particles with all their physical characteristics (say, magnetic momentum) can give only the interacting eld theory. So one can say that individual free (linear) elds present themselves as the basis to describe these particles in the framework of interacting eld theory. As elementary particle is a uantum object, so the attempts to develop a classical model of particle remain preliminary but necessary stage of study for transformation to uantum theory. In this paper the system of interacting scalar and electromagnetic elds are being considered in the Robertson-Walker Universe with the metric [2]: dr 2 ds 2 = dt 2 R 2 (t) 1 kr + 2 r2 d 2 + r 2 sin 2 d 2 ; (1.1) where R(t) denes the size of the Universe, and k takes the values 0 and 1. Droplet: it is some kind of soliton-like solutions to the eld euations possessing sharp boundary. Similar solution was rst obtained by Werle [3]. Further, a series of work was done where the solutions with sharp boundary to the nonlinear eld euations were being found and studied in external gravitational eld as well as in the selfconsistent one [4-10]. Present paper generalizes the partial results obtained by the authors earlier. Moreover here the uestion of stability is considered which presents a growing interest. 2 Fundamental euations and their solutions We will choose the Lagrangian of interacting scalar (') and electromagnetic (F ) elds in the form [4]: L =(1=2)' ; ' ; (1=4)F F ('); (2.1) where the function (') = 1 + (') characterizes the interaction ( (') = 1 corresponds to the system of free elds). We will seek the static spherically symmetric solutions assuming that the scalar eld ' is the function of r only, and the vector eld A possesses one component A 0 (r), i.e. ' = '(r); A = 0 A 0 (r) = 0 A(r): 2

3 It means that (F ) also possesses one component i.e. F =( )F 01 (r) =A 0 (r); where 0 denotes dierentiation with respect to r. The euations to scalar and electromagnetic elds ( p gg ' ; )+( p g=2) F F '(') =0; '(')=@ =@'; ( p g F ('))=0: (2.3) In accordance with the assumption, made above, the euation (2.3) is easily integrated at r >0: F 01 (r) =P(')= g 0 = P(') p 1 kr 2 =R 3 r 2 ; (2.4) where = const, P (') =1= (') and g 0 = g=sin 2 = R6 r 4 (1 kr 2 ). The euation (2.2) for '(r) in this case metamorphoses to the euation with "induced nonlinearity" [5]: g 0 g 0 g 11 ' 0 0 = 2 g 11 P ' : (2.5) Let us make the following assuption of the cosmological character of time in Robertson- Walker Universe. Let us suppose that the cosmological time scale is much greater than the usual time scale. In other words in the case considered R(t) can be interpreted as a constant. Then it is also easy to nd the rst integral and solution in uadrature for the euation (2.5): The regularity condition of T 0 0 P (') in the form ' 0 = p 2 p P + C=R r 2 p 1 kr 2 ; C = const; (2.6) Z d'= p P + C = p 1 kr 2 =R r + C 3 : (2.7) at the center leads to the fact that C = 0. Choosing P (') =1= (') =J 2 4= 1 J 2= 2 ; (2.8) where J = '; =2n+1; n =1;2, for '(r) one gets: '(r) = 1 It is obvious that at r! 0 r = r c 1 exp 2 p 2 R 1=r 2 k + C 3 =2 : (2.9) '(0)! 1=; and beginning with some =2 p 2= (R 2 2 C k 2 2 ); '(r) becomes totally imaginary as in this case the suare bracket possesses negative value. As far as we are dealing with a real scalar eld, '(r) atr > r c becomes nonphysical. So without losing the generality wemay write that at r! r c ; '(r c )! 0: (An illustration of the inverse interaction function P (') and the scalar eld obtained is given in Figure 1 and Figure 2.) 3

4 Let us write the energy-momentum tensor for the interacting elds: T = ' ; ' ; F F (') L: (2.10) From (2.10) we nd the density of eld energy of the system: and total energy E f = Z T 0 0 T 0 0 = P R 4 r 4 (2.11) 3 gd 3 x = 3p 2 2( 1) : (2.12) Thus, we came to the conclusion that energy density T 0 0 and total energy of the congurations obtained do not depend on the conventional values of the parameter k =0;1. 3 Stability problem To study the stability of the congurations obtained we will write the linearized euations for the radial perturbations of scalar eld. Assuming that '(r; t)='(r)+(r;t); '; (3.1) from (2.2) in view of (2.5) we get the euation for (r; t): +3 _ R R _ 1 kr kr 2 R 2 rr P '' =0: (3.2) R 4 4 r As far as according to the assumption the external gravitational eld is cosmological one, we can consider that R(t) is a slowly varying time-function: _R(t) 0. Assuming that (r; t) v(r) exp( i t); =!= R; (3.3) from (3.2) we obtain (1 kr 2 )v 00 +(2=r 3kr)v 0 2 P +(! 2 '' ) v =0: (3.4) R 4 4 r Let us rst consider the case when k = +1. Then substituting v(r) = y(x), where x =1 1=r 2 ; from (3.4) we get the euation 4 xy xx +2y x +! 2 2 P '' y =0; (3.5) (1 x) 2 R 4 which for y(x) =u(z); x=z 2 ;takes the form u zz +! 2 2 P '' u =0: (3.6) (1 z 2 ) 2 R 4 Further substitution () =u(z)= p 1 z 2 ; z= th ; 4

5 leads the euation (3.6) to the normal form of Liouville [11] +! P '' R 4 sech 4 =0: (3.7) In case of k = 1 the euation (3.4) can analogously be transferred to the form (3.7) doing the following substitutions: x =1+1=r 2 ; x = z 2 and z =th: At last in case of k = 0 from (3.4) we get W 00 +! 2 2 P '' R 4 r 4 W =0; (3.8) where W (r) =rv(r): Using the form of P '' from (2.8), we come to the conclusion that for 5 the expressions of the potentials V (') =1+ 2 P '' sech 4 and V R 4 0 (') = 2 P '' R 4 r 4 tends to +1 at r! 0aswell as at r! r c = p 2 2 p(r 2 2 C k2 2 ). It means that for 5 for P (') given by (2.8) the conguration obtained is stable for the class of perturbation, vanishing at r = 0 and r = r c. In stability can be assured in general introducing the variable = Z r dr p g 0 g = 1 Z r 11 R and rewriting the euation for perturbation in the form dr r 2 p 1 kr 2 d 2 d 2 +(2 2 P '' ) =0: The euation mentioned possesses at = 0 nonnegative solution = d'= d, which according to the Sturm theorem corresponds to the absence of "coupled" state with 2 < 0: 4 Conclusion Thus, we obtain the object with sharp boundary, described by the regular function '(r). In the center of the system r = 0 '(0)! 1=, and at some critical value of radius r = r c function '(r) possesses trivial value. The conguration obtained, possesses limited energy density and nite total energy. From (2.12) it is explicit that the expression for energy does not contain r, dening the size of droplet. It means that the droplets of dierent linear sizes up to the soliton with r c!1share one and the same total energy. For dierent values of k, the eld function '(r) changes it's form. It is noteworthy to notice that at r c!1for k = 0 droplet transfers to usual solitonian solution, while in case of k = 1 this type of transition remains absent. It should also be emphasized that the values k = 1 enforce the stability of the congurations obtained, which isobvious from the expressions of V 0 (') and V ('). 5

6 References [1] Bogoliubov N. N., Shirkov D. V. An introduction to the theory of uantized elds. Moscow, Nauka, 1976, 479p. [2] Weinberg S.Gravitation and Cosmology. John Wiley and Sons, Inc., NY, [3] Werle J. Phys. Lett. 1977, B 71, No. 2, p [4] Shikin G. N. Prob. Theor. Grav. and Elem. Chast. Moscow, Energoizdat, 1984, issue 14, p [5] Bronnikov K. A., Rybakov Yu. P., Shikin G. N. Izvestia VUZov, Phys., 1991, No. 5, p [6] Bronnikov K. A., Shikin G. N. Proc. of Sir A. Eddington centenary symp. v. 2, On the Relativity Theory, Singapore, WS, 1985, p [7] Rybakov Yu. P., Saha B., Shikin G. N. Izvestia VUZov, Phys., 1992, No. 10, p [8] Rybakov Yu. P., Saha B., Shikin G. N. XIII International Conference on General Relativity and Gravitation, Abstracts, Cordoba, Argentina, 1992, p.66. [9] Rybakov Yu. P., Saha B., Shikin G. N. VIII Russian Grav. Conf. Abstracts, Moscow, RGA 1993, p [10] Rybakov Yu. P., Saha B., Shikin G. N. Communications in Theoretical Physics (Allahabad Mathematical Society) 1993, Vol. 3, No 1, p [11] Kamke E. Dierentialgleichungen Losungsmethoden und Losungen. Leipzig,