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Exact Solutions to the Interacting Spinor and Scalar Field Equations in the Godel Universe

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Exact Solutions to the Interacting Spinor and Scalar Field Equations in the Godel Universe XJ9700082 E2-96-367 A.Herrera 1, G.N.Shikin2 EXACT SOLUTIONS TO fHE INTERACTING SPINOR AND SCALAR FIELD EQUATIONS IN THE GODEL UNIVERSE 1 E-mail: [email protected] ^Department of Theoretical Physics, Russian Peoples ’ Friendship University, 117198, 6 Mikluho-Maklaya str., Moscow, Russia ^8 as ^; 1996 © (XrbCAHHeHHuft HHCTtrryr McpHMX HCCJicAOB&Htift, fly 6na, 1996 1. INTRODUCTION Recently an increasing interest was expressed to the search of soliton-like solu ­ tions because of the necessity to describe the elementary particles as extended objects [1]. In this work, the interacting spinor and scalar field system is considered in the external gravitational field of the Godel universe in order to study the influence of the global properties of space-time on the interaction of one ­ dimensional fields, in other words, to observe what is the role of gravitation in the interaction of elementary particles. The Godel universe exhibits a number of unusual properties associated with the rotation of the universe [2]. It is ho ­ mogeneous in space and time and is filled with a perfect fluid. The main role of rotation in this universe consists in the avoidance of the cosmological singu ­ larity in the early universe, when the centrifugate forces of rotation dominate over gravitation and the collapse does not occur [3]. The paper is organized as follows: in Sec. 2 the interacting spinor and scalar field system with £jnt = | <ptp<p'PF(Is) in the Godel universe is considered and exact solutions to the corresponding field equations are obtained. In Sec. 3 the properties of the energy density are investigated. In Sec. 4 this field system is considered in flat space-time in order to determine the role of gravitation for the interacting fields. In Sec. 5 interacting spinor and scalar fields with Cmt = 5 in the Godel universe are considered and exact solutions to the corresponding field equations are obtained. In Sec. 6 the analysis of the energy density distribution is made for three different functions G(Ip). In Sec. 7 we study this field system in the Minkowski space-time. A discussion and interpretation of our results is presented in Sec. 8, together with a summary of our research. 2. INTERACTION £int = ± ip^F(Is) IN THE GODEL UNIVERSE The Lagrangian of the system of interacting spinor and scalar fields reads £ = ^(#7* V,,# - V^*) - + 2 + 2 C1) where_F(/s) represents an arbitrary function of the spinorial invariant Is = . S = <p is a massless scalar field, 7^ are the spinorial matrices in curve space-time defined by the tetradic vectors and the metric tensor components in the following form [4] 7" = e?(* w, (2) t)ab = diag(+l, -1, -1, -1); 1 where e® form a set of tetradic 4-vectors which is given by the expression [5] 9pv — (3) and 7® are the flat space-time Dirac matrices; we have chosen them according to Ref. [6]. Vp are the covariant spinor, derivatives; they are defined as follows [7]: = dp - r„(a), (4) where rp(x) are the spinorial affine connection matrices of curve space-time, determined by the relation = (5) F£„ being the Christoffel symbols. The metric of Godel universe is represented in the following form [8] ds2 = dt2 — dx2 + ^ e2v^n* dy2 + 2 e'^2Clx dydt — dz2 \ (6) here fi is the rotating angular velocity of the universe. For the 7^ and 7^ we have To = To; Ti = Ti; T2 = --^e1 >/2nT(\/27o + 72); Ta = Tai Ts = %; 70 = 70 — V272; 71 = 71; 72 = \/2e-v^0*7 2; 7s = 7®; { t5 = iwWtV = t5; (7) 1 where E^xr = y/^gSfiv\ T and e0i23 = 1. Consequently, the spinorial affine connection matrices become r« = 17‘r2; r, = |fV; r2 = A^n^i-c. ^ = 0 (8) From (1) we obtain the following set of field equations: 1 dR •V* = 0, _ _ 1 dR i V/i^r7p + m# + - - -g* = 0, (9 ) 1 = 0, x/-? 5x' 2 where R(S) = 1 4- F(Is) and g = det \gftV\. We consider # and y? depending only on x. Then, from the last equation we have /=§e-^n‘- / dV tp = , C = const. (10) C/Z On the other hand, the first equation of (9) takes the form T1 dx* 4- + V^1)* 4- i(m - $(x))$ = 0, (11) where $(z) = 1 • dQ/dS and Q(5) = 1/R Assuming that $ = and \ta(z, y, z, t) = va(x), a = 1, 2, 3, 4, we get V4 + ^ V4 + t(m - § - $)ui = 0, V3 + ^ v3 + i(m 4* § — $)vg = 0, (12) . v2 + ^ *>2 “ *(™ + 9 — $)v3 = 0, + ^5 Vi i(m — 9 — $)f4 = 0. From (12) we obtain the following equation for S — dS 4- \/2 QS = 0, (13) dx which has the solution S = C1e-V2nr, Ci = const. (14) With the transformation utt(x) = va(x)e ar^v^2, the set of equations (12) becomes . u\ 4- i(m - 9 - $)«i = 0, «3 + *(m 4- j - $)ti2 = 0, . tt2 - i(m -h $ - $)tf3 = 0, u[ — t(m — Y — $)«4 = 0. The solutions which satisfy this system are «i = ±01 ch[0i(x)], u2 = ±a2 ch[02(z)j, (16) u3 = ±»a2sh[02(x)], «4 = ±«oi sh[0i(x)j; x, where $i = (9 — m) x 4- f $dx + 6% and 02 = — (9 4- m) x 4- f $ dx + b2 . 3 The general solution to (12) has the form vi = ±aie-flz/v'5ch[0i(z)], v2 =±o 2e"nr/V5ch[^2(ar)], v3 = ±ia2 e~nx^, sh[fl2(a:)], V4 = ±«aie-nz/v^sh[0i(z)], Thus we have obtained exact solutions to the set of field equations (12); the general solution contains four integration constants: a\, a2 , bi, b2 . For the invariant S =t>i t>i+ v2 v2 — v3 i>3— v4 v4 we have 5 = (o; + o?)e-' /5n*. (18) 3. ENERGY DENSITY DISTRIBUTION In this section we study the energy density distribution of our system along the x axis and determine the properties of the solutions in the Godel universe. The energy-momentum tensor (EMT) which corresponds to (1) is T" = V"»-V"T7„*-V„*7"»)+y> i„v='"R(S)-^£. (19) By using the first two equations from (9), the Lagrangian can be written as follows c = (-5§s + fi)- .(20) Substituting. (20) to the zero-component of the EMT, one dbtains ^ = (2,) In order to determine whether the solution is a soliton one, it is necessary to analize the distribution of the energy density per unit invariant volume, i. e., e = Toy/\3g\, where 3g is the determinant of the matrix containing only the spatial components of the metric tensor. Since \/\3g\ = e^nx/y/2 = (a? -f 02)/x /2S, for £ we have: C2 S £ = iz(Qs)- (22) 2S(a\ + alf ds The total energy of the field system is defined by the formula +oo E,= f Tg\ffi\dV. (23) — OO 4 To obtain a finite value of Ej it is necessary to choose F(S) in an appropriate way. For example, if (A S)"+1 F(S) = n 1,2, 3, (24) 1 + (AS)" ’ where A is a coupling constant, for Ef (by integrating within the finite limits of the y and z axes) one gets the following expression: oo C2 c2 Ef = (QS) (25) 4fi(af + a|) 4AQ(aj + al) From (25) it follows that when A = 0 (absence of interaction), Ej —► oo. The same result we obtain when Q —► 0 (transition to Minkowski space-time). £ has the following asymptotical behaviour f 1/S —► 0, x —► —oo, (26) \ S —*• 0, x —► +oo, i. e., the energy density per unit invariant volume is localized. Thus, from (25) and (26) one can deduce that the solutions of the field system under consideration possess a localized energy density and finite total energy, i. e., are soliton-like. The qualitative dependence of e on the x coordinate is plotted in Fig. 1. Fig.l Qualitative dependence of the energy density per unit of invariant volume on the x coordinate: a deformed soliton due to the rotation of the universe. 5 Thus we have obtain an asymmetric (deformed) soliton due to the rotation of the universe. 4. INTERACTING SPINOR AND SCALAR FIELDS IN FLAT SPACE-TIME In this section we consider the interacting spinor and scalar fields in flat space- time in order to understand the role of the external gravitational field of the Godel universe. It is of interest to know, whether or not this interaction in Minkowski space-time leads to soli ton-like solutions. With the transition to flat space-time, when fi —► 0, all F matrices vanish. At the same time, reduce to and 7^ to 7^. The corresponding set of field equations is v\ -f i{m - $)%i = 0, V3 + *( m - §)"2 = 0, f27) v'2 — t(m — $)v3 = 0, v[ — *(m — $>)v4 = 0 where $ = (C2/2) dQ/dS. From (27) we get an equation for the spinorial invariant S: S' = 0, then S =const. and $ is also constant. Let us put M = m — $. Then from (27) we obtain the set of equations t," - M2 va = 0, a= 1,2, 3, 4. (28) with the solution Va = CaieM* + Ca2 e-u‘, where CQ\ and Ca2 are integration constants. Thus, the solutions of the equation system (27) adopt the form »i = Cue"* +C21e-"*, »i = C12e"* + C22e-"*, «3 = -<(C12e"*-C 22f"*), V4 — —i(Cue"* — C2ic-"*), where C11, C12, C21, C22 are constants. In this case the energy density is constant in the whole Minkowski space- time (see (21)—(22)), so the total energy of the field system is infinite for any function F(S). This means that for the interacting spinor and scalar field system there is no soliton-like solution, since the energy density is not localized and the total field energy is infinite.
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