The dynamics of the full class of multidimensional cosmological models with topology FRW $\times$ T$^{D}$, (where T$^{D}$ is a D-dimensional torus) near the singularity is investigated. Phase portraits show possible evolutions of FRW $\times$ T$^{D}$ models with hydrodynamical energy-momentum tensor. The problem of stability of solutions with a "crack-of-doom" singularity is also discussed

Szydłowski, Marek

Łapeta, A.

Łapeta, B.

Acta Physica Polonica B

English

Jagiellonian Univeristy Repository

VoL B21 (1990) ACTA PHYSICA POŁOWICA No 8MULTIDIMENSIONAL COSMOLOGICAL MODELS WITH HYDRODYNAMICAL ENERGY-MOMENTUM TENSOR.PART II. ANALYSIS OF DYNAMICAL SYSTEMS AT INFINITY*By M. Szydłowski, A. ŁapetaAstronomical Observatory of the Jagellonian University, Cracow**aNd B. ŁapetaInstitutc of Meteorology and Water Management, Cracow***( Receiυed June 12, 1989; reυised version receiυed January 15, 1990)The dynamics of the fuli class of multidimensional cosmological models with topology FRW × TD, (where TD is a D-dimensional torus) near the singularity is investigated. Phase portraits show possible evolutions of FRW×Ts models with hydrodynamical energy- -momentum tensor. The problem of stability of solutions with a ”crack-of-doom” singularity is also discussed.PACS numbers: 98.80.Dr0. IntroductionThis paper is a continuation of our previous work [4]. In the present part we disscussed trajectories at infinity.In Section 1 analysis of the dynamics near singularities is performed with the help of the method of dynamical systems. It has been possible to reduce the problem to a 2- -dimensional phase space [Part I]. The conclusions are summarized in Section 2.1. Dynamics of cosmological models near singularityTo have a complete analysis of the phase portrait of a dynamical system one must know how the system,s trajectories behave at infinity. To this purpose one should perform the transformation to projective variables (z = x~1, u = yx~1)', (v = y~1, w — xy~1)♦ The work was supported by the Polish Interdyscyplinary Project CPBP 01.03. ♦* Address: Obserwatorium Astronomiczne UJ, Orla 171, 30-244 Kraków, Poland.♦** Address: Instytut Meteorologii i Gospodarki Wodnej, Borowego 14, 30-215 Kraków, Poland.(643)645Fig. 1.1-1.3. Phase portraits for model FRW×Td with dust matterin projective coordinates (z, u), for D= 1, and K= 1, 0 -1and conformal time t: dτ = xdt (for z, u) and dt = y dt (for v, w). In these variables the straight lines z = 0 -oo<w<oo and υ = 0 -oo < w < oo correspond to infinitely distant points of the piane (x, y), and studying of this system in projective variables is analogical to investigating the system on finite regions. As previously, we will consider five cases.A. Vacuum modelSystem of equations (see Eq. (15)-Part I) in projective coordinates (z, u) assumes the form:(0where dtl = Hdt, with a condition:K(-6-6Du-D(D-1)u2) ≥ 0. (2)In projective variables (υ, w) we havedt? dw D(D—1)---- = Dv+3vw, —— = 2Dw+2w+ —------dτ1 dτl 3(3)647Fig. 1.4-1.6. Phase portraits for model FRW×Tfl with dust matter in projective coordinates (z, u), for D = 6, and K = 1, 0 — 1with the condition:(4)where dτ2 = h dt.B. Model with dust matterDynamical system (19) (see Part I) in variables (z, u) assumes the fonn:dz 2D+1 D(D-1) D(D-1) , (2D+1)dτl (D+2) Z+ (D+2) ZU 2(D+2) u Z+ ~D+Γκz,4« 3 3<d~1) D(D-T) 2 2Kdτ1 D+2 + D+2 “+ 2(D+2) “ + (D+2) ZJ>(Z>-1) 2(D+2)u≡+<^⅛>l⅛D+2 (5)649Fig. 1.7-1.9. Phase portraits for model FRW×Td with dust matter in projective coordinates (o, h>), for .0=1, and K = 1, 0 -1in the domain3+3Du + D(D~— u2+3Kz2 ≥ 0. (6)2For coordinates (», w) we have:dv 0(0+5) O—4 3 2 3K 3dτ2 .2(0+2) D+2 D+2 D+2dw D(D-1) D(D-7) 3(0-1) . 2 3---- —----------------------------- w----------- -— w---- >-----wdτ2 2(0+2) 2(0+2) D+2 D+220+1D+2Kυ2-D+23K , wv ,in the domain:3w2 + 3Dw+ — ------ - +3KtΛ≥ 0.OThe phase portraits of systems (5) and (7) prestnted in Figs. 1.1-1.11:(7)(8)651Fig. 1.10-1.12. Phase portraits for model FRW×Tfl with dust matter in projective coordinates (», w), for D = 6, and K = 1, 0 -1After transformation to variables (z, u) and time τ1: dτ1 = xdτ, system (23) (see Part I)C. Model with a massless scalar fieldassumes the form:dz , du ,— = 2z+Dzu+2Kz3, — ≈ -2Kz2m, (9)dτ1 dτ1with the condition:D(D-l) , ,3+3Du + 2 - - u2+3Kz2 ≥ 0, (10)while for variables (v, w) and time t2; dτ2 = ydτ we obtain:— = 2vw+Dv, — = — 2Kv2, dτ2 dτ2with the condition:, D(D-l) ,3w2+3Dw+ -i------ - +3Kv2 > 0.In Figs. 2.1-2.12 phase portraits of the above systems are presented.(H)(12)653Fig. 2.1-2.3. Phase portraits for model FRW×Td with massless scalar field in projective coordinates (z, u), for D = 1, and K = 1, 0 — 1D. Model with radiative matterDynamical system (26) (see Part I) in projective variables (z, u) assumes the form:dzdτ1dudτ13(0+2) --------- z + 0+3O2------ uz —0+33(0-1)^sτr0+D(β-1) 2 ι (2O+3)K ,2(0+3) 0+3D(D-7) _ 0(0-1) 32(0+3) “ 2(0+3) "2 ι (2O+3)K 2„ _z“H----- -—— zu.0+3 0+33X+z , (13)in the domain30+3+ u.(14)For variables (y, w) we have:dv D(D+7)^dτ ~ 2(0+3) r+9 3 2 3K 3------ -------------Σ OW V , 0+3 0+3 0+3655Fig. 2.4-2.6. Phase portraits for model FRW×Td with massless scalar field in projective coordinates (z, u), for D = 6, and K = 1, 0 -1dw 0(0-1) 0(0-7) 3(D — 1) , 3 3 3K 2----- —----------------------------w — -— ------- w —   - w-------- - V wdτl 2(0 + 3) 2(0 + 3) 0 + 3 0+3 0+32(O+3)K ,------------ v ,D + 3(15)in the domain:, 0(0-1) , „ 3w2 + 30w + —-------- + 3Kυ2 ≥ 0. (16)E. Model with the cosmological constantFor variables (z, w) system (28) (see Part I) assumes the form:dzdxz — Duz — 0(0-1)32(0-1)3(0+2)du dτ-2u-2Du2 + 2A0 + 20(0-1)32(0-1)3(0+2)Auz2,(17)657Fig. 2.7-2.9. Phase portraits for model FRW×Td with massless scalar field in ρrojective coordinates (p, w), for 2>= 1, and K = 1, 0 -1in the domain:(18)In variables (r, w) the system is:dw dτ1dv dτ1= Dt>+3t>w+' D(D-1)= 2Dw+2w2 + ■■ --2.4D+22(P-1)3(D+2)Λv2 -- 24D+2v2w, (19)in the domain:(20)As it can be seen, from the phase portraits obtained for the considered dynamical system in ρrojective variables (z, u) and (v, w), the structure of the phase piane is in prin- ciple independent of dimension D, except for the case D = 1. Another dimension explicitly inchided in the portraits is D = 6, which is distinguished by the superstring theories.659Fig. 2.10-2.12. Phase portraits for model FRW×Tfl with massless scalar field in projective coordinates(v, w), for D = 6, and K = 1, 0 — 1Furthermore, one should notice that all typical critical points are situated at the boundaries of the corresponding physical consitions. (A critical is called — typical, for t → oo, if it is an attracting point. A solution, for t → 0 (τ → — oo), is typical if the corresponding critical point is repelling.) [1, 2].Another property of typical critical points, which can be seen from the phase portraits, is that they lie at or below the straight linę u = 0, w = 0.Among five cases considered here, the phase portraits of the FRW x TD model with a mssless scalar field (p = p' = ρ) are peculiar in a sense. They are represented not by isolated critical points, but by infinitely many of them, constituting straight lines z = 0, — oo < u < oo and v = 0, —oo<w<oo (“singular straight lines”). Points situated on and above the non-physical region are unstable repelling points, whereas those situated below the non-physical region are stable attracting points. It should be also noticed that the straight linę z = 0 corresponds to the singularity of the physical space (H → oo), ■and the striaght linę v = 0 — to the singularity of the intemal space (A → oo).For a model with negative curvature (K = — 1) in variables z, u, besides the singular straight linę there are also iwo attracting knots z0 = ± / — —, «0 = 0, which correspond√ *to stable solutions with a static microśpace (j = 0).6602. ConclusionsIn this paper, the dynamics of the complete class of multidimensional cosmological modela with the topology FRW×Td and a source in the form of hydrodynamic energy- -momentum tensor was investigated.The phases portraits have been (as in the Part I), madę with the help of the computer program “Dynamie”.As in the Part I of the present paper, we shall not discuss particular cases, but rather point to some generał conclusions.1) For models with the closed intemal space and D > 1, the typical state of the metric, for large times is R → oo, r → 0, and t → t0. In other words, after a finite period of cosmo­logical time the model attains a singular state, in which the size of the physical space is much greater than that in the internal space. Rosenbaum, Ryan, Urrutia and Matzner in their paper of 1987 [3], showed that in the model with dust matter there are singularities of the kind which they called “crack-of-doom” singularities. The authors considered the case of one-dimensional internal space (D = 1) and stated that the dust models can reach a state in which, after a finite time t0, the internal space attains the zero size, while the physical space exhibits a regular behaviour (a singularity appears in the total space, but it is “hidden” from an observer in the physical space). Our considerations show that such states can actually occur, but they imply the existence of one additional dimension and are independent of the form of matter (for the dust case D = 1, dynamical equations are integrable; this was pfobabaly a reason that such singularities had been noticed at all). The models for D = 1 are very pećuliar ones, sińce D = 1 is a bifurcation parameter and the phase portraits for D > 1 have a quite different topological structure. To sum up, typical states of the metric are those in which R → ∞, r → 0 and t → t0 (D > 1); therefore, the crack-of-doom singularities mayappearonly after the physical space has grown in size to infinity. It should be noticed that in multidimensional cosmological theories the closed character of physical space doeś not imply that it is of a finite size (R < Rmax, as it is in the case of classical (1 + 3)-dimensional closed models).2) Let us consider models with a source in the form of hydrodynamic energy-momen- tum tensor. It can be shown [3] that quantum effects due to a massless scalar field in an extemal gravitational field at high temperatures, give rise to the hydrodynamic energy-scalar fields correspond to the case p = ρ discussed in the present paper.APPENDIXGenerał remarks useful in determining possible evolution patterns from phase portraits1. JPhase trajectories are scaled with different time parameters: Time parameter τ dt = — , dτ1 = x dτ) is a strictly monotonie function of time rf — = Λ>θ). The The effects of classical massless-momentum tensor of radiative matter661new time parameters τ1, τ2 are strictly monotonie functions of τ, for x >0 and y > 0, respectively, and of —r, for x < 0 and y < 0, respectively. Accordingly, the attracting critical points occuring in the region x < 0 and y < 0 are repelling critical points in time τ or t.2. In asymptotie states, typical states of the metric are always situated at boundaries of the relevant constraint conditions and are represented by Kasner asymptotics:3p1+Dp2 = 3pl+Dpl = I,- Pi± =3±√3D(D+2)3(D+3)P2± ~D±√3D(D+2)3(0+3)The asymptotics of corresponding solutions, in tim,e t, have the form: near the initial singularityR∞tpι∖ rκtfl', .R ∞ tp'-, r∞rpι∖ t → 0. (Al)R∞tpt-, rxtpιt, t→0. (A2)and near the finał singularityR ∞ Go ~ 0p,+, r oc (t0 - t)p*-, (BI)R oc (t0—t)pι^, r oc(to-t)p2+, t→t0. (B2)3. Asymptotie states, in which the macrospace and the microspace are comparable in size, are represented by non-stable saddle points. For instance, asymptotie solution for radiations are of the form:2R oc r oc (0d+4, near the initial singularity,2R oc r oc (t0- <)d+4, near the finał singularity.4. Singularities of the crack-of-doom type are always connected with the presence of the critical point v0 = 0, w0 = θ> i∙e∙ a singularity occurs in the intemal space, and thus in the total space, for arbitrary values x. The corresponding asymptotie solutions near singularities of this type are of the form (BI) and (B2). If t → t0, R → const for D = 1, and R → oo for D > 1, while r oc (t0—t) for D = 1, and r oc (t0-t)pι+ for D > 1 (p2+ < 0). The asymptotie B2 is valid for a space-time with an expanding physical space and a contracting intemal space, and the asymptotie BI for symmetrical solutions with a contracting physical space and an expanding internal space. Let us notice that the initial singularity in the FRW(R = — 1) × T1 model has the structure of a crack-óf-doom singu­larity. Because of a dynamics of additional dimensions, although the physical space has no initial singularity, the singularity in the total space results from that of the intemal space (which expands from a singularity to a constant size). From the phase portraits 662it can be seen that tbe finał singularities of the crack-of-doom type are typical features of models with closed physical space; this models exħibit the following behaviour R → const for D = 1, and R → ∞ for D > 1, wħen t → t0∙ For models with fiat physical space, this state is attained for t → co.5. Let us notice that the phase trajectories in the system (z, u) and (v, w) are sym- metrical with respect to the axes z = 0 and v ≈ 0, respectively. In the region z < 0 and v < 0, times τt and τ2 are monotonie functions of —t. From the phase portraits it can be seen that the crack-of-doom singularities are stable finał singularities (in time r) within the class of solutions with contracting intemal spaces, and stable initial singularities within the class of solutions with expanding intemal spaces.We are gratefull to Prof. M. Heller for reading the manuscript and discussion.REFERENCES[1] D. Errousmin, K. Pleys, Obyknovennye differentsiyalnye uraυnenniya, Moscow 1980 (in Russian)’[2] A. M. Samoyleńko, S. A. Krivasheiya, N. A. Perestiuk, Differentsiyalnye uravneniya, Kiev 1984 (in Russian).[3] M. Rosenbaum, M.Ryan, L. Urrutia, R. Matzner, Phys. Rev. D36, 1032 (1987).[4] M. Szydłowski, B. Łapeta, A. Łapeta, Acta Phys. Pol. B21, 627 (1990).

Multidimensional cosmological models with hydrodynamical energy-momentum tensor : part II : analysis of dynamical systems at infinity

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Multidimensional cosmological models with hydrodynamical energy-momentum tensor : part II : analysis of dynamical systems at infinity

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