A closed-form solution in a dynamical system related to Bianchi IX

A closed- form solutio n in a dyn amical s ystem related to Bianc hi IX R. Con te ∗ Service de ph ysiq ue de l’ ´ etat condens ´ e (URA 2464) CEA–Sacla y , F–911 91 Gif-sur-Yv ette Ce dex, F r ance E-mail: Rob ert.Con te@cea.fr 7 Septem b er 2007 Abstract The Bianc hi I X cosmolog ical mod el in v acuum ca n b e rep resented by sev eral six-dimensional dynamical systems. In one of them we present a new closed form solution expressed by a third P ainlev´ e function. Keywor ds : Bianchi IX in v acuum, third Painlev ´ e equation. P ACS 19 95 : 0 2 .30.-f, 05 .45.+b, 4 7 .27.-i, 98.8 0.Hw The Bianchi IX cosmological mo del in v acuum can b e defined by the metric [4] d s 2 = σ 2 d t 2 − γ αβ d x α d x β , (1) γ αβ = η ab e a α e b β , η = diag ( A, B , C ) , (2) in which e a α are the comp onents of the three frame vectors, and σ 2 = ± 1 according as the metric is Minko vskian or Euclidean. Introducing the lo g arithmic time τ by the hodog raph transfo r mation d τ = d t √ AB C , (3) this giv es rise to the six-dimensional sy s tem o f three second o rder ODEs σ 2 (log A ) ′′ = A 2 − ( B − C ) 2 and cyclically , ′ = d / d τ , (4) or equiv alently σ 2 (log ω 1 ) ′′ = ω 2 2 + ω 2 3 − ω 2 2 ω 2 3 /ω 2 1 and cyclically , (5) under the change of v ariables A = ω 2 ω 3 /ω 1 , ω 2 1 = B C a nd cyclica lly . (6) If o ne introduce s the six v aria bles y 1 = A σ , z 1 = d 2d τ log( B C ) , and cyclically , (7) the dy namical s ystem (4) can b e alternatively represented b y [2] d y j d τ = − y j ( z j − z k − z l ) , d z j d τ = − y j ( y j − y k − y l ) , (8) ∗ Preprint S2007/085. T o app ear, Physics Letters A. 1 in w hich ( j, k, l ) is any p ermutation of (1 , 2 , 3). This system a dmits the first integral σ − 2 K 1 = y 2 1 + y 2 2 + y 2 3 − 2 y 2 y 3 − 2 y 3 y 1 − 2 y 1 y 2 − ( z 2 1 + z 2 2 + z 2 3 − 2 z 2 z 3 − 2 z 3 z 1 − 2 z 1 z 2 ) . (9) All the single v alued so lutio ns of (4) are kno wn in closed form [1, 8], except a four-para meter solution [5 ] whic h would extrap ola te the three-parameter elliptic solution [1] ω j = σ q ℘ ( τ − τ 0 , g 2 , g 3 ) − e j , j = 1 , 2 , 3 , K 1 = 0 , (10) in w hich ℘, g 2 , g 3 , e j is the classica l notation of W eierstr ass, ℘ ′ 2 = 4 ℘ 3 − g 2 ℘ − g 3 = 4( ℘ − e 1 )( ℘ − e 2 )( ℘ − e 3 ) , ( g 2 , g 3 , e j ) co mplex , (11) and g 2 , g 3 , τ 0 are arbitra ry . The solution (10) also represents the motion o f a rigid b o dy ar o und its center o f ma ss (E ule r , 17 50), σ ω ′ 1 = ω 2 ω 3 , a nd cyclica lly . (12) An y hin t to find the a b ove mentioned missing four-parameter so lution would be welcome, and some indicatio ns can b e found in Ref. [5]. In the pr esent Letter, we pre s ent such a hint , a s a five-parameter solution of (8). Despite its lack of physical meaning, it could share some analytic structure with the unkno wn solutio n a nd therefor e provide a useful insight. When one co o rdinate y i v anis hes, say y 1 = 0, the corr e sp o ndence (3) b etw een the physical time t and the lo garithmic time τ breaks do wn, but the system (8), whose in vestigation was then started in Ref. [6], can be in tegrated in closed for m. T a k ing account of the tw o a dditional fir st integrals [6], c = − z 1 , K 2 = y 2 y 3 e − 2 cτ , (13) the s ystem reduces to        ( z 2 − z 3 ) ′ = − ( y 2 + y 3 )( y 2 − y 3 ) , ( y 2 + y 3 ) ′ = − ( y 2 − y 3 )( z 2 − z 3 ) − c ( y 2 + y 3 ) , ( y 2 − y 3 ) ′ = − ( z 2 − z 3 )( y 2 + y 3 ) − c ( y 2 − y 3 ) , ( z 2 + z 3 ) ′ = − ( y 2 − y 3 ) 2 . (14) F or c = 0, the system for z 2 − z 3 , y 2 + y 3 , y 2 − y 3 is a no ther Euler top, whose g eneral solution is            y 1 = 0 , z 1 = 0 , z 2 − z 3 = p ℘ ( τ − τ 1 , g 2 , g 3 ) − e 1 , y 2 + y 3 = p ℘ ( τ − τ 1 , g 2 , g 3 ) − e 2 , y 2 − y 3 = p ℘ ( τ − τ 1 , g 2 , g 3 ) − e 3 , z 2 + z 3 = ζ ( τ − τ 1 , g 2 , g 3 ) + e 3 ( τ − τ 1 ) + 2 z 0 , (15) in w hich g 2 , g 3 , τ 1 , z 0 are the four arbitrary constants. F or c 6 = 0 , the elimination of ( y 3 , z 2 , z 3 ) b etw een the t w o first in tegrals and the o riginal system yields the genera l solution                        y 1 = 0 , z 1 = − c 6 = 0 , − y ′′ j + y ′ j 2 y j + y 3 j − K 2 2 e − 4 cτ y − 1 j = 0 , j = 2 or 3 , y 2 y 3 = K 2 e 2 cτ , z 2 − z 3 = y ′ 3 2 y 3 − y ′ 2 2 y 2 , z 2 + z 3 = ( y 2 − y 3 ) 2 − ( z 2 − z 3 ) 2 + σ − 2 K 1 − c 2 2 c , (16) 2 and the second or der ordinary differen tial equation for y 2 (or for y 3 as w ell) is a third Painlev´ e equation [7], with the corr esp ondence d 2 w d ξ 2 = 1 w  d w d ξ  2 − d w ξ d ξ + αw 2 + γ w 3 4 ξ 2 + β 4 ξ + δ 4 w , (17) w = y 2 or y 3 , ξ = e − 2 cτ , α = 0 , β = 0 , γ = 1 , δ = − K 2 2 c − 4 . (18) In the generic case cK 2 6 = 0, this so lution is a meromorphic function of τ , with a tr a nscendental depe ndence on the tw o constants of in tegra tion other than ( c, K 1 , K 2 ). What is remark a ble is that the unknown four-par ameter solutio n of (4) and the P ainlev´ e II I solution (16) o f (8) ar e b oth extra po lations of an Euler top. This s uggests lo ok ing for ano ther po ssible three- dimensional E ule r top in the six -dimensional physical sy stem (4). Suc h a three- dimensional subsystem would necessar ily corr esp ond to a no n self-dual curv ature [3]. References [1] V. A. Belins kii, G. W. Gibb o ns, D. N. Page, and C. N. Pope, Asymptotically Euclidean Bianchi IX metrics in quantum gr avit y , P hys. Lett. A 76 (19 78) 433 –435 . [2] G. Con top oulos, B. Grammaticos, a nd A. Ra mani, The mixmaster universe mo del, revisited, J. P hys. A 27 (199 4) 535 7 –536 1. [3] G. W. Gibbo ns and C. N. Pop e, The p ositive action conjecture and asymptotically Euclidean metrics in quantum gravit y , C o mm un. Math. P hys. 66 (1979) 26 7 –290 . [4] L. D . Landau et E. M. 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