KdV solitons in Einsteins vacuum field equations

KdV solit o ns in E i nstein’s v acuum field equati ons Deb o jit Sarma 1 and Mahadev Patgiri 2 Dep artment of Physics, Cotton Col le ge, Guwahati-781001, India Abstract W e present a me tric for which Einstein’s field equations in v acuum generate the Kort weg-de V ries (KdV) equation and th us its N -soliton solutions solv e the v acuum equa- tions. Th e metric of th e one soliton s olution has b een inv estigated and is a non-singular, Loren tzian metric of t yp e N in the P etrov classification. Keywor ds: Einstein’s equations, gra vitational w av es, solitons, KdV equation, in t egrabilit y P ACS numb ers: 04.2 0 .Jb, 04.30.-w, 02.30.Ik, 0 5.45.Yv 1 In tro duction Einstein’s field equations are inheren tly nonlinear a nd hence exact solutions for them a re not easy to obtain. Because of their nonlinearity , they a re exp ected to hav e soliton solutions, at least fo r some spacetimes. These are pulse-lik e tra v elling w a v e solutions having fascinating prop erties, one b eing that they propag ate without noticeable disp ersion and inte ract without c hange of form altho ug h they do not ob ey linear sup erp o sition. Einstein’s equations admit wa v e solutions whic h is eviden t on linearization of these equations for some metrics. The problem of obtaining suc h solutions for the field equations without linearization is ho w ev er a more difficult one, one reason b eing that no general solution of these equations is known. The tec hniques of extracting soliton solutions of no nlinear ev o lutio n equations ha ve b een a pplied to the task of obtaining exact solutions o f Einstein’s equations. Belinski and Z akharo v [1, 2, 3 ] (a nd [4] for a comprehensiv e ov erview) mo dified the inv erse scattering tra nsform (IST) for application to Einstein’s equation in v acuum, and since then there has been a consisten t searc h fo r soliton solutions of the v acuum equations based on their tec hnique. Differen t metrics are taken as the seed metric, from whic h solito n solutions are generated b y the IST. In this manner, a n umber of solutions o f the Einstein equations ha ve b een found. 1 debo jitsarma @yahoo.c o m 2 mahadev@scie ntist.com 1 In this paper, we approac h the problem from a differen t direction. W e attempt to construct a metric for whic h the v acuum Einste in’s equations pro duce a w ell-kno wn, nonlinear, ev o lution equation, namely the Kort weg-de V r ies (KdV) equation [5], whic h is kno wn to b e integrable, so that the existence of N -soliton solutions and integrabilit y of the sy stem is assured. Substitution of an y of the soliton solutions in the metric will result in a spacetime solving the v acuum Einstein’s equations. W e sp ecifically c hec k the metric with one soliton solution and study its prop erties, part icularly , the curv ature and classification sc heme. The metric is Lorentzian and using the Pere s alg o rithm [6], w e find that the metric obtained here b elongs t o the ty p e N in the P etro v classification sc heme and is therefore a metric with radiat ion. Although the metric apparen t ly has a singularit y a t t = 0, we sho w that this is not a phy sical singularit y and app ears only due t o the c ho ice of the co ordinate system. The plan of this pap er is as follows: in section t wo, we presen t the metric that generates the KdV equation. In section three, the metric with one soliton solution is discussed in detail. The fourth section is the concluding one. 2 V acuum s pacetime and the KdV equation W e in tro duce the line elemen t ds 2 = − h af 2 ( x, t ) − 2 f xx ( x, t ) i dt 2 + 2  3 2 t  4 3 dx 2 − 2 f ( x, t ) dtdx + dy 2 + 2  3 2 t  2 3 dxdy + dtdz (1) where a is a constan t and the subs cripts denote partial deriv ativ es. The only non v anishing elemen t of the Ricci tensor for the metric ( 1 ) is R tt and the corresponding v acuum equation for it, R tt = 0 yields the follow ing nonlinear equation f xt ( x, t ) − af ( x, t ) f xx ( x, t ) − af 2 x ( x, t ) + f xxxx ( x, t ) = 0 . (2) Setting a = 6 and p erforming one inte gration ov er x leads to one of the b est-kno wn in tegrable nonlinear partial differen tial equation in the literature, namely , the KdV eq uation in its standard form (the in tegratio n constant v anishes for soliton solutions [7]) f t ( x, t ) − 6 f ( x, t ) f x ( x, t ) + f xxx ( x, t ) = 0 . (3) It is w ell kno wn that the KdV equation has all the b eautiful prop erties that characterize an integrable, no nlinear system including an infinite n umber of conserv ed quan tities in in- v olutio n (whic h defines Lio uville in tegrability), N -soliton solutions, a bi-hamiltonian structure among other prop erties ( f or a review of solito n equ ations, sp ecially the KdV equation, and their prop erties see [7, 8, 9, 10]). Apart fro m the source of muc h new mathematics, this equation app ears in a wide v ariety of ph ysical problems. In in tegrable systems, lik e the KdV, disp ersion 2 is compensated by nonlinearity giving rise to soliton solutions of all o r ders called N soliton solutions. The one soliton solution (1SS) of the KdV equation is f ( x, t ) = − k 2 2 sec h 2 η 2 (4) where η = k x − k 3 t , k b eing the wa v e num b er. Equation (4) represe n ts a in v erted pulse of sech 2 profile t r a velling in the p ositive x direction. The KdV equation b eing nonlinear, linear com binatio ns of the one soliton solution do not pro vide new solutions. Inste ad, t wo soliton solutions (2SS) ma y be constructed, whic h represen t a nonlinear interaction of t wo solitons. F or the KdV, the 2SS is f ( x, t ) = 1 2  k 2 1 − k 2 2  " k 2 2 cosec h 2 ( η 2 / 2) + k 2 1 sec h 2 ( η 1 / 2) ( k 2 coth( η 2 / 2) − k 1 tanh( η 1 / 2)) 2 # (5) where η 1 = k 1 x − k 3 1 t and η 2 = k 2 x − k 3 2 t . That (4) and (5) satisfy the KdV equation (3) can b e v erified b y direct substitution. Existence of soliton solutions alone do es not imply integrabilit y , although it is relev an t to men tion that existence of at least t hree soliton solutions is a necessary condition for it. It is w ell established tha t t he KdV equation passes a ll tests of integrabilit y . 3 Prop erties of th e spacet ime with one sol iton solut ion If w e substitute f ( x, t ) from (4) in to the metric (1), we obtain the following f o rm for it. ds 2 = − k 4 sec h 2 η 2 dt 2 + 2  3 2 t  4 3 dx 2 + k 2 sec h 2 η 2 dtdx + dy 2 + 2  3 2 t  2 3 dxdy + dtdz (6) whic h satisfies the v acuum Einstein equations. Substituting the t wo soliton solution (5) in (1) w ould again lead to a metric satisfying the v acuum equations. In this manner, a hierarc hy of solutions of the v acuum equations can b e constructed. The metric (6) is Lor entzian with determinan t detg = − 1 4  3 2 t  4 3 . The nonzero comp onents of the Riemann-Christoffel curv ature tensor R µ ν ρσ corresp onding to the metric (6) are R 1 441 = − 2 9 t 2 (7) R 1 442 = (2 / 3) 2 / 3 3 t 8 / 3 (8) R 2 441 = − (2 / 3) 1 / 3 3 t 4 / 3 (9) R 2 442 = 2 9 t 2 (10) 3 R 3 141 = 2 2 / 3 3 1 / 3 t 2 / 3 (11) R 3 142 = 4(2 / 3) 1 / 3 3 t 4 / 3 (12) R 3 241 = 4(2 / 3) 1 / 3 3 t 4 / 3 (13) R 3 242 = 2 9 t 2 (14) R 3 441 = 2 k 2 sec h 2 ( η / 2) 9 t 2 (15) R 3 442 = ( 2 3 ) 2 / 3 k 2 sec h 2 ( η / 2) 3 t 8 / 3 (16) It app ears from the curv ature comp o nen ts a b o v e, as w ell as the determinan t of g µν , that the metric is singular at t = 0. Ho w ev er, this is merely a co o rdinate singularity . T o sho w this, w e first note t hat the curv ature scalars R µν ρσ R µν ρσ and R µν ρσ R ρσλτ R µν λτ do not sho w any singularit y b eing constants equal to zero. Moreov er, a co ordinate transformation t = 2 3 e − 3 τ (17) allo ws us to write (6 ) as ds 2 = − 4 k 4 e − 6 τ sec h 2 ξ 2 dτ 2 + 2 e − 4 τ dx 2 − 2 k 2 e − 3 τ sec h 2 ξ 2 dτ dx + dy 2 + 2 e − 2 τ dxdy + 2 e − 3 τ dτ dz (18) where ξ = k x − 2 3 k 3 e − 3 τ . The metric, in the form giv en by (1 8) remains Loren tzian with its determinan t b eing − e − 10 τ . F or the spacetime (18), the comp onen ts of the curv ature tensor are R 1 441 = − 2 (19) R 1 442 = − 3 e 2 τ (20) R 2 441 = − 2 e − 2 τ (21) R 2 442 = 2 (22) R 3 141 = − 6 e − τ (23) R 3 142 = − 4 e τ (24) R 3 241 = − 4 e τ (25) R 3 242 = − e 3 τ (26) R 3 441 = 2 k 2 sec h 2 ξ 2 (27) R 3 442 = 3 e 2 τ k 2 sec h 2 ξ 2 (28) 4 The curv ature comp onen t s abov e are clearly non-singular. In addition, the curv ature scalars R µν ρσ R µν ρσ and R µν ρσ R ρσλτ R µν λτ are nonsingular. Consider the one soliton metric in fo r m giv en b y (6). Under the asymptotic condition η → ∞ , sec h 2 η 2 → 0, a nd w e a lso find that the v alue of the determinant of the metric tensor do es no t c hang e and only the curv ature comp o nen ts R 3 441 and R 3 242 v anish, other comp onen ts remaining unc hang ed. Hence, the metric remains non-singular under the ab ov e-men tioned asymptotic condition. F or v acuum solutions, the W eyl conformal tensor C µν ρσ and the curv ature tensor R µν ρσ b ecome iden tical. W e apply t he Pere s algorithm to determine the P etro v classification of the conformal tensor. In the presen t case, C µν ρσ 6 = 0 (29) ho w ev er C 2 = ⋆C 2 = 0 and C 3 = ⋆C 3 = 0 (30) where C 2 = C µν ρσ C µν ρσ (31) ⋆C 2 = ⋆C µν ρσ C µν ρσ (32) C 3 = C µν λξ C λξ ρσ C µν ρσ (33) ⋆C 3 = ⋆C µν λξ C λξ ρσ C µν ρσ (34) where ⋆ C µν ρσ = 1 2 η µν λξ C λξ ρσ (35) η µν λξ b eing the fully an ti- symmetric tensor in four dimens ions. The P eres algorithm states that equations (29) and ( 3 0) imply that the spacetime is ty p e N or I II. Additionally , in our case, the following con traction of the W eyl tensor C µν λξ C λξ ρσ = 0 whic h means that the metric is of t ype N. The geo desic equations for a mat eria l particle in the metric (6) are found to b e du 0 dτ = 0 (36) du 1 dτ = − 2    u 1 u 0 t +  2 3  2 / 3 u 2 u 0 3 t 5 / 3    (37) du 2 dτ = 2  2 1 / 3 (3 t ) 2 / 3 u 1 + u 2  u 0 3 t (38) du 3 dτ = 2 9 t 5 / 3 h k 2 sec h 2 ( η / 2)  9 t 2 / 3 u 1 − 9 t 5 / 3 k 3 u 1 tanh( η / 2 ) + 2 2 / 3 3 1 / 3 u 2  u 0 +3 1 / 3 (2 t ) 4 / 3 u 1  3 × 2 1 / 3 t 2 / 3 u 1 + 3 1 / 3 u 2  +36 t 5 / 3 k 3 cosec h 3 ( η )sinh 4 ( η / 2)  ( u 1 ) 2 + k 4 ( u 0 ) 2 i (39) 5 The g eo desic equations ab ov e are coupled differen tial equations whic h cannot b e easily solv ed b y insp ection. How ev er, the symmetries of the spacetime may help in obtaining their solutions and will b e in ve stigated in the next communic ation. 4 Conclus ion It is clear, therefore, that w e ha v e a metric for whic h the v acuum Einstein’s equations reduce to the KdV equation whic h is known to b e completely in tegrable. Th us the solutions o f t he Einstein equations, for this metric, are the soliton solutions of the KdV equations. F rom this metric, we get the one soliton metric whic h is no n- singular and of Petro v t yp e N. In this con text, we note that many of the soliton solutions found by the IST do not ha v e pro p erties that characterize solitons suc h as p ermanenc e of form and amplitude [4]. F or solito n solutions generated by the metric studied in this pap er, ho w ev er, w e exp ect t hese properties to be presen t since they are the KdV solitons. 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