Abel olmayan Kaluza-Klein teorisinin klasik çözümleri

Abstract

ÜZET 3 Bu çalışmada iç uzayı S olan yedi boyutlu Kaluza-Klein teori sini göz önüne aldık. Genelleştirilmiş Einstein denklemlerinin kozmo lojik ve kütle çekimse! çözümlerinin nümerik ve analitik bir incele mesini yaptık. Dış uzay radyasyon ile dolu ve iç uzay boş olduğu durumda ge nelleştirilmiş Einstein denklemlerinin evrenin evriminin büyük kısmın da, iç uzayı sabit kalan bir çözümünün bulunduğu gösterildi. Alan denklemleri elde edildikten sonra bazı özel haller için analitik çö zümler bulundu. Euler-Newton ve Runge-Kutta yöntemleri kullanılarak, alan denklemlerinin nümerik çözümleri elde edildi. Nümerik çözümle rin tutarlılığı analitik olarak da incelendi. Beş, altı ve yedi boyutlu abel olmayan Kaluza-Klein teorisinde Schwarzschild çözümlerinin varlığı araştırıldı. Altı ve yedi boyutlu Hallerdi Schwarzschild çözümü nün olmadığı gösterildi.

CLASSICAL SOLUTIONS OF NONABELIAN KALUZA- KLEIN THEORY SUMMARY In four dimensional space time the most natural gravitational action is the Einstein-Hil bert action on which all standard cosmological models are based. The Kaluza-Klein compactifi cation of additional dimensions requires that a suitable theory should be based on dimensional ly continued Euler forms [1-11]. In cases where the radius of the internal space is constant or variable, the investigation of cosmological solutions has been the subject of various papers [2,4,5,12]. In this framework, it has been pointed out that in cases where the internal space radius is not constant, numerical methods are necessary [5]. In this thesis we have made a numerical investigation of the cosmological solutions of Kaluza-Klein theory, with a three dimensional maximally symmetric internal space, based on a dimensionally continued, four dimensional Euler form. In such a theory the four dimensional reduced action, and equations of motion are free of the cosmological term [11]. The topics covered in this thesis include a short review of the Cartan structure equations in differential geometry, calculation of the connection and curvature of the spheres §2and S3. These topics are not results of original research but are necessary for an introduction into nonabelian Kaluza-Klein theory and are extensively used throughout the thesis. The main part of this thesis starts with introducing the model, the equations of motion of the model which are given by PQ+2S2+S(P+Q) ¦vi-(2Q+2S) + g (P+Q+4S)+(Q-P)S = -p-3p (P+Q+4S)+ £ (2P+2S)+(P-Q)S = -p-3q where p = 'i. + i- r 9 2 a a o- £- + 1-.2 u2 S = b` b£ a b and the dot denotes derivative with respect to t, a(t) is the space radius, b(t) is the internal space radius,, p is the energy-matter density, p is the pressure in space and q is the pressure in internal space. The equations are linear in the second derivatives of the variable a and b. As has been shown by [13], this is the general but unique property of equations of motion derived from dimensionally continued EuTer form actions. These equations lead to p+3(p+p) +3(p+q) =o which corresponds to the conservation of the seven dimensional energy momentum tensor. With p = p/3, q = o this conservation equation yields -vn-4 3 pa b = m = Const. We find analytical solutions of the equations of motion for some special cases where the internal space radius b is constant. We investigate the consequences of the equations of motion p = p/3, q = o. Equations of motion are given by (P+Q+4S) +2(P+S) £ +PQ+2SP+2S2 = o ¦(2P+6S) +(-3P+Q) £ +3S (Q-P) = o where P,Q and S are given above, and can not be solved analytically. We look for a solution such that the internal space b remains small and constant. We assume that at t = t0, a = a0, b = b0«' a0 whereas a = b = o. We solve equations of motion for a and b and iterate backwards in t using the Euler-Newton method. To increase the reliability of.our numerical solutions we write the system of equations of motion as four first order equations by introducing the variables a = â and M b. Then we use the Runge-Kutta method. The numerical results indicate that b approaches zero before `a`. To investigate this point more clearly, we cast the equations of motion into the form of a single second order differential equation where `a` is the independent variable and b = b(a) is the dependent variable. Denoting derivatives with respect to `a` by primes we have b - b'a b a2+b'a -4 -3 We use p = ma b from equations. of motion and. solve the equation for â obtaining -V11Va = a(a,b,b' ) We solve equations of motion for a and b to obtain a = a(a,a,b,6) b = b(a,â,b,b) By,.using these five equations, we eliminate the four `unknows` a, b, J i a,b to obtain b as a function of a, b,b' b`= F(a,b,b') This equation can be used to compute b as a function of `a` except at t = t0 where â = o and b1 is not well-defined. Therefore in order to check the numerical solution obtained by iterating in the variable t by comparing it with the method just described, for ti<t <t0 we use the t-iteration method but at t = tx we start the a-iteration procedure by matching the new initial conditions to the result obtained by. the t-itsration program. Thus a1 s a(t) b' b â ll '1 Thus numerical results which are obtained using two different methods of iteration for solving the differential equations have been given. For each method of iteration, the differential equations themselves are considered in two different forms depending on the choice of the independent variable. At the end of the section IV, the consistency of the numerical solutions of the equations of motion are demonstrated using analytical methods..ix-Investigation of the existence of the Schwarzschild soluton in nonabelian Kaluza-Klein theory is an important topic. In section V the Schwarzschild solution is reviewed for four dimensions and is investigated for five, six and seven dimensions. In four dimensions the Schwarzschild solution is unique. In five dimensional Kaluza-Klein theory the Schwarzschild solution depends on one parameter. In this theory this parameter shows that the inertia! mass and the gravitational mass of the matter which causes the gravitational field can be different. In six dimensions we find that the Schwarzschild solution does not exist. In seven dimensions we obtain a third degree equation which is given by 34 ( - )3t9 ( 21 )2+ 252 ( - ) - 177 = 0 M M M where m is the inertial mass and M is the gravitational mass. This equation has to be satisfied. Hence for a consistant solution to this order m^M. In higher order calculations even if there is a consistent n,. there is no solution in which the'inertial mass is equal gravitational mass. ı his is not compatible with observations. In the appendix we present the computational programs and their results connected with the methods of iteration used in section IV. -x-