Sci.Int.(Lahore),29(3),573-575, 2016 ISSN 1013-5316;CODEN: SINTE 8 573 CONFORMAL MINKOWSKI SPACETIME AND THE COSMOLOGICAL CONSTANT Farhad Ali and Wali Khan Mashwani Department of Mathematics, Kohat University of Science and Technology, Kohat, KPK, Pakistan. email: [email protected]; [email protected] ABSTRACT: This paper introduces the use of Noether symmetry equation for Lagrangian of conformal plane symmetric static Minkowski spacetime to find all those Minkowski spacetimes which admit the conformal factor. We also discuss the cosmological constant in these spacetimes.
Keywords: Static Conformal Mikowski Spacetime, Cosmological Constant, Einstein Field Equations.
1 INTRODUCTION equation (2) which admit the conformal factor e(x) for Symmetries play an important role in different areas of some particular values of the function . The research, including differential equations and general relativity (x) [4, 6, 7, 10, 12]. The Einstein field equations(EFE) [9], cosmological constant is also discuss for the obtained 1 spacetimes. R Rg g = T. (1) 2 The Noether Symmetry Governing Equation 2 A symmetry X is the Noether symmetry if it satisfies the are the building blocks of the theory of general relativity. following equation, These are Highly non-linear second order partial differential 1 equations and it is not easy to obtain exact solutions of these X L (D)L = DA, (4) equations. Symmetries help a lot in finding solutions of these where equations. These solutions (spacetimes) have been classified 1 0 1 2 3 by using different spacetime symmetries. Among different X = X ,s ,s ,s ,s , (5) spacetime symmetries isometries or the Killing vectors (KVs) t x y z have the importance that they help in understanding the is the first order prolongation of geometric properties of spaces also corresponding to each X = 0 1 2 3 , (6) isometries there is some conserved quantity. They are also s t x y z subset of the Noether symmetry (NS) [1-3] i.e. D is differential operator define as KVs NS. This relation shows that the KVs do not lead to all the D = t x y z , (7) conserved quantities or the first integrals. Therefore, it looks s t x y z reasonable to look for Noether symmetries from the i Lagrangian of spacetimes. Instead of taking the Lagrangian and A is a gauge function. , and A are functions from the most general form of the spacetime and solving a set i of s,t, x, y, z, and ,s are functions of of partial differential equations involving unknown metric coefficients one may adopt an easy approach and directly look s,t, x, y, z,t,r, y, z , where ``." denotes differentiation for the NS from Lagrangian of all known spacetimes obtained with respect to s . From differential geometry we know that through classification by KVs. However, we have adopted for the general cylindrically symmetric static space-times here the longer route, so that we also have a counter check on given by eq. (2), the usual Lagrangian is the spacetimes obtained through the classification by the KVs. (x) 2 2 2 2 L = e (t r y z ), (8) Using the Lagrangian of plane and spherical symmetric static spacetimes, a complete list of Noether symmetries have been Using eq.(8) in eq.(4) we get the following system of 19 obtained and new first integral have been found. Some new partial differential equations (PDEs) solutions have also been found in the case of spherical t = 0, x = 0, y = 0, z = 0, As = 0, symmetry. Here, we want to use the technique of Noether (x) 0 (x) 1 symmetries and find all those Minkowski spacetimes which 2e s = At , 2e s = Ax , admit the conformal factor [8]. We take the following form of 2 (x) 2 (x) 3 2b e s = Ay , 2e s = Az , the conformal Minkowski spacetime [11] (x)1 2 2 = 0, 2 3 = 0, 2 (x) 2 2 2 2 (2) x y s z y ds = e (dt dx dy dz ), 1 2 1 3 (9) For spacetime (2), the Einstein field equations (1) read y x = 0, z x = 0, 0 1 0 2 1 2 (x) x t = 0, y t = 0, xx (x) x (x) e = kT00 = kT22 = kT33, 0 3 1 0 4 z t = 0, x (x) 2t s = 0, 3 2 (x) 1 3 1 1 x (x) e = kT11. x (x) 2z s = 0, x (x) 2x s = 0. 4 This system consists of seven unknowns, (3) i We will find all those classes of the spacetime given in , (i = 0,1,2,3) , , and A . Solutions of this system May-June 574 ISSN 1013-5316;CODEN: SINTE 8 Sci.Int.(Lahore),29(3),573-575, 2016 give the Lagrangian along with the Noether symmetries. We discuss the value of for a < 2 and a > 2 here. Corresponding to these Lagrangian one may easily write The value of for a = 2 will discuss in solution 4. The spacetimes, which are the exact solutions of EFE. In the following sections different solutions of the system (9) are value of the cosmological constant as x for discussed. a < 2. While for a > 2 the value of 0 as 3 Solution of the System (9) x . In this section we discuss all the possible conformal Solution 3: The third solution of (9) is as follows Minkowski plane symmetric static spacetimes x Solution 1: The first solution of the system (9) is ds2 = e (dt 2 dx2 dy 2 dz 2 ), (16) ds2 = e(x) (dt 2 dx2 dy 2 dz 2 ), (10) x 2 2 0 As,t, x, y, z = 2 C1e C,s,t, x, y, z =1/2C1 s C2 s C3, As,t, x, y, z = C,s,t, x, y, z = C1, s,t, x, y, z = C3 y zC4 C5 , 0 1 1 2 3 s,t, x, y, z = C5 y C6 z C7 , s,t, x, y, z = C1 s C2 , s,t, x, y, z = 0, s,t, x, y, z = C3t C6 z C7 , s,t, x, y, z = C4 t C6 y C2 , 2 3 (11) s,t, x, y, z = C5 t C8 z C9 , s,t, x, y, z = C6 t C8 y C4. we see that in equation (11) there are seven parameters which (17) implies that there are seven Noether symmetries, four The spacetime given in equation (16) admit two additional translations, along s , t , y and z . Two boasts one along Noether symmetrie along with symmetries given in solution 1. th y axis and the other along z axis. One rotation in the yz The 8 symmetry is the mix symmetry of scaling and plane. The Einstein field equation (15) are the same for translation which corresponds to C2 in equation (17) and spacetime given in equation (10) in which the cosmological 9th symmetry is corresponds to the parameter C in the constant depends upon the values of function (x) . 1 above solution (17), which also carry a gauge function Solution 2: The second solution of (9) is x 2 2 x a 2 2 2 2 2 e . The Einstein field equations (15) for spacetime ds = ( ) (dt dx dy dz ), (12) (16) are 1 x e = kT = kT = kT , 0 2 C1t 00 22 33 As,t, x, y, z = C,s,t, x, y, z = C1 s C2 , s,t, x, y, z4= C4 y C5 z C6 , (18) a 2 x 3 1 C1 x 2 C1 y 2 e = kT11. s,t, x, y, z = , s,t, x, y, z = C4 t C7 z 4C8 , a 2 a 2 Which can also be written as 3 C1 z x s,t, x, y, z = C t C y C 5 7 3 1 a 2 = e (kT00 2 ), (13) 4 x (19) the Lagrangian of the spacetime given in equation (12) admit 3 = e (kT ). eight Noether symmetries which are given in equation (13). 11 4 2 These symmetries have one additional symmetry along with symmetries discussed in solution 1. The additional Noether The system (19) give us one equation in three variables T00, symmetry is the scaling symmetry. The Einstein field T and for x = 0 which has many solutions. And the equations for spacetime (12) are 11 a(a 4) x value of the cosmological constant tends to zero as x a tends to infinity. 2 ( ) = kT00 = kT22 = kT33, 4x (14) Solution 4: The following is the fourth solution which has the 3a2 x maximum Noether symmetries for the plane symmetric static ( )a = kT . 4x2 11 conformal Minkowski spacetime x Rearranging the system (14) we have ds2 = ( )2 (dt 2 dx2 dy 2 dz 2 ), (20) x a a(a 4) = ( ) (kT00 2 ), 4x (15) x 3a2 = ( )a (kT ). 11 4x2
May-June Sci.Int.(Lahore),29(3),573-575, 2016 ISSN 1013-5316;CODEN: SINTE 8 575
As,t, x, y, z = C,s,t, x, y, z = C1, 0 2 2 2 2 s,t, x, y, z = 1/2t x y z C3 C4 y C5 z C6 t C8 z C7 y C9 , 1 s,t, x, y, z = C3 t C4 y C5 z C6 x, 2 2 2 2 2 s,t, x, y, z = 1/2t x y z C4 C3 t C5 z C6 y C7 t C10 z C11, 3 s,t, x, y, z = 1/2 t 2 x2 y2 z 2 C C t C y C z C t C y C . 5 3 4 6 8 10 2 (21
The solution (21) consist of eleven Noether symmetries in admit either 6 isometries or 10 isometries. The cosmological which ten are the isometries of the spacetime given in equation constant is discussed in each solution. In solution 4 the (20) which there are 11 conservation laws corresponding to cosmological constant is negative for some region and positive different Noether symmetries. This is like the De Sitter other. Its value become very large for very large x . This spacetime which has also ten isometries. The Einstein field mean that the cosmological constant change the sign when we equations for spacetime given in equation (21) is go through the spacetime. Somewhere it is positive and 3 x 2 somewhere its value is negative and there is certain region 2 ( ) = kT00, where the value of the cosmological constant is zero. x (22) 3 x 2 REFERENCES 2 ( ) = kT11, x [1] Ali, F. and Feroze, T., Classification of plane symmetric adding the equations we have T = T in that case the static spacetimes according to their Noether’s 00 11 symmetries, Int. J. Theo. Phys., 52, 3329-3342, (2013). system reduce to equation [2] Ali, F. Conservation Laws of Cylindrically Symmetric x2 3 Vacuum Solution of Einstein Field Equations, Applied = kT . (23) 2 00 2 Mathematical Sciences 8, 4697-4702, (2014). [3] Al-Kuwari, H. A. and Taha, M. O., Noether’s theorem and 3 At x = 0 we have = which shows that is local gauge invariance, Am. J. Phys., 59, 363-365, 2 (1991). negative for all values of . And as x . [4] Bluman G. and Kumei S. Symmetries and differential equations, (Springer-Verlag, New York, 1989). 3 3 It mean that for T = , = 0 , for T < , [5] Frolov V. P. and Novikov I. D., Black Hole Physics Basic 00 kx2 00 kx2 Concepts and New Developments (Kluwer Academic 3 Publishers 1998). < 0 and for T00 > 2 > 0 . That is for the manifold [6] Ibragimov N. H., Elementary Lie Group Analysis and kx Ordinary Differential Equations, JohnWiley & describe by the metric given in equation (20) the cosmological Sons,(1999). constant is negative, positive and zero in different regions of [7] Ibragimov N. H. and Kovalev V. F., Approximate and the given spacetime. Which can be interpreted as it is attractive Renormgroup symmetries, Higher education press, in some region and repulsive in some other region. There is a Beijing and Springer-Verlag GmbH Berlin Heidelberg place somewhere in the spacetime where the value of the (2009). cosmological constant is zero. [8] Lester J. A., Conformal Minkowski space-time. I - Relative infinity and proper time, Il Nuovo Cimento B, 72, 4 CONCLUSION 261-272, (1982). In this paper we classify the static conformal Minkowski [9] Misner C. W., Thorne K. S. and Wheeler J. A., spacetimes according to Noether symmetries. It has been Gravitation, W. H. Freeman, and Company, San shown here that the static conformal Minkowski spacetimes Francisco, (1973). admit, at least 7 Noether symmetries and at most 11 Noether [10] Olver P. J., Application of Lie groups to differential symmetries. We obtain four classes of the static conformal equations, Graduate Text in Mathematics, 107, Minkowski spacetimes. They are discussed as solution 1, 2, 3 (Springer-Verlag, New York, 1993). and 4 respectively. The first solution has 7 Noether [11] Stephani, H., Kramer, D., MacCallum, M. A. H., symmetries and 6 isommetries, the second solution has 8 Hoenselaers, C. and Herlt, E., Exact Solutions of Noether symmetries in which 6 are the isometries, one Einstein’s Field Equations, Cambridge University Press, homothety and one pure Noether symmetry. In solution 3 we Cambridge, (2003). obtained 9 Noether symmetries, in which 6 are isometries of [12] Stephani H., Differential Equations: Their Solutions the corresponding spacetime and the remaining are purely Using Symmetry, Cambridge University Press, New Noether symmetries. In solution 4 we obtained 11 Noether York, NY, USA, 1989. Symmetries in which 10 are the isometries of the spacetime given in equation (20) and only one is purely Noether symmetry. This shows that the static conformal Minkowski
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