the unizednatin abdus salam educational, scientiffic .a a eqinS I organhzation itraonlXA0203566 centre enrpaen, for theoretical physics COLLINEATIONS OF TILE RICCI TENSOR FOR CYLINDRICAL SPACETIMES Asghar Qadir K. Saifuliah and M. Ziad 3 4~ Available at: http://ww.ictp. trieste. it-pub- of f IC/2002/65 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS COLLINEATIONS OF THE RICCI TENSOR FOR CYLINDRICAL SPACETIMES Asghar Qadir' Department of Mathematics, Quaid-i-Azamn University, Islama bad, Pakistan and Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia, K. Saifullah 2 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan and The Abdus Salam International Centre for Theoretical Physics, Tries te, Italy and M. Ziad' Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. MIRAMARE - TRIESTE July 2002 1 aqadirs~comsats.net.pk 2 saifulahqauedupk 3 mziad~qau.edu.pk Abstract A complete classification of cylindrically symmetric static Lorentzian manifolds according to their Picci collneations (M) is provided. It is found that the Lie algebras of the RCs of these manifolds (for non-degenerate Ricci tensor) has dimension ranging from 3 to 10 excluding 8 and 9. The comparison of the RCs with the Killing vectors (KVs) and homothetic motions (HMs) is made. Corresponding to each Lie algebra there arise highly non-linear differential constraints on the metric coefficients. These constraints are solved to construct examples which include some exact solutions admitting proper R~s. Their physical interpretation is given. The classification of plane symmetric static spacetimes (Farid et al., J. Math. Phys. 36 (1995) 5812) emerges as a special case of this classification when the cylinder is unfolded. 2 1 Introduction Spacetime symmetries not only make it possible to obtain exact solutions of the Einstein field equations (EFEs) Rab - - Rg~b = STab,(1 2 but also provide an invariant basis for their classification. Isometries (KVs), homothetic mo- tions (HMs), Ricci collineations (RCs) and matter collineations (MCs) are some of these symme- tries [1]. A vector field V on a Lorentzian Manifold M is called an isometry if the Lie derivative of g along V is zero, i.e. £vgO= . (2) For HMs the right side is merely replaced by Ag, where is a constant. Similarly, B is an RC for a Ricci tensor R, if LER = 0, (3) which reduces, in component form, to BcRab. ± R..c ± RbcBe =b0. (4) The R~s, like the KVs, ae purely geometrical in construction, but like the MCs give physical information by virtue of the EFEs [2]. Ever since the first investigations of cylindrically symmetric spacetimes by Levi-Givita [3] and Weyl [4] and, later by T. Lewis [5], these spacetimes have been studied extensively for their mathematical and physical properties. These have recently been studied particularly in the con- text of black holes [6], gravitational waves and cosmic strings [7, 8, 9. Some examples of well known cylindrically symmetric astrophysical and cosmological solutions discussed in the litera- ture include Einstein-Maxwell fields [10], the gravitational field inside a rotating hollow cylinder [11], vacuum solutions [12], dust solutions 13], perfect fluid solutions with and without rigid rotations [14], gravitational waves [15, 7], magnetic strings [16], static gravitational fields [17] and a large number of cosmic string solutions. This paper classifies cylindrically symmetric static spacetimes according to their RCs. As a result of this classification, cases of 10, 7, 6, 5, 4 and 3 RCs for non-degenerate Ricci tensor i.e. when det (Rab) $ 0 have been obtained . The corresponding metrics are implicitly given in the form of constraints on the Ricci tensor components. Concrete examples have been constructed by solving these constraints. Physical nature of the spaces thus obtained is also discussed where possible. The cases of proper (i.e. non-isometric) RCs are 10, 7, 6, 5, 4 and 3 dimensional. It may be worth mentioning here that plane symmetry may locally be thought of as a special case of cylindrical symmetry [18]. As such, the classification of plane symmetric static metrics [19] can be obtained as a special case of this classification. The RC equations ae given in Section 2. 3 The plan of the paper is as follows. The RCs for the non-degenerate Ricci tensor, are given in Section 3 while the degenerate cases, where det (Rab) = 0, ae discussed in Section 4. While the RCs for the non-degenerate cases are always finite dimensional, those for the degenerate cases are mostly, but not always, infinite dimensional. The comparison of RCs with KVs is made in Section 5 where the non-trivial RCs are discussed in detail. Concluding remarks axe given in Section 6. 2 The Ricci Collineation Equations The most general cylindrically symmetric static metric can be written as [18] 2 2 2 2 ds = evpdt- dp - a e\(P)d9 - ,PdZ (5) where a has the dimensions of length. For this metric the surviving components of the Ricci tensor are = ~~ + ,2 + VA' + '' 2 42 2 4 ~~~~~~~~~~~~(6) 2 = i - (A ±A + V'A' + I Here ( 0xlx 2 X3 ) (t, p,0, z) and prime and dot will denote differentiation w.r.t. p and t respectively. For the sake of brevity &4 will be written as R, for i = 0, 1, 2, 3. The RC equations [1] then can be written as R'B + 2RaB a = 0, (7) RaBa±+RbB' = 0, (a, b =0,1,2,3). (8) Eq. (7) (which drops the summation convention) gives four equations and Eq. (8) are six equations. These constitute together ten highly non-linear partial differential equations involving four components of the arbitrary RC vector B -(B 0 , B', B 2 , B 3 ), four components of the Ricci tensor, R0 , R,, R 2, R3 and their partial derivatives. Here the B's ae functions of t, po, 0 and z; and R, depend on p only. The minimal symmetry is given by 49t, a96, 9,,, which has the algebra 3 Ricci Collineations for the Non-Degenerate Ricci Tensor In this section Eqs. (7) and (8) are solved for the RCs of the non-degenerate Ricci tensor i.e., when R, 0 0, i = 0, 1, 2,3. In the beginning the calculations are given in some detail in order to explain the procedure, but later only the R~s and the metrics corresponding to the constraints on Rjs are given. Metrics for the non-trivial RCs are discussed in Section 5. 4 Taking a = 0, 2 and b = 2,3 in Eqs. (8) and differentiating these relative to z, and t, one obtains 3 equations. Now, solving them simultaneously yields RB1B 3 = 0. This gives rise to two cases: (A) R' #4 0 or (B) R' = 0. 3.1 Case (A) R' $~0, (.B1 3 =0) 3 Here, taking a = 1 and b = 2,3 in Eq. (8), one gets B23 = B j.UigtsinE.()wh a = 2 and b =3 gives (2) B2 = 0. This gives rise to further two cases: A(I) (R2 #0 or A(II) (R)' =0. Case A(I) ( ) 0 (which implies that B2 = 0) Eq. (8) implies that 23= 0 and so from Eq. (7) with a = 2 and 3, R~Bl = 0= MB1 This further gives rise to three possibilities depending on whether one or both of Rand R3' is/are non-zero. Case AI(a) R = , R' 760 In this case note that the RC Eqs. (7) and (8) imply that B = B = B 2= B 2 =B2 B2= B' = 0. Thus B 2 = c, a constant. Integrating Eq. (7) for a = 1 w.r.t. p gives v1 where A, (t, z) is an arbitrary function of t and z. Using Eq. (9) in Eq. (7) for a = 0, and Eq. (8) for a = 0 and b = 1, and differentiating w.r.t. p and t respectively, and comparing gives A1 (t, z) KRVj A, (t,z) = 0. (10) Similarly from Eq. (7) for a = 3 and Eq. (8) for a = 1 and b = 3 one gets A,~ R 3 / R 3 N' A,33 (t, Z)- VRk1 2R VfR-) A,(t,z) =O. (11) Eqs. (10) and (11) imply that and (EL. fl"adI'en separating constants. Here again aise four cases depending upon whether none, one or two of these constants is zero. Case Ala(l) ce 0, /3= 0 In hiscas l R =kian 2RIV =k 2, where k,and k2 are nonzero constants. The solution of Eqs. (10) and (11) in this case is A, (t, z)= C2 tZ +Cat +C4 Z +C 5 . (12) Putting this value of A, in Eq. (9) one gets from Eq. (7) for a =0 and 3 respectively 0 B = -ki C24-Z+ C3-+ C4 Zt C5 ] + A2 (P,Z) ,(13) 3 B = -k2[C2t--+C 3tZ +C 4 - + C5ZJ+ A3(A Z) .(14) 5 Using Eqs. (13) and (14) in Eqs. (7) and (8), and integrating w.r.t. p yields A 2 ( ) = - (C2 Z +C 3 ) 1 Ro dp +B 3 (z), (15) A 3 (t, ) = - (C2 t+ C4 ) f ~dp +B 4 (t). (16) With these values in Eqs. (13) and (14), one gets from Eq. (8) for a =0, b =3 kR' [ki (2- + c4t) - B3 ,3 (z] + 2 ~~C2-- ~=0 L 2 L- k k 2 ](7 where R0 kR3"2, k being a constant of integration, has been used. Clearly c2 =0 and one is left with kR k2 [c4k,t - B3,3 (z) + C3 k2 Z- h4 (t) = 0. (18) Nowif k - 1= 0, Eq. (18) gives B4 (t) - c4kkit = CAk2 Z- kB 3,3 (z) C6 (say) .(19) On integrating and substituting in Eqs.