THE GLOBALLY PATHOLOGIC PROPERTIES OF A STATIC PLANARY SYMMETRIC EXACT SOLUTIONS

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1 NLELE ŞTIINŢIFICE LE UNIVERSITĂŢII "L.I.CUZ" DIN IŞI Tomul XLIII-XLIV, s.i.b.fasc. Fizica Solidelor - Fizică Teoretică, THE GLOBLLY PTHOLOGIC PROPERTIES OF STTIC PLNRY SYMMETRIC EXCT SOLUTIONS BY C. DRIESCU, Marina-ura DRIESCU, Iordana ŞTEFĂNOEI BSTRCT Since researches on global pathological manifolds are very active now, the goal of the present paper is to investigate the class of the static planary symmetric exact solutions with " g = sinh ( z) ". fter discussing the geometry supported by a combined matter-source made of dust stuck on a cosmic string, we analyse the structure of the non-spacelike geodesics and display the Penrose diamond and the embedding of the - dimensional Lorentzian submanifold into a -dimensional pseudo-euclidean one. Finally, the behaviour of electrostatic and magnetostatic fields in this Universe has been under consideration, emphasizing the global pathology and stresing significant differences from the Minkowskian case.. INTRODUCTION In this paper, we analyse the globally pathological properties of the exact solution sinh( z) that admits a G 6 - group of motion acting on the decomposition of M space-time given by the flat Euclidian R times a curved Lorentzian manifold M covered by the path (z, t) ( I R) R, i.e. M = R M R R {} R.. On the base manifold, we choose the metric ds = δ dx dx B dz sinh ( z)dt B + () having {z = } as a singular point. Starting with the general Killing equation X l g l g X l ik,l + g lk X, i + li, k = () written on components as "lexandru Ioan Cuza" University, Faculty of Physics, ISI, 66, ROMNI, MRIN@UIC.RO Received December 5, 998. Downloaded from

2 9 C. DRIESCU, Marina-ura DRIESCU, Iordana ŞTEFĂNOEI X, µ + ν Xµ, ν = ; sinh ( z)x, µ + X µ, = ; coth ( z) X + X, µ = ; ( µ, ν =, ) () we get the following six Killing vectors fields: X () = x ; X () = y ; X () = y x x y ; X t () = e [ z coth( z) t ] t X( 5) = e ( z + coth( z ) t ). () For the G subgroup generated by X () X (6) the invariant properties are characterized by C. µν µ βν µνϑ µ = = and N = C. β ε ε ϑ meaning N =, N =, N =. Hence, the rank of N is and also it can be shown that the modulus of the signature of N is σ =. Therefore, our G group of motion acting on M does concretely belong to the Bianchi type VIII and so, with E = VII for X () X (), we get a G 6 = VII VIII group of motion []. For some z R {}, X () and X (5) are always non-spacelike vectors because g( X t sinh (), X () ) = e ( z ) (5) and g( X t sinh (5), X (5) ) = e ( z ) (6) Concerning the orbits of X (), (5) given by t + (z) = ln[sinh( z)] + l (T ) t (z) = ln[sinh( z)] + l (T ) (7) with z (, ), one can notice that t + (z) goes from the past infinity at the singular point {z = } to the future infinity reached at the spatial infinity {z = }, while t - (z) comes from the past infinity at {z = } and reaches the future infinity in the {z = } singular point.. THE GEOMETRY OF THE MODEL

3 THE GLOBLLY PTHOLOGIC PROPERTIES OF STTIC... 9 Considering the mentioned decomposition of M, we introduce the dual pseudo-orthonormal tetrads: ω = dx ; =, ω = dz ; ω = sinh ( z)dt. (8) Employing the Cartan formalism in the analyse of (), we get the -form connection Γ coth( z) = ω and the essential Riemann tensor component R =, which is obviously free of singularities. The nonvanishing components of the Ricci tensor read R, R = = (9) leading to a constant negative curvature of M namely R = -. s the only non-trivial components of the Einstein tensor are G = G =, we come to a suitable matter-sources that can support such geometries [].. THE GEOMETRY SUPPORTING MTTER SOURCE Since G =, the only conventional source that can be branched is the vacuum itself. However, because the dust-like matter-source has T = ρ >, it is necessary to engage a cosmological constant like vacuum state w v < such that together with the dust, one gets w s + w v =. Defining the energymomentum tensor for the cosmological constant contribution as T (v) ab = λ g ab (s) with λ > and the dust-like energy-momentum tensor Tab = ρ ua u b in order to (s) get T T v + = with the dust at rest, i.e. u a a = δ, it yields ρ = λ. To also (c) get G =, it follows that we need an extrasource with T ab = µ XaXb and Xa = ηa for µ = λ. The total energy-momentum tensor (with respect to a b { ω ω } a,b =, ) will be given by T ab = λ ( η a η b η a η b + η ab) () and does clearly describes a combined matter-source made of dust, with ρ = λ, stuck on a z-directed global cosmic string of tension µ = λ imbedded in a

4 9 C. DRIESCU, Marina-ura DRIESCU, Iordana ŞTEFĂNOEI static Universe of negative cosmological constant Λ = k λ. Finally, the Einstein equations and the conservation law T ab ; b = of the energy-momentum tensor does necessarily lead to a constant λ.. TIMELIKE ND NULL GEODESICS Let us analyse the structure of the non-spacelike (null and timelike) geodesics using the Euler- Lagrange approach... THE NORML NULL GEODESICS. For constant x and y we get from ds =, the null geodesics[] e z t+ = t + ln e z () + e z t = t ln e z () + traveling from z = to z and forming the light cone. They get imprisoned at the spatial infinity z = as they cannot be future or past extended respectively, beyond t = t.. THE TIMELIKE GEODESICS. Concerning the normal ones, with x, y = const., the metric () reads dσ = (ds ) sinh ( z)(dt) (dz) x,y = cst. = () pointing out the Lagrange associated function Φ = sinh ( z)(t) & (z) & (5) The Euler-Lagrange equations lead to & t sinh ( z) = k, k = const. (6) and k z& = ± (7) sinh ( z) Choosing σ = as the initial value of the affine parameter when t =, z = z

5 THE GLOBLLY PTHOLOGIC PROPERTIES OF STTIC and dz = u z = we get for k the expression k = sinh(z ). Thus, (7) dσ σ= has the solution z ( σ ) = ln cosh( z ) cos( σ) + cosh ( z ) cos ( σ) (8) for σ R which inserted in (6) leads to the following cosmic time dependence on σ sinh( z ) + tan( σ) t ( σ ) = ln (9) sinh( z ) tan( σ) s it can be seen pastly extending the already obtained geodesic, the test particle runs it on the whole, from past to future infinities, in a finite proper interval σ = arctan(sinh( z )). dz If one chooses a non-zero initial velocity = u z, at t =, z = z dσ σ = then it yields k (u = sinh( z ) z ) + and the following solutions z ( σ ) = F + F, F + F > z ( σ ) = F F, F F > () where cosh( z) F = (u z ) sinh ( z) + cosh ( z) sin± σ + arcsin M M = (u z ) sinh ( z) + cosh ( z), and cosh( z ) cosh( z ) cot ± σ + arcsin k cot arcsin + k M M t( σ ) = ± ln cosh( z ) cosh( z ) cot ± σ + arcsin + k cot arcsin k M M () In the figure is represented the whole embedding of M into the pseudoeuclidian manifold

6 96 C. DRIESCU, Marina-ura DRIESCU, Iordana ŞTEFĂNOEI ds = (dz) (dt) (dw), with Z = sinh( z) cosh( t) ; T = sinh ( z)sinh( t ) ; W = cosh( z) () In the (z, t) parametrization, it clearly covers only a part of the R S covering manifold W + T Z =. Using the geodesics () we set the advanced and respectively retarded null coordinates Fig. The embedding of M into the pseudo-euclidean manifold of signature, in the (z, t) parametrization (). e z e z u = t ln z ; v t ln e = + e z () + + e z with the essential restriction u v = ln z > e + showing the allowed region on M in the plane (u, v). With the (corresponding) compact Penrose null coordinates

7 THE GLOBLLY PTHOLOGIC PROPERTIES OF STTIC z z = e e u arctan t ln ; v = arctan t + ln () z e + e z + we get the Penrose diamond, in figure, depicting the conformal structure at infinity of the considered space-time. Fig. The Penrose diagram 5. ELECTRIC ND MGNETIC FIELDS On the considered manifold, the source-free Maxwell equations bc R b a = g a;bc = ab where a are the components of the -electromagnetic potential and ; states for the covariant derivate, read: B + coth( z) = ; B=, (5) B B + coth( z) coth ( z) = coth( z) ; (6)

8 98 C. DRIESCU, Marina-ura DRIESCU, Iordana ŞTEFĂNOEI + coth( z) coth ( z) = (7) coth( z) Using (5-7) and the Lorentz condition a ; a =, namely + + B B coth ( z) = (8) we are going to obtain the closed form solutions for particular configurations of magnetostatic and/or electrostatic fields. Firstly, let us consider the case = = and B, B =, i.e. = (y, z) and = (x, z), corresponding to the following nonvanishing components of the magnetostatic field B = -,, B = -,, B = (, -, ). In these assumptions, (5) becomes + + coth( z) = y z z (9) and + + coth( z) x z z = () with the corresponding solutions = = + (y, z) cn exp i n y cn exp i n y n P ( cosh( z), Q ( cosh( z)) () and respectively = = + (x,z) c n exp i n x cn exp i n x n () P (cosh( z),q (cosh( z)) Here, the z-dependence is a combination of two linear independent thorus functions, which can be expressed in terms of hypergeometric functions as []

9 THE GLOBLLY PTHOLOGIC PROPERTIES OF STTIC P cosh η (cosh η) = F n, + n; ; () π Γ(n + ) n n F, ; n ; cosh Q (cosh η) =, n n η + + Γ(n + ) cosh η () where η = z, and are respectively divergent at η and η =. In figure representing (y, z) we notice that, unlike the Minkowskian case, the global pathology of this manifold has a significant influence on the magnetostatic field distribution. Secondly, by considering, =, let us turn to the electrostatic field E, with the components E =, ; E =, ;E =, + coth( z) Setting = = = in (7) we come to the equation: coth( z), = (5) x y z sinh ( z) which, employing the cylindrical coordinates x = ρcosφ ; y =ρsin φ ;z = z (6) gets the following general solution φ φ ρ + = ρ φ = im im (,, z) Jm n cme cme n m= (7) P (cosh( z)), Q (cosh( z)) containing the thorus functions coshη coshη (coshη) = F n, + n;; (8) coshη + Q P (coshη) = π n+ Γ(n + ) tanh η n 5 n F ; + + n+ Γ(n + ) cosh η ;n + ;cosh η, η= z (9)

10 C. DRIESCU, Marina-ura DRIESCU, Iordana ŞTEFĂNOEI which, as it can be seen in figure, are respectively divergent in z = and z =. Fig. The two linear independent expressions of (y, z) given by (), for n = 5. REFERENCES Fig. The two linear independent expressions of (ρ, ϕ, z) given by (7), for n = 5 and m =. [] Dariescu, C., Dariescu, M.. and Gottlieb, I. Gravity, Theoretical Physics and Computers, Eds. D. Vulcanov and I. Cotaescu (Mirton Publishing House, Timisoara), (997). [] Hawking, S.W. and Ellis, G.F.R. The Large Scale Structure of Space-time, Cambridge Monographs on Mathematical Physics (Cambridge University Press), (976). [] Kramer, D., Stephani, H., Herlt, E. and MacCalum, M. Exact Solutions of Einstein s Field Equations, Ed. E. Schmutzer (Cambridge University Press), (98). [] Rijic, I.M. and Gradstein, I.S. Tabele de integrale, sume, serii si produse, (Editura Tehnica, Bucuresti), (955).