A reinterpretation of the Taub singularity

We reinterpret the well known Taub-singularity in terms of a cylinder symmetric geometry. It is shown that a cylindrical analog to the Einstein-Rosen bridge as well as a cosmic string will be present in the geometry. PACS numbers: 02.40.tm , 04.20.Jb Typeset using REVTEX Electronic address: [email protected] Electronic address: [email protected] and [email protected] 1 It is a somewhat curious fact that the static spherical symmetric vacuum solution of Einstein’s field equations is unique while no unique solution exist for the corresponding cylinder symmetric problem. Even though the most general form of the cylinder symmetric vacuum solution, the Levi-Civita metric [1], has been known for a long time physical interpretations has only been achieved for a relatively restricted class of solutions [2]. The aim of these notes is to provide a reinterpretation of the well known Taub singularity [3] in terms of a cylinder symmetric geometry. Along the way we give by this approach an interpretation of two previously ill-understood solutions of the Levi-Civita metric. The structure of these notes is as follows. We will first use a well known connection between plane and cylinder symmetry in order to construct the space-time manifold of a cylinder symmetric shell of finite thickness which is filled with an incompressible perfect fluid. This connection has not, as far as we know, been utilized before in the study of cylinder symmetric systems in the literature. A cylinderical analog to the Taub-singularity resides in the vacuum on one side of the shell while the other vacuum is flat. These vacuum solutions have appeared in previous studies but have so far defied interpretation [2,4]. We match this construction using the Israel-formalism to the conic flat space which appear outside cosmic gauge strings. The resulting structure can be interpreted either as a cosmic gauge string with a naked singularity in the interior region or as a naked cylinder-symmetric singularity with a cosmic string in the interior. It is possible to travel from the conic region and into the singular region after having been traveling through a flat part which we interpret as a cylinderical analog to the Einstein-Rosen bridge. In [5,6] a static solution of Einsteins equations was obtained which was interpreted as an infinite plane wall of finite thickness composed of an incompressible ideal perfect fluid with a constant energy density. The pressure inside the wall is everywhere greater than zero and the boundaries of the wall are derived from the physical condition of vanishing pressure