On Cosmological String Backgrounds with Toroidal Isometries

A large class of cosmological solutions (of the Einstein equations) in string theory, in the presence of Maxwell fields, is obtained by O(d, d) transformations of simple backgrounds with d toroidal isometries. In all the examples in which we find a (closed) expanding universe, such that the universe admits a smooth, complete initial value hypersurface, a naked singularity may form only at the time when the universe collapses. The discrete symmetry group O(d, d, Z) identifies different cosmological solutions with a background corresponding to a (relatively) simple CFT, and therefore, may be useful in understanding the properties of naked singularities in string theory. e-mail address: [email protected] Address after September 15: Racah Inst. of Physics, The Hebrew University, Jerusalem 91904, ISRAEL. e-mail: [email protected] To find realistic cosmological solutions in string theory, one represents a classical solution by M × K, where M is a 2 − d conformal field theory (CFT) with a four dimensional space-time, and with a central charge c = 4, and K is some internal space represented also by a CFT. Moreover, though the possible formation of singularities is one of the questions of interest, we do not want the singularities to be “built in” in the initial conditions. We want M to admit a smooth, complete initial value hypersurfaces. In ref. [1], Nappi and Witten presented a CFT that describes a closed expanding universe in string theory. This universe is a cosmological solution with two toroidal isometries, namely, the background M is independent of two coordinates. In this paper, we use the method of O(d, d) transformations [2, 3] in order to generate a large class of cosmological solutions in string theory. In particular, the solutions of [1] are just a one parameter sub-family, of the backgrounds obtained by O(2, 2) transformations of (a suitable analytic continuation of) a product of two D = 2 black holes [4]. Moreover, when the internal space K also has toroidal isometries, one can turn on non-trivial gauge fields. In the examples we discuss, we find an anisotropic expanding and contracting closed universe, in the presence of Maxwell fields. The topology of a spatial slice is that of a three sphere. In the family of solutions we study, a naked singularity can appear only at the initial time, when the volume of the universe vanishes, and/or at the time when the universe recollapses. The cosmic censorship is therefore protected, within a large class of anisotropic cosmological solutions. We begin by establishing our notation following [3]. The two dimensional action describing a string propagating in a (curved) background (of a metric, an antisymmetric tensor, and a dilaton) inD-dimensions, that is independent of d coordinates is S = 1 2π ∫ dz[ΣIJ (x)∂X I ∂̄X − 1 4 φ(x)R] (1) = 1 2π ∫ dz[Eij(x)∂θ ∂̄θ + F 2 ia(x)∂θ ∂̄x + F 1 ai(x)∂x ∂̄θ