Cosmological string solutions in 4 dimensions from 5d black holes

Abstract

We obtain cosmological four dimensional solutions of the low energy effective string theory by reducing a five dimensional black hole, and black hole-de Sitter solution of Einstein gravity down to four dimensions. The appearance of a cosmological constant in the five dimensional Einstein-Hilbert action produces a special dilaton potential in the four dimensional effective string action. Cosmological scenarios implemented by our solutions are discussed.To describe strings propagating in nontrivial background there are two different approaches. First, one can start with a conformal field theory, e.g. (gauged) WZW, FeiginFuchs or Liouville theory, and rewrite this as a string σ model. Although the gauged WZW model is an exact conformal field theory it is possible to obtain the σ model background only perturbatively. The reason is that in general the 2d gauge field can be eliminated only in an α ′ expansion. One example for which one got an exact result is the SL(2, R)/U(1) WZW model [1,2]. In the second approach one starts with the low energy effective action and tries to find solutions in an α ′ expansion. In terms of both approaches some cosmological solutions have been obtained so far. Exact solutions, e.g. , are given by the gauged WZW model based on SL(2, R)/SO(1, 1) × R 2 [3] or the combination of the SU(2) WZW model with a Feigin-Fuchs part [4,5]. In the second or "phenomenological" approach cosmological solutions have been discussed in [6] and [7]. In all these solutions the metric is either time independent like the SU(2) WZW in the string frame a or spatially flat or the dilaton is constant. In the present paper we use the second approach and find time dependent solutions (metric and dilaton) which are spatially not flat. We use the following procedure. We neglect all terms O(α ′2 ) and consider only curvature and dilaton terms. Then we will rewrite the effective string action in four dimensions (4d) as *

Cosmological solutions in M and string theory

“…(37)). Then, in a manner similar to that discussed by Behrndt and Förste in [28], we go one dimension down to four, obtaining an action of a scalar-tensor theory of gravity in a Brans-Dicke frame:…”

String Kaluza-Klein cosmology

“…In a recent paper [5] we have obtained a four dimensional cosmological solution containing a dilaton and a graviton. There we started with a five dimensional black hole solution of Einstein gravity and reduced it to a four dimensional cosmological solution.…”

Erratum: Cosmological solutions in M and string theory [Phys. Rev. D57, 7340 (1998)]

“…Finally, we can obtain solutions with spatial curvature, either by directly solving differential equations for models with spatially curved sections [8,15,21] or by using a Wick rotation and a dimensional reduction of the 5D Schwarzschild black hole solutions [28]. The equations of motion including spatial curvature (but for simplicity excluding the stringy cosmological term and the axion contribution) arė…”

“…(37)). Then, in a manner similar to that discussed by Behrndt and Förste in [28], we go one dimension down to four, obtaining an action of a scalar-tensor theory of gravity in a Brans-Dicke frame:…”