Dilaton spacetimes with a Liouville potential

TL;DR

This work analyzes D-dimensional Einstein–dilaton gravity with a Liouville potential under D-2 planar symmetry, deriving static and time-dependent solutions that fall into three one-dimensional classes and applying them to non-supersymmetric string backgrounds. The study reveals widespread singularities in static backgrounds, non-asymptotically flat geometries, and a breakdown of Birkhoff's theorem in the presence of scalar matter, while showing that allowing time dependence yields regular two-dimensional backgrounds reminiscent of thick domain walls. It connects these geometries to $SO(9)$-symmetric limits and to explicit backgrounds for USp(32) Type I and SO(16)×SO(16) heterotic strings, highlighting the role of the Weyl parameter $d$ and the discriminant-driven class structure. The results provide a framework for brane cosmology in nontrivial dilaton backgrounds and motivate further study of two-dimensional solutions that extend beyond the one-dimensional sector. Overall, the paper identifies the general one-dimensional solutions and demonstrates, through explicit string-theory examples, how scalar fields with Liouville potentials shape the global structure of dilatonic spacetimes.

Abstract

We find and study solutions to the Einstein equations in D dimensions coupled to a scalar field source with a Liouville potential under the assumption of D-2 planar symmetry. The general static or time-dependent solutions are found yielding three classes of SO(D-2) symmetric spacetimes. In D=4 homogeneous and isotropic subsets of these solutions yield planar scalar field cosmologies. In D=5 they represent the general static or time-dependent backgrounds for a dilatonic wall-type brane Universe of planar cosmological symmetry. Here we apply these solutions as SO(8) symmetric backgrounds to non-supersymmetric 10 dimensional string theories, the open USp(32) type I string and the heterotic string SO(16)XSO(16). We obtain the general SO(9) solutions as a particular case. All static solutions are found to be singular with the singularity sometimes hidden by a horizon. The solutions are not asymptotically flat or of constant curvature. The singular behavior is no longer true once we permit space and time dependence of the spacetime metric much like thick domain wall or global vortex spacetimes. We analyze the general time and space dependent solutions giving implicitly a class of time and space dependent solutions and describe the breakdown of an extension to Birkhoff's theorem in the presence of scalar matter. We argue that the solutions described constitute the general solution to the field configuration under D-2 planar symmetry.

Figures (8)

• Figure 1: Plots of the Ricci scalar for the numerical value $d=-5$. From left to right for spacelike Class I $\eta=-1,+1$ and, timelike Class I with $\eta=-1$ respectively
• Figure 2: Plot of the Ricci scalar and $\rho(p)$, $P(p)$ (dotted line) for the $SO(9)$ time-dependent solution.
• Figure 3: Plot of the dilaton field $\phi(p)$ and the Ricci scalar curvature $R(p)$ for the "compact" $\eta=-1$ case in Class II
• Figure 4: Plot of the dilaton field $\phi(u)$ and the Ricci scalar $R(u)$ for $\eta=1$ (infinite proper distance) for Class II.
• Figure 5: Plot of the dilaton field $\phi(q)$ for $d=-9/10$ (dotted line) and $d=-7/2$ for Class III.