Planary Symmetric Static Worlds with Massless Scalar Sources

TL;DR

This paper explores exact solutions of static, plane-symmetric spacetimes generated by a massless scalar field, revealing their algebraic properties, singularities, and potential for constructing new models through junction conditions.

Contribution

It provides a detailed analysis of the algebraic classification and global properties of scalar field-generated static spacetimes, including their singularities and geodesic structures.

Findings

Solutions are algebraically special, type D.

Spacetimes contain naked timelike singularities.

Models are globally pathological with no Cauchy surfaces.

Abstract

Motivated by the recent wave of investigations on plane domain wall spacetimes with nontrivial topologies, the present paper deals with (probably) the most simple source field configuration which can generate a spatially planary symmetric static spacetime, namely a minimally coupled massless scalar field that depends only upon a spacelike coordinate, $z$. It is shown that the corresponding exact solutions $({\cal M}, {\bf{\rm g}}_{\pm})$ are algebraically special, type $D - [S - 3T]_{(11)}$, and represent globally pathologic spacetimes with a $G_{4}$ - group of motion acting on ${\bf{\rm R}}^{2} \times {\bf{\rm R}}$ orbits. In spite of the model simplicity, these $\phi$ - generated worlds possess naked timelike singularities (reached within a finite universal time by normal non-spacelike geodesics), are completely free of Cauchy surfaces and contain into the $t$ - leveled sections points which can not be jointed by ${\rm C}^{1}$ - trajectories images of oblique non-spacelike geodesics. Finally, we comment on the possibility of deriving from $({\cal M}, {\bf{\rm g}}_{\pm})$ two other physically interesting ^^ ^^ $\phi$ - generated'' spacetimes, by appropiate jonction conditions in the $(z = 0)$ - plane.