String or branelike solutions in four-dimensional Einstein gravity in the presence of cosmological constant

TL;DR

This paper explores string and branelike solutions in four-dimensional Einstein gravity with cosmological constant, revealing unique static and dynamic solutions, including black strings, solitons, and Kasner-de Sitter spacetimes, with implications for singularities and matter collapse.

Contribution

It provides the first comprehensive classification of static and time-dependent string and branelike solutions in Einstein gravity with positive and negative cosmological constants.

Findings

Banados-Teitelboim-Zanelli black string is the only warped solution with negative cosmological constant.

General nonwarped solutions characterized by mass density and tensions, with tensions summing to minus the mass density.

All static solutions other than black strings and solitons have naked singularities.

Abstract

We investigate string or branelike solutions for four-dimensional vacuum Einstein equations in the presence of cosmological constant. For the case of negative cosmological constant, the Banados-Teitelboim-Zanelli black string is the only warped stringlike solution. The general solutions for nonwarped branelike configurations are found and they are characterized by the Arnowitt-Deser-Misner mass density and two tensions. Interestingly, the sum of these tensions is equal to the minus of the mass density. Other than the well-known black string and soliton spacetimes, all the static solutions possess naked singularities. The time-dependent solutions can be regarded as the anti-de Sitter extension of the well-known Kasner solutions. The speciality of those static regular solutions and the implication of singular solutions are also discussed in the context of cylindrical matter collapse. For the case of positive cosmological constant, the Kasner-de Sitter spacetime appears as time-dependent solutions and all static solutions are found to be naked singular.