Exact solutions of Einstein and Einstein-scalar equations in 2 + 1 dimensions

TL;DR

This paper presents new exact solutions to Einstein and Einstein-scalar equations in 2+1 dimensions, including a nonstatic generalization of the BTZ metric and a static scalar field solution with singularities.

Contribution

It introduces novel exact solutions in lower-dimensional gravity, extending the BTZ metric and exploring scalar field configurations with singularities.

Findings

Derived a nonstatic, circularly symmetric Einstein solution with a cosmological constant and null fluid.

Obtained a static Einstein-scalar solution with a curvature singularity at r=0.

Analyzed the properties and invariants of the solutions, including the Kretschmann invariant.

Abstract

A nonstatic and circularly symmetric exact solution of the Einstein equations (with a cosmological constant $\Lambda$ and null fluid) in $2+1$ dimensions is given. This is a nonstatic generalization of the uncharged spinless BTZ metric. For $\Lambda = 0 $, the spacetime is though not flat, the Kretschmann invariant vanishes. The energy, momentum, and power output for this metric are obtained. Further a static and circularly symmetric exact solution of the Einstein-massless scalar equations is given, which has a curvature singularity at $r =0$ and the scalar field diverges at $r=0$ as well as at infinity .