Exact solutions of Einstein and Einstein-scalar equations in 2 + 1 dimensions
A nonstatic and circularly symmetric exact solution of the Einstein equations (with a cosmological constant $Λ$ and null fluid) in $2+1$ dimensions is given. This is a nonstatic generalization of the uncharged spinless BTZ metric. For $Λ= 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 .
gr-qc↗