Robinson-Trautman spacetimes in (2+1) dimensions

TL;DR

This paper introduces a (2+1)-dimensional Robinson-Trautman evolution model with a null fluid, demonstrating that generic initial data relaxes to stationary black hole solutions, serving as a simplified model for gravitational radiation dynamics.

Contribution

It develops a novel (2+1)-dimensional Robinson-Trautman framework with a fourth-order evolution equation, linking geometric flow to black hole solutions and gravitational radiation analogs.

Findings

Numerical solutions show relaxation to stationary black holes.

The model mimics dissipative gravitational radiation phenomena.

Stationary configurations correspond to boosted BTZ black holes.

Abstract

We propose a Robinson-Trautman evolution in $(2+1)$-dimensional spacetime that retains key structural features of the four-dimensional case. We consider a recently studied exact family of metrics to select a nonstationary geometry with a cosmological constant, sourced by a null fluid. The metric is completely determined by a single positive function $P(u,\phi)$, while the corresponding matter content is encoded in a null-fluid density. Motivated by the role of the area-preserving Calabi flow in four dimensions, we introduce a fourth-order length-preserving evolution equation for $P(u,\phi)$ whose stationary configurations correspond, for negative cosmological constant, to boosted BTZ black holes. Numerical solutions strongly support the relaxation of generic regular initial data $P(0,\phi)$ toward the stationary sector. The resulting system provides a simple toy model for dissipative dynamics driven by null radiation in lower-dimensional gravity, with several structural similarities to phenomena associated with genuine gravitational radiation.