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Robinson-Trautman spacetimes in (2+1) dimensions

Alberto Saa

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 $P(u,φ)$, 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 $P(u,φ)$ whose stationary configurations correspond, for negative cosmological constant, to boosted BTZ black holes. Numerical solutions strongly support the relaxation of generic regular initial data $P(0,φ)$ 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.

Robinson-Trautman spacetimes in (2+1) dimensions

Abstract

We propose a Robinson-Trautman evolution in -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 , 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 whose stationary configurations correspond, for negative cosmological constant, to boosted BTZ black holes. Numerical solutions strongly support the relaxation of generic regular initial data 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.
Paper Structure (4 sections, 17 equations, 1 figure)

This paper contains 4 sections, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The RT flow in (2+1) dimensions
  3. The numerical solutions
  4. Final remarks

Figures (1)

  • Figure 1: Representative numerical evolutions of the flow (\ref{['rtf1']}). In the left panel the initial data (\ref{['P1']}) relax toward the isotropic stationary state with $v=0$. In the right panel, on the other hand, the initial condition (\ref{['P2']}) relaxes toward an anisotropic stationary state (\ref{['rocket']}) with $v\approx0.3$. The successive insets illustrate, from top to bottom, the smoothing of the angular profile towards a stationary solution as the flow approaches its late $u$ limit. In both panels, the green (external) curves correspond to the respective initial conditions, while the internal curves correspond to the right-hand side of the flow (\ref{['rtf1']}) at $u=0$, without scale. Blue (continuous) lines are the negative values, which makes $P(u,\phi)$ decrease locally, while the red (dashed) are the positive values, associated with the local increase of $P(u,\phi)$. For an animation and further computational details, see RT3Site.