Spinning $\sigma$-model solitons in $2+1$ Anti-de Sitter space

Abstract

We obtain numerical solutions for rotating topological solitons of the nonlinear $\sigma$-model in three-dimensional Anti-de Sitter space. Two types of solutions, $i)$ and $ii)$, are found. The $\sigma$-model fields are everywhere well defined for both types of solutions, but they differ in their space-time domains. Any time slice of the space-time for the type $i)$ solution has a causal singularity, despite the fact that all scalars constructed the curvature tensor are bounded functions. No evidence of a horizon is seen for any of the solutions, and therefore the type $i)$ solutions have naked singularities. On the other hand, the space-time domain, along with the fields, for the type $ii)$ solutions are singularity free. Multiple families of solutions exhibiting bifurcation phenomena are found for this case.

arXiv comment15 pages, 6 figures

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• Growing Hair on the extremal $BTZ$ black hole
• Exactly solvable gravitating perfect fluid solitons in (2+1) dimensions
• Rotating AdS Einstein universes from constrained metrics