We construct G\"odel-type black hole and particle solutions to Einstein-Maxwell theory in $2+1$ dimensions with a negative cosmological constant and a Chern-Simons term. On-shell, the electromagnetic stress-energy tensor effectively replaces the cosmological constant by minus the square of the topological mass and produces the stress-energy of a pressure-free perfect fluid. We show how a particular solution is related to the original G\"odel universe and analyze the solutions from the point of view of identifications. Finally, we compute the conserved charges and work out the thermodynamics.
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Bañados et al. (2006) studied this question.
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