Black String and Gödel type Solutions of Chern-Simons Modified Gravity

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Chern-Simons (CS) modified gravity with a prescribed CS scalar field does not admit rotating black hole solutions with spherical topology of the horizon. In this paper, we show that it does admit rotating black hole/string solutions with cylindrical topology of the horizon and present two intriguing physical examples of such configurations. First, we show that the Banados-Teitelboim-Zanelli (BTZ) stationary black string, that is obtained by adding on a spacelike flat dimension to the BTZ black hole metric of three-dimensional gravity, solves the field equations of CS modified gravity with a specific source term and irrespective of the choice of CS scalar field. Next, we consider the Lemos solution for a rotating straight black string in general relativity and show that for the CS scalar field being a function of the radial coordinate alone, this solution persists in CS modified gravity. We also discuss two examples of Gödel type metrics in CS modified gravity by uplifting to four dimensions a general one-parameter family of Gödel type solutions of three-dimensional gravity. The first example is the usual Gödel solution of general relativity which also survives in CS modified gravity with the CS scalar field depending on two variables, the radial and the azimuthal coordinates. The second example represents a new nontrivial (non general relativity) Gödel type solution to the vacuum field equations of CS modified gravity. This solution originates from the respective vacuum solution of topologically massive gravity when extending it to four dimensions by adding on an extra spatial coordinate and choosing the CS scalar field as a linear function of this coordinate.