Rotating Carroll black holes: A no go theorem

Recently, there has been a lot of interest in Carroll black holes and in particular whether or not one could find a Carrollian analog of a rotating black hole spacetime. Here we show that every stationary and axisymmetric solution (and thence also a black hole) of Carrollian general relativity in any number of $d>3$ dimensions is necessarily also static (up to a ``topological rotation''). The case of $d=3$ dimensions is special. There, the topological rotation is important and one can have a rotating Carroll Ba\ifmmode \acute{n}\else \'{n}\fi{}ados-Teitelboim-Zanelli black hole, obtained from a static one by the Carroll boost accompanied by the reidentification of the angular coordinate, similar to what happens in the Lorentzian case. We also find a Carrollian analog of an accelerating black hole, showing that Schwarzschild is not the only possible stationary and axisymmetric Carroll black hole in four dimensions. A generalization of the no go theorem to include Maxwell, dilatonic, and axionic matter fields is also discussed.