Abstract
No Kerr-like exact solution has yet been found in Chern-Simons modified gravity. Intrigued by this absence, we study stationary and axisymmetric metrics that could represent the exterior field of spinning black holes. For the standard choice of the background scalar, the modified field equations decouple into the Einstein equations and additional constraints. These constraints eliminate essentially all solutions except for Schwarzschild. For noncanonical choices of the background scalar, we find several exact solutions of the modified field equations, including mathematical black holes and $pp$-waves. We show that the ultrarelativistically boosted Kerr metric can satisfy the modified field equations, and we argue that physical spinning black holes may exist in Chern-Simons modified gravity only if the metric breaks stationarity, axisymmetry, or energy-momentum conservation.
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