Twisted doughnuts: Thick disk torus around equatorial asymmetric black hole

The Kerr black hole spacetime is symmetric with respect to a well-defined equatorial plane. When such a symmetry is broken, for instance, by some putative effects beyond general relativity, the Keplerian circular orbits around the black hole are distorted vertically away from the equatorial plane by an amount depending on the orbital radius. As a result, the Keplerian thin disk acquires a curved surface. In this work, we extend such results to thick tori configurations by considering non-self-gravitating Polish doughnut models. We show that due to the equatorial asymmetry of the spacetime, the centers and the cusps of tori are distorted away from the original equatorial plane toward the same direction as that experienced by the stable Keplerian orbits, and the entire tori configurations are twisted toward that direction as well. The shape of the distorted tori is demonstrated explicitly using a constant specific angular momentum profile $\ell(r,y)=\ell_0$ of the disk fluid. However, the result also applies to non-constant profiles of $\ell(r,y)$ generically in the sense that any asymmetric profile of $\ell(r,y)$ that attempts to produce a symmetric tori configuration either turns out to be ill-defined near the equatorial plane or suffers from fine-tuning issues.