Aschenbach effect: Unexpected topology changes in the motion of particles and fluids orbiting rapidly rotating Kerr black holes

Newtonian theory predicts that the velocity $\mathcal{V}$ of free test particles on circular orbits around a spherical gravity center is a decreasing function of the orbital radius $r$, $\mathrm{d}\mathcal{V}/\mathrm{d}r<0$. Only very recently, Aschenbach [B. Aschenbach, Astronomy and Astrophysics, 425, 1075 (2004)] has shown that, unexpectedly, the same is not true for particles orbiting black holes: for Kerr black holes with the spin parameter $a>0.9953$, the velocity has a positive radial gradient for geodesic, stable, circular orbits in a small radial range close to the black-hole horizon. We show here that the Aschenbach effect occurs also for nongeodesic circular orbits with constant specific angular momentum $\ensuremath{\ell}={\ensuremath{\ell}}_{0}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. In Newtonian theory it is $\mathcal{V}={\ensuremath{\ell}}_{0}/\mathcal{R}$, with $\mathcal{R}$ being the cylindrical radius. The equivelocity surfaces coincide with the $\mathcal{R}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ surfaces which, of course, are just coaxial cylinders. It was previously known that in the black-hole case this simple topology changes because one of the ``cylinders'' self-crosses. The results indicate that the Aschenbach effect is connected to a second topology change that for the $\ensuremath{\ell}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ tori occurs only for very highly spinning black holes, $a>0.99979$.

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