**Publication date:** Dec 2004

**Abstract:**

Aschenbach [Aschenbach, B. (2004). Measuring mass and angular momentum of black holes with high-frequency quasi-periodic oscillations. Astronomy and Astrophysics, 425:1075-1082] has shown that in Kerr black-hole spacetimes with rotation parameter a>0.9953, the Keplerian orbital velocity measured in locally non-rotating frames (LNRF) has a positive radial gradient in a small region in the vicinity of the event horizon and proposed that excitation of oscillations in Keplerian thin discs can be related to this fact. Similarly, we show that in the equatorial plane of marginally stable thick discs (with uniformly distributed specific angular momentum ℓ(r,θ)=const) the orbital velocity relative to the LNRF has a positive radial gradient in the vicinity of black holes with a>0.99979. The change of sign of the velocity gradient occurs just above the centre of the thick toroidal discs, in the region where stable circular geodesics of the Kerr spacetime are allowed. Therefore, the same mechanism as in the Keplerian discs could trigger oscillations in thick discs, but the rotational parameter of the Kerr spacetime must be much closer to the extreme-hole state with a=1. The global character of the phenomenon is given in terms of topology changes of the von Zeipel surfaces (equivalent to equivelocity surfaces in the tori with ℓ (r,θ)=const). Toroidal von Zeipel surfaces exist around the circle corresponding to the minimum of the equatorial LNRF velocity profile, indicating a possibility of development of some vertical instabilities in those parts of marginally stable tori with positive gradient of the LNRF velocity.

**Authors:**

Stuchlík, Zdeněk; Slaný, Petr; Török, Gabriel;