**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;