**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;