Abstract I report the discovery of a new effect of General Relativity which is important to understand very rapidly rotating(Kerr) black holes. The orbital velocity of a test particle is no longer a monotonic function of the orbit radius when the spin of the black hole is >0.9953, but displays a local minimum-maximum structure for radii smaller than 1.8 gravitational radii. There the rate of change of the orbital velocity per radius unit equals the radial epicyclic frequency and is exactly one third of the polar epicyclic frequency, suggesting a 3:1 resonant oscillatory motion of the particle. If associated with the most recently observed quasi-periods the mass of the supermassive black hole Sgr A* in the centre of our Galaxy is determined to 3.3×106M⊙, and the spin is 0.99616.
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