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We study the dynamics of spiral waves and oscillation modes in relativisticrotating discs around black holes. Generalizing the Newtonian theory, we showthat wave absorption can take place at the corotation resonance, where thepattern frequency of the wave matches the background disc rotation rate. Wederive the general relativistic expression for the disc vortensity (vorticitydivided by surface density), which governs the behaviour of densityperturbation near corotation. Depending on the gradient of the generalized discvortensity, corotational wave absorption can lead to the amplification ordamping of the spiral wave. We apply our general theory of relativistic wavedynamics to calculate the non-axisymmetric inertial-acoustic modes (also calledp-modes) trapped in the inner-most region of a black hole accretion disc.Because general relativity changes the profiles of the radial epicyclicfrequency and disc vortensity near the inner disc edge close to the black hole,these p-modes can become overstable under appropriate conditions. We presentthe numerical results of the frequencies and growth rates of p-modes forvarious black hole spin and model disc parameters (the surface density profileand sound speed), and discuss their implications for understanding theenigmatic high-frequency quasi-periodic oscillations observed in black holeX-ray binaries.

Corotation resonance and overstable oscillations in black-hole accretion discs: general-relativistic calculations