  We study the dynamics of spiral waves and oscillation modes in relativistic
rotating discs around black holes. Generalizing the Newtonian theory, we show
that wave absorption can take place at the corotation resonance, where the
pattern frequency of the wave matches the background disc rotation rate. We
derive the general relativistic expression for the disc vortensity (vorticity
divided by surface density), which governs the behaviour of density
perturbation near corotation. Depending on the gradient of the generalized disc
vortensity, corotational wave absorption can lead to the amplification or
damping of the spiral wave. We apply our general theory of relativistic wave
dynamics to calculate the non-axisymmetric inertial-acoustic modes (also called
p-modes) trapped in the inner-most region of a black hole accretion disc.
Because general relativity changes the profiles of the radial epicyclic
frequency and disc vortensity near the inner disc edge close to the black hole,
these p-modes can become overstable under appropriate conditions. We present
the numerical results of the frequencies and growth rates of p-modes for
various black hole spin and model disc parameters (the surface density profile
and sound speed), and discuss their implications for understanding the
enigmatic high-frequency quasi-periodic oscillations observed in black hole
X-ray binaries.
