1.1 Wave evolution and quasinormal modes in the Kerr spacetime

Using mode expansion one can construct a retarded Green's function as it was done for the Schwarzschild spacetime. By studying the analytical properties of the modes and the Green's function as functions of the complex frequency ω one can demonstrate that in the general case of a rotating black hole the time evolution of radiation from a source of the perturbation is qualitatively the same as for the Schwarschild black hole. Namely, the emitted radiation consists of the following three components:

(i) an initial wave burst that contains radiation emitted directly by the source of the perturbation,

(ii) exponentially damped ‘ringing’ at frequencies that do not depend on the source of the perturbation at all and

(iii) a power law ‘tail’ that arises because of backscattering by the long-range gravitational field.

Quantitive differences which exist between non-rotating and rotating black hole cases are of the most interest since they, in principle, might allow an observer receiving radiation from a black hole to determine its angular velocity. Let us discuss first quasinormal modes in the Kerr spacetime.

When the black hole has nonzero angular momentum, a, the azimuthal degeneracy is split. For a multipole ℓ there are consequently 2ℓ + 1 distinct modes that approach each Schwarzschild mode in the limit a → 0. These modes correspond to different values of m, where −ℓ ≤ m ≤ ℓ.

Quasinormal modes for Kerr black holes were first calculated by Detweiler. In the limit of the extremal black hole (a → M), complex frequencies of quasinormal modes possess the following properties:

It is interesting that some quasinormal modes become very long lived for rapidly rotating black holes. This could potentially be of great importance for gravitational wave detection.

1.2 Gravitational radiation from a particle plunging into the black hole

In general, the equation governing a black hole perturbation is not homogeneous. One must typically also include a source term appropriate for the physical situation under consideration. Perhaps the simplest relevant problem is that of a test particle moving in the gravitational field of a black hole. When the mass m of the particle is sufficiently small compared to that of the black hole (m << M), the problem can be viewed as a perturbation problem. The radiation emitted by a test particle of mass m which falls radially into a black hole is one of the astrophysical applications of the perturbation equations.

Simple dimensional arguments show that the total energy ΔE, emitted by the particle of mass m plunging into the black hole of mass M, is proportional to m²/M. When the black hole is rotating, ΔE is an asymmetric function of ˜L . It has a minimum at the negative value of ˜L. This can be understood in the following way. Positive values of ˜L correspond to a particle that corotates with the black hole, whereas negative values are for counter-rotation. When a particle that was initially counter-rotating reaches the vicinity of the black hole, it will be slowed down because of frame-dragging. Thus, fewer gravitational waves are radiated. Similarly, an initially corotating particle is speeded up, and the number of gravitational waves that emerges increases.

1.3 Superradiant scattering

For wave scattering by an absorbing non-rotating body, the amplitude of the reflected wave is always less than the amplitude of the infalling wave. In the presence of a ergosphere, that is, the region around a rotating black hole where ξ²_{(t )} > 1, some of the impinging waves can be amplified. This effect is known as superradiance. The condition for superradiant modes is

 (1.1)

The maximum amplification of an incoming wave is 0.3% for scalar waves, 4.4% for electromagnetic waves, and an impressive 138% for gravitational waves.