1.1 Field equation

The electromagnetic field and gravitational perturbations are of most interest in astrophysical applications. Both fields are massless and carry spin. For simplicity, we consider first a massless scalar field with zero spin and discuss later effects caused by spin. Moreover, we consider the simpler case of a non-rotating black hole.

A massless scalar field evolves according to the Klein-Gordon equation

 (1.1)

where g is the determinant of the metric g_μν and J a scalar charge density.

1.2 Spherical reduction

In a general spherically symmetric spacetime with metric

 (1.2)

one can decompose a general solution into the spherical modes

 (1.3)

where Y_ℓm(θ, φ) are the spherical harmonics. The functions u_ℓm obey the two dimensional wave equation

 (1.4)

where ²□ = (−γ)^−1/2∂_A[(−γ)^1/2γ ^AB∂_B] is the two-dimensional ‘box’ operator for the metric γ_AB, and

 (1.5)

For the Schwarzschild metric, we find

 (1.6)

1.3 Radial equation and effective potential

We focus now on solutions of the homogeneous equations with j_ℓm = 0. Since the Schwarzschild geometry is static we can decompose them into monochromatic waves

 (1.7)

where the radial function û_ℓ(r, ω) is a solution of the equation

 (1.8)

Here r_∗ is the so called tortoise coordinate

 (1.9)

Figure 1.1. The effective potential V_ℓ for ℓ = 0, 1, 2 as a function of r .

Figure 1.2. The same potentials as functions of r_∗. The constant in the definition of r_∗ is fixed so that r_∗ = 0 for r = 3M.

and V_ℓ(r ) is the effective potential

 (1.10)

The effective potential for different values of ℓ is shown in figures 1.1 and 1.2. The maximum of the effective potential V_ℓ(r ) is roughly at the location of the unstable circular photon orbit (r = 3M).

The form of the radial wave equation is similar to the quantum mechanical equation for one-dimensional potential scattering and hence most problems concerning perturbed black holes involve elements familiar from potential scattering in quantum mechanics. One would, for example, expect waves of short wavelength λ << r_S to be easily transmitted through the barrier. Waves with λ ≈ r_S will be partly transmitted and partly reflected, and finally waves with λ >> r_S should be completely reflected by the black hole barrier.

1.4 Basic solutions

Let us consider two linearly independent solutions of the radial wave equations which have the following asymptotic forms

 (1.11)

 (1.12)

They are known as IN and UP modes, respectively. The complex conjugates of these solutions are also solutions of the radial wave equation. They are known as OUT and DOWN modes, respectively. For any two solutions u₁ and u₂ the Wronskian

 (1.13)

is constant. Calculating the Wronskian for solutions ûⁱⁿ_ℓ and û^up_ℓ and their complex conjugates and using their asymptotics, one obtains the following relations:

 (1.14)

 (1.15)

 (1.16)

and

 (1.17)

Since |A_in| ≥ 1, the solutions ûⁱⁿ_ℓ and û^up_ℓ are linearly independent.

1.5 Interpretation of basic solutions

Let us discuss the physical meaning of the basic solutions. By combining the radial solutions with exp(-iωt) we get functions describing wave propagation. They have simple physical interpretations. The DOWN solution satisfies the boundary condition that there is no radiation escaping to infinity. This means that exactly the right amount of radiation with just the right phase must emerge from the past horizon H⁻ in order to cancel any radiation that might otherwise be scattered back to infinity from a wave originally incoming from past infinity.

Thus, in this solution, there is radiation coming in from infinity, radiation emerging from H⁻ to meet it, and radiation going down the black hole at H⁺. The amplitudes of the various waves are such that down is an acceptable solution to the radial wave equation. The UP mode is defined analogously by the boundary condition that there be no incoming radiation from infinity. In a similar way the IN solution does not contain radiation outgoing from H⁻, while the OUT mode has no radiation going down the black hole at H⁺. The situation is presented graphically in figure 1.3.

Figure 1.3. IN, UP, OUT and DOWN modes.

One can use the diagrams presented in figure 1.3 as mnemonic rules for the definition of the basic functions. The regions inside the squares represent the spacetime in the exterior of the eternal version of the black hole. The straight lines at the angle of π/4 represent null rays. Two boundaries ⁺ and ⁻ correspond to asymptotic future and past infinities. The other two boundaries H⁺ and H⁻ are the event horizon and the past horizon, respectively. This type of diagram can be obtained by special conformal transformations that bring infinitely distant points of the spacetime to a finite distance. The corresponding Penrose-Carter conformal diagram proved to be a very powerful tool for the study of the global structure of spacetime. Asymptotic values of massless fields in an asymptotically flat physical spacetime are related to the boundary values at the null surfaces ⁺ and ⁻, representing the so-called future and past null infinities.