Scattering and Absorption by Black Holes

In this example we will investigate the scattering and absorption of radiation by black holes.

For simplicity, we will assume a monochromatic plane wave produced by a distant source incident along the symmetry axis on an isolated Kerr black hole. Dolan [Class.Quant.Grav.25:235002,2008] calculated the scattering and absorption cross sections in this case. We will explore how the Black Hole Perturbation Toolkit can be used to reproduce his results. All equation numbers refer to Dolan’s paper.

Absorption cross section

The absorption cross section is given by [Eq. (19)]:
(Subscript[σ, a]) = ((4 (π^2))/(ω^2)) (∑_(l=2)^∞) |(Subscript[, -2])(Subscript[S, l2]) (0, 0; a ω)((|)^2)(Subscript[𝕋, l2])
where (Subscript[, -2])(Subscript[S, l2]) is a spin-weighted spheroidal harmonic and [Eq. (20)]
(Subscript[𝕋, lm])=1-|(e^(2i (Subscript[δ, lmω]^±)))((|)^2)
is the transmission factor for a given mode, which can be determined from radial solutions of the Teukolsky equation via [Eq. (30)]
(e^(2i (Subscript[δ, lmω]^P)))=(((-1))^(l+1))((Subscript[𝒞, lmω])/(16(ω^4)))((Subscript[B, lmω]^ref)/(Subscript[B, lmω]^inc)),
where (Subscript[B, lmω]^inc) and (Subscript[B, lmω]^ref) are the incidence and reflection coefficients as defined by the asymptotic behaviour of the “in” solutions of the radial Teukolsky equation,
(Subscript[, s])(Subscript[R, lmω])(r)~({{{, ({{(Subscript[B, lmω]^trans)(Δ^(-s))(e^(-i κ (SubStar[r]))), (SubStar[r])->-∞}, {(Subscript[B, lmω]^inc)(r^(-1))(e^(-i ω (SubStar[r])))+(Subscript[B, lmω]^ref)(r^(2s+1))(e^(+i ω (SubStar[r]))), (SubStar[r])->+∞}})}})
We can compute everything that is needed for this calculation using the Teukolsky and SpinWeightedSpheroidalHarmonics packages.

Calculation of (Subscript[σ, a])

Load packages

First, we load the packages Teukolsky and SpinWeightedSpheroidalHarmonics packages:

Computation for a single l, ω mode

Now, we start by considering just a single l, ω mode.

Spin-weighted spheroidal harmonic

First, we compute the spin-weighted spheroidal harmonic for the mode:

This can be treated as a function which can be numerically evaluated for any value of θ and ϕ. We are particularly interested in the case θ=0, ϕ=0:

Note that we have obtained an exact result since we are considering the case a=0. For a≠0 we will not be able to obtain an exact answer, but will have an approximate value.

Asymptotic amplitudes

Next, we need the asymptotic amplitudes. We obtain these by constructing a solution to the radial Teukolsky equation. In this case, an analytic solution is not available so we need to specify the precision (this is achieved by setting the precision of the input frequency):

Note that this produces a warning that machine precision may be inadequate in some cases. In this case it is fine, so let’s turn off the warning:

We now extract the asymptotic amplitudes from this solution:

Teukolsky-Starobinsky constant

Next, we need the Teukolsky-Starobinsky constant
((Subscript[𝒞, lmω])^2)=((((λ+2))^2)+4a m ω-4(a^2)(ω^2))((λ^2)+36 a m ω-36 (a^2)(ω^2))+(2λ+3)(96 (a^2)(ω^2)-48 a m ω)-144(a^2)(ω^2)
This requires the eigenvalue, which can be computed using SpinWeightedSpheroidalEigenvalue.

Phase factor

Finally, we construct the phase factor (e^(2i (Subscript[δ, lmω]^±))):

Absorption cross section

We now put everything together to compute the absorption cross section, (Subscript[σ, a]):

Dependence of (Subscript[σ, a]) on mode parameters

Now, it would be interesting to explore how this absorption cross section depends on the mode parameters.

Package calculation into a function

To do so, we package everything we have done so far into a function :

Explore behaviour as a function of frequency (Schwarzschild case)

We can now explore how the absorption cross section depends on the frequency and mode number by tabulating (Subscript[σ, a]) for a set of values of ω and l.

Let’s now look at a wider range of the parameter space and try to reproduce Fig. 1 from Dolan’s paper. For this, we need to work at higher precision in order to obtain an accurate answer in some cases.
[Aside: To speed things up, we can compute do this calculation in parallel. We just need to make sure the relevant packages are loaded on each of the sub-kernels and then use ParallelTable instead of Table]

Kerr case

So far, we have only considered the Schwarzschild case (a=0). However, it is just as easy to study the Kerr case. Let’s try a highly rotating Kerr black hole with a=0.99M, reproducing Fig. 3 from Dolan’s paper.

We see an interesting feature in the l=2 mode which is associated with superradiance.

Exercises

Dolan’s paper also gives results for other spin fields (specifically, s=0, s=(1/2),  s=1, s=2). Reproduce those results using the Teukolsky and SpinWeightedSpheroidalHarmonics packages.

Compute the scattering cross section, given in Eq. (14).