5.2 Macroscopic greybody factors

The problem of scattering of a general spin field from a Kerr black hole was solved in a series of classic papers by Starobinsky [248], Starobinsky and Churilov [249] and Press and Teukolsky [255, 256, 235Jump To The Next Citation Point, 257Jump To The Next Citation Point] in the early 1970s (see also [145, 4, 122Jump To The Next Citation Point]). The scattering of a spin 0 and 1/2 field from a Kerr–Newman black hole has also been solved [257Jump To The Next Citation Point], while the scattering of spins 1 and 2 from the Kerr–Newman black hole has not been solved to date.

Let us review how to solve this classic scattering problem. First, one has to realize that the Kerr–Newman black hole enjoys a remarkable property: it admits a Killing–Yano tensor [269, 232, 142]. (For a review and some surprising connections between Killing–Yano tensors and fermionic symmetries, see [148].) A Killing–Yano tensor is an anti-symmetric tensor f = − f μν νμ, which obeys

∇ (λfμ)ν = 0. (153 )
This tensor can be used to construct a symmetric Killing tensor
K = f λf , ∇ K = 0, (154 ) μν μ λν (λ μν)
which is a natural generalization of the concept of Killing vector K μ (obeying ∇ (μK ν) = 0). This Killing tensor was first used by Carter in order to define an additional conserved charge for geodesics [65Jump To The Next Citation Point]
Q = K Λ™xμΛ™xν, (155 ) μν
and thereby reduce the geodesic equations in Kerr to first-order equations. More importantly for our purposes, the Killing tensor allows one to construct a second-order differential operator K μν∇ μ∇ ν, which commutes with the Laplacian ∇2. This allows one to separate the solutions of the scalar wave equation 2 s=0 ∇ Ψ = 0 as [65]
Ψs=0 = e−iωt+im Ο•S (πœƒ)R (r), (156 ) ω,A,m ω,A,m
where A is the real separation constant present in both equations for S (πœƒ) and R (r). The underlying Killing–Yano tensor structure also leads to the separability of the Dirac equation for a probe fermionic field. For simplicity, we will not discuss further fermionic fields here and we refer the interested reader to the original reference [160Jump To The Next Citation Point] (see also [41]). The equations for spin 1 and 2 probes in Kerr can also be shown to be separable after one has conveniently reduced the dynamics to a master equation for a master scalar Ψs, which governs the entire probe dynamics. As a result, one has
Ψs = e−iωt+im Ο•Ss (πœƒ)Rs (r). (157 ) ω,A,m ω,A,m
The master scalar is constructed from the field strength and from the Weyl tensor for spin 1 (s = ±1) and spin 2 (s = ±2) fields, respectively, using the Newman–Penrose formalism. For the Kerr–Newman black hole, all attempts to separate the equations for spin 1 and spin 2 probes have failed. Hence, there is no known analytic method to solve those equations (for details, see [70]). Going back to Kerr, given a solution to the master scalar field equation, one can then in principle reconstruct the gauge field and the metric from the Teukolsky functions. This non-trivial problem was778ikm solved right after Teukolsky’s work [89, 87]; see Appendix C of [122] for a modern review (with further details and original typos corrected).

In summary, for all separable cases, the dynamics of probes in the Kerr–Newman geometry can be reduced to a second-order equation for the angular part of the master scalar Ssω,A,m(πœƒ) and a second-order equation for the radial part of the master scalar Rs (r) ω,A,m. Let us now discuss their solutions after imposing regularity as boundary conditions, which include ingoing boundary conditions at the horizon. We will limit our discussion to the non-negative integer spins s = 0,1,2 in what follows.

The angular functions Ssω,A,m(πœƒ) obey the spin-weighted spheroidal harmonic equation

[ 2 -1--∂πœƒ (sin πœƒ∂πœƒ) − a2 (ω2 − μ2δs,0)sin2πœƒ − 2aωs cos πœƒ − (m-+-s-cosπœƒ)-- sin πœƒ sin2 πœƒ ] +A Ssω,A,m (πœƒ) = 0. (158 )
(The Kronecker δs,0 is introduced so that the multiplicative term only appears for a massive scalar field of mass μ.) All harmonics that are regular at the poles can be obtained numerically and can be classified by the usual integer number l with l ≥ |m | and l ≥ |s|. In general, the separation constant s A = A aω,l,m depends on the product aω, on the integer l, on the angular momentum of the probe m and on the spin s. At zero energy (ω = 0), the equation reduces to the standard spin-weighted spherical-harmonic equation and one simply has As = l(l + 1 ) − s2 0,l,m. For a summary of analytic and numerical results, see [44].

Let us now take the values As aω,l,m as granted and turn to the radial equation. The radial equation reduces to the following Sturm–Liouville equation

[ ] −s ( s+1 ) s s Δ ∂r Δ ∂r − V (r) R (r) = 0, (159 )
where Δ (r) = (r − r )(r − r ) = r2 − 2M r + a2 + Q2 + − in a potential Vs(r). The form of the potential is pretty intricate. For a scalar field of mass μ, the potential 0 V (r) is real and is given by
2 V0(r) = − H--(r) − 2am ω + A0 + μ2(r2 + a2), (160 ) Δ (r) aω,l,m
where 2 2 H (r) = ω(r + a ) − qeQr − am. For a field of general spin on the Kerr geometry, the potential is, in general, complex and reads as
H2 (r) − 2is(r − M )H (r) V s(r) = − -------------------------− 4isωr − 2am ω + Asaω,l,m − s(s + 1), (161 ) Δ(r)
where H (r) = ω(r2 + a2) − am. This radial equation obeys the following physical boundary condition: we require that the radial wave has an ingoing group velocity – or, in other words, is purely ingoing – at the horizon. This is simply the physical requirement that the horizon cannot emit classical waves. This also follows from a regularity requirement. The solution is then unique up to an overall normalization. For generic parameters, the Sturm–Liouville equation (159View Equation) cannot be solved analytically and one has to use numerical methods.

For each frequency ω and spheroidal harmonic (l,m ), the scalar field can be extended at infinity into an incoming wave and an outgoing wave. The absorption probability σabs or macroscopic greybody factor is then defined as the ratio between the absorbed flux of energy at the horizon and the incoming flux of energy from infinity,

dEabsβˆ•dt- σabs(ω, l,m, s;M, a,Q ) = dE βˆ•dt . (162 ) in
An important feature is that in the superradiant range (13View Equation) the absorption probability turns out to be negative, which results in stimulated as well as spontaneous emission of energy, as we reviewed in Section 2.1.
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