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, 235
, 257
]
in the early 1970s (see also [145, 4, 122
]). The scattering of a spin 0 and 1/2 field from a Kerr–Newman
black hole has also been solved [257
], 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
, which obeys
This tensor can be used to construct a symmetric Killing tensor
which is a natural generalization of the concept of Killing vector
(obeying
). This Killing
tensor was first used by Carter in order to define an additional conserved charge for geodesics [65
]
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
, which
commutes with the Laplacian
. This allows one to separate the solutions of the scalar wave equation
as [65]
where
is the real separation constant present in both equations for
and
. 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 [160
] (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
, which governs the entire probe dynamics. As a result, one has
The master scalar is constructed from the field strength and from the Weyl tensor for spin 1 (
) and
spin 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
and a second-order
equation for the radial part of the master scalar
. 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
in what
follows.
The angular functions
obey the spin-weighted spheroidal harmonic equation
(The Kronecker
is introduced so that the multiplicative term only appears for a massive scalar field of
mass
.) All harmonics that are regular at the poles can be obtained numerically and can be classified by
the usual integer number
with
and
. In general, the separation constant
depends on the product
, on the integer
, on the angular momentum of the probe
and on the
spin
. At zero energy (
), the equation reduces to the standard spin-weighted spherical-harmonic
equation and one simply has
. For a summary of analytic and numerical results,
see [44].
Let us now take the values
as granted and turn to the radial equation. The radial equation
reduces to the following Sturm–Liouville equation
where
in a potential
. The form of the potential is
pretty intricate. For a scalar field of mass
, the potential
is real and is given by
where
. For a field of general spin on the Kerr geometry, the potential is,
in general, complex and reads as
where
. 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 (159) cannot be solved analytically and one has to use
numerical methods.
For each frequency
and spheroidal harmonic
, the scalar field can be extended at infinity
into an incoming wave and an outgoing wave. The absorption probability
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,
An important feature is that in the superradiant range (13) 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.