Several choices of coordinate will lead to independent
symmetries.
For the Kerr black hole, there is only one meaningful choice:
. For the Reissner–Nordström black
hole, we identify
, where
is the Kaluza–Klein coordinate that allows one to lift the gauge
field to higher dimensions, as done in Section 4.3. For the Kerr–Newman black hole, we use, in general, a
coordinate system
parameterized by a
transformation
Let us define locally the vector fields
and These vector fields obey the We will now match the radial wave equation around the Kerr–Newman black hole in the near region
(209) with the eigenvalue equation
For simplicity, let us first discuss the case of zero probe charge and non-zero probe angular
momentum
. The matching equations then admit a unique solution
For probes with zero angular momentum , but electric charge
, there is also a unique
solution,
Finally, one can more generally solve the matching equation for any probe scalar field whose probe angular momentum and probe charge are related by
In that case, one chooses the coordinate system (211 In conclusion, any low energy and low mass scalar probe in the near region (206) of the Kerr black hole
admits a local hidden
symmetry. Similarly, any low energy, low mass and low charge
scalar probe in the near region (206
) of the Reissner–Nordström black hole admits a local hidden
symmetry. In the case of the Kerr–Newman black hole, we noticed that probes
obeying (205
) also admit an
hidden symmetry, whose precise realization depends on
the ratio between the angular momentum and the electric charge of the probe. For a given ratio (224
),
hidden symmetries can be constructed using the coordinate
. Different choices
of coordinate
are relevant to describe different sectors of the low energy, low mass and
low charge dynamics of scalar probes in the near region of the Kerr–Newman black hole. The
union of these descriptions cover the entire dynamical phase space in the near region under the
approximations (205
) – (206
).
http://www.livingreviews.org/lrr-2012-11 |
Living Rev. Relativity 15, (2012), 11
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