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
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,[81, 77].
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 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).
Living Rev. Relativity 15, (2012), 11
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