### 6.4 Local symmetries

We will now make explicit the local symmetries of the near-horizon scalar field equations (209). For this purpose it is convenient to define the “conformal” coordinates defined in terms of coordinates by (see [68] and [210] for earlier relevant work)
The change of coordinates is locally invertible if . We choose the chirality , as it will turn out to match the chirality convention in the description of extremal black holes in Section 4.4.

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

with so that

Let us define locally the vector fields

and
These vector fields obey the Lie bracket algebra,
and similarly for . Note that
In terms of the coordinates , the Casimir becomes
where .

We will now match the radial wave equation around the Kerr–Newman black hole in the near region (209) with the eigenvalue equation

The scalar field has the following eigenvalues and . In the case where an electromagnetic field is present, one can perform the uplift (2) and consider the five-dimensional gauge field (6). In that case, the eigenvalue of the five-dimensional gauge field under is the electric charge . Let us denote the eigenvalue along as . Eqs. (209) and (220) will match if and only if the two following equations are obeyed
where has been defined in (200).

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

upon choosing (and ). This shows in particular that the Kerr black hole has a hidden symmetry, as derived originally in [68].

For probes with zero angular momentum , but electric charge , there is also a unique solution,

upon choosing (and ). This shows, in particular, that the Reissner–Nordström black hole admits a hidden symmetry, as pointed out in [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 that case, one chooses the coordinate system (211) and the unique solution is then
When and or and , one recovers the two previous particular cases. The condition (224) is equivalent to the fact that the scalar field has zero eigenvalue along . Since and are quantized, as derived in (7), there is always (at least) one solution to (224) with integers and .

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).