From the expression of the entropy in terms of the charges , one can define the chemical potentials

Note that electromagnetic charges are quantized, but when the charges are large one can use the continuous thermodynamic limit. These potentials obey the balance equationAnother way to obtain these potentials is as follows. At extremality, any fluctuation obeys

where is the angular potential at extremality and are electric and magnetic potentials at extremality; see Section 2.1 for a review of these concepts.One can express the first law at extremality (58) as follows: any variation in or is accompanied by an energy variation. One can then solve for . The first law for a non-extremal black hole can be written as

Let us now take the extremal limit using the following ordering. We first take extremal variations with . Then, we take the extremal limit of the background configuration. We obtain (57) with where the extremal limit can be practically implemented by taking the limit of the horizon radius to the extremal horizon radius .The interpretation of these chemical potentials can be made in the context of quantum field theories in curved spacetimes; see [47] for an introduction. The Hartle–Hawking vacuum for a Schwarzschild black hole, restricted to the region outside the horizon, is a density matrix at the Hawking temperature . For spacetimes that do not admit a global timelike Killing vector, such as the Kerr geometry, the Hartle–Hawking vacuum does not exist, but one can use the generator of the horizon to define positive frequency modes and, therefore, define the vacuum in the region where the generator is timelike (close enough to the horizon). This is known as the Frolov–Thorne vacuum [144] (see also [128]). One can take a suitable limit of the definition of the Frolov–Thorne vacuum to provide a definition of the vacuum state for any spinning or charged extremal black hole.

Quantum fields for non-extremal black holes can be expanded in eigenstates with asymptotic energy and angular momentum with and dependence as . When approaching extremality, one can perform the change of coordinates (23) in order to zoom close to the horizon. By definition, the scalar field in the new coordinate system reads in terms of the scalar field in the asymptotic coordinate system as . We can then express

and the near-horizon parameters are When no electromagnetic field is present, any finite energy in the near-horizon limit at extremality corresponds to eigenstates with . When electric fields are present, zooming in on the near-horizon geometry from a near-extremal solution requires one to perform the gauge transformation with gauge parameter given in (19), which will transform the minimally-coupled charged scalar wavefunction by multiplying it by . Finite energy excitations in the near-horizon region then require . Invoking (classical) electromagnetic duality, the magnetic contribution has the same form as the electric contribution. In summary, the general finite-energy extremal excitation has the formFollowing Frolov and Thorne, we assume that quantum fields in the non-extremal geometry are populated with the Boltzmann factor

where are the electric and magnetic charge operators. We also assume that modes obey (64) at extremality. Using the definitions (60) – (61), we obtain the non-trivial extremal Boltzmann factor in the extremal and near-horizon limit where the mode number and charges in the near-horizon region are equal to the original mode number and charges . This completes the argument that the Frolov–Thorne vacuum is non-trivially populated in the extremal limit.Now, as noted in [4], there is a caveat in the previous argument for the Kerr black hole and, as a trivial generalization, for all black holes that do not possess a global timelike Killing vector. For any non-extremal black hole, the horizon-generating Killing field is timelike just outside the horizon. If there is no global timelike Killing vector, this vector field should become null on some surface at some distance away from the horizon. This surface is called the velocity of light surface. For positive-energy matter, this timelike Killing field defines a positive conserved quantity for excitations in the near-horizon region, ruling out instabilities. However, when approaching extremality, it might turn out that the velocity of light surface approaches asymptotically the horizon. In that case, the horizon-generating Killing field of the extreme black hole may not be everywhere timelike. This causes serious difficulties in defining quantum fields directly in the near-horizon geometry [183, 229, 228]. However, (at least classically) dynamical instabilities might appear only if there are actual bulk degrees of freedom in the near-horizon geometries. We will argue that this is not the case in Section 2.9. As a conclusion, extremal Frolov–Thorne temperatures can be formally and uniquely defined as the extremal limit of non-extremal temperatures and chemical potentials. However, the physical interpretation of these quantities is better understood finitely away from extremality.

The condition for having a global timelike Killing vector was spelled out in (34). This condition is violated for the extremal Kerr black hole or for any extremal Kerr–Newman black hole with , as can be shown by using the explicit values defined in (2.4). (The extremal Kerr–Newman near-horizon geometry does possess a global timelike Killing vector when and the Kerr–Newman–AdS black holes do as well when , which is true for large black holes with . Nevertheless, there might be other instabilities due to the electric superradiant effect.)

The extremal Frolov–Thorne temperatures should also be directly encoded in the metric (25). More precisely, these quantities should only depend on the metric and matter fields and not on their equations of motion. Indeed, from the derivation (60) – (61), one can derive these quantities from the angular velocity, electromagnetic potentials and surface gravity, which are kinematical quantities. More physically, the Hawking temperature arises from the analysis of free fields on the curved background, and thus depends on the metric but not on the equations of motion that the metric solves. It should also be the case for the extremal Frolov–Thorne temperatures. Using a reasonable ansatz for the general black-hole solution of (1), including possible higher-order corrections, one can derive [83, 20] the very simple formula

From similar considerations, it should also be possible to derive a formula for in terms of the functions appearing in (25). We propose simply that While we do not have a direct proof of the equivalence between (68) and (61), the formula is consistent with the thermodynamics of (AdS)–Kerr–Newman black holes as one can check from the formulae in Section 2.4. It would be interesting to generalize the arguments of [83, 20] to prove the equivalence.Similarly, one can work out the thermodynamics of five-dimensional rotating black holes. Since there are two independent angular momenta , , there are also two independent chemical potentials , associated with the angular momenta. The same arguments lead to

where and are defined in the near-horizon solution (36).When considering the uplift (2) of the gauge field along a compact direction of length , one can use the definition (69) to define the chemical potential associated with the direction . Since the circle has a length , the extremal Frolov–Thorne temperature is expressed in units of ,

where is defined in (68).

The entropy of the extremal Kerr black hole is . Integrating (57) or using the explicit near-horizon geometry and using (67), we find

and is not defined.The entropy of the extremal Reissner–Nordström black hole is . Integrating (57), we obtain

while is not defined.For the electrically-charged Kerr–Newman black hole, the extremal entropy reads as . Expressing the entropy in terms of the physical charges and , we obtain

Using (57) and re-expressing in terms of the parameters we find We can also derive from (67) and the explicit near-horizon geometry (39). is consistent with (68).For the extremal Kerr–Newman–AdS black hole, the simplest way to obtain the thermodynamics at extremality is to compute (60) – (61). Using the extremality constraint (46), we obtain

where we used the definitions (48). The magnetic potential can then be obtained by electromagnetic duality. The expressions coincide with (67) – (68). These quantities reduce to (74) in the limit of no cosmological constant when there is no magnetic charge, . The extremal entropy is given by .
Living Rev. Relativity 15, (2012), 11
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