From the expression of the entropy in terms of the charges , one can define the
chemical potentials
Another way to obtain these potentials is as follows. At extremality, any fluctuation obeys
where 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
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
Following Frolov and Thorne, we assume that quantum fields in the non-extremal geometry are populated with the Boltzmann factor
where 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
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
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
,
The entropy of the extremal Kerr black hole is . Integrating (57
) or using the explicit
near-horizon geometry and using (67
), we find
The entropy of the extremal Reissner–Nordström black hole is . Integrating (57
), we
obtain
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
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
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Living Rev. Relativity 15, (2012), 11
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