2.6 Temperature and chemical potentials

Even though the Hawking temperature is zero at extremality, quantum states just outside the horizon are not pure states when one defines the vacuum using the generator of the horizon. Let us review these arguments following [156Jump To The Next Citation Point, 159Jump To The Next Citation Point, 83Jump To The Next Citation Point]. We will drop the index I distinguishing different gauge fields since this detail is irrelevant to the present arguments.

From the expression of the entropy in terms of the charges 𝒮ext(𝒥 ,𝒬e, 𝒬m ), one can define the chemical potentials

( ) ( ) ( ) 1-- ∂-𝒮ext 1-- ∂-𝒮ext 1-- ∂𝒮ext T = ∂𝒥 , T = ∂𝒬 , T = ∂𝒬 . ϕ 𝒬e,m e e 𝒥 ,𝒬m m m 𝒥 ,𝒬e
Note that electromagnetic charges are quantized, but when the charges are large one can use the continuous thermodynamic limit. These potentials obey the balance equation
1 1 1 δ𝒮ext = ---δ𝒥 + --δ𝒬e + ---δ𝒬m. (57 ) Tϕ Te Tm

Another way to obtain these potentials is as follows. At extremality, any fluctuation obeys

0 = TH δ𝒮 = δℳ − (ΩeJxtδ𝒥 + Φeextδ 𝒬e + Φexmtδ𝒬m ), (58 )
where ext ΩJ is the angular potential at extremality and ext Φ e,m 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 (58View Equation) as follows: any variation in 𝒥 or 𝒬m,e is accompanied by an energy variation. One can then solve for ℳ = ℳ (𝒥 ,𝒬 ,𝒬 ) ext e m. The first law for a non-extremal black hole can be written as

-1- δ𝒮 = TH (δ ℳ − (ΩJ δ𝒥 + Φe δ𝒬e + Φm δ𝒬m )). (59 )
Let us now take the extremal limit using the following ordering. We first take extremal variations with δℳ = δℳext (𝒥 ,𝒬e, 𝒬m ). Then, we take the extremal limit of the background configuration. We obtain (57View Equation) with
| TH ∂TH ∕∂r+ | Tϕ = Tlim→0 Ωext−-Ω---= − ∂Ω--∕∂r--|| , (60 ) H J J J + r+=r|ext ----TH------ -∂TH-∕∂r+--|| Te,m = TlHim→0 Φext − Φ = − ∂Φ âˆ•∂r | , (61 ) e,m e,m e,m + r+=rext
where the extremal limit can be practically implemented by taking the limit of the horizon radius r+ to the extremal horizon radius r ext.

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 ρ = e− ω∕TH at the Hawking temperature TH. 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 ˆω and angular momentum mˆ with ˆt and ϕˆ dependence as ˆ e− iˆωˆt+iˆmϕ. When approaching extremality, one can perform the change of coordinates (23View Equation) in order to zoom close to the horizon. By definition, the scalar field ϕ in the new coordinate system xa = (t,ϕ,𝜃,r ) reads in terms of the scalar field ˆÏ• in the asymptotic coordinate system a ˆ ˆ ˆx = (t,ϕ,𝜃, ˆr) as a ˆ a ϕ (x ) = ϕ(ˆx ). We can then express

− iωt+imϕ −iˆωˆt+imˆÏ• e = e , (62 )
and the near-horizon parameters are
ˆω − m Ω m = ˆm, ω = --------J. (63 ) λ
When no electromagnetic field is present, any finite energy ω in the near-horizon limit at extremality λ → 0 corresponds to eigenstates with ˆω = mˆΩext J. When electric fields are present, zooming in on the near-horizon geometry from a near-extremal solution requires one to perform the gauge transformation A (x) → A(x) + dΛ (x) with gauge parameter given in (19View Equation), which will transform the minimally-coupled charged scalar wavefunction by multiplying it by eiqeΛ(x). Finite energy excitations in the near-horizon region then require ˆω = m ΩeJxt+ qeΦexet. 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 form
ˆω = m ΩexJt + qeΦeext+ qm Φemxt. (64 )

Following Frolov and Thorne, we assume that quantum fields in the non-extremal geometry are populated with the Boltzmann factor

( ) ˆω-−-ˆm-ΩJ-−-qˆeΦe-−-qˆm-Φm-- exp ℏ TH , (65 )
where ˆqe,m are the electric and magnetic charge operators. We also assume that modes obey (64View Equation) at extremality. Using the definitions (60View Equation) – (61View Equation), we obtain the non-trivial extremal Boltzmann factor in the extremal and near-horizon limit
( m qe qm ) exp − ℏ ---− ℏ---− ℏ--- , (66 ) Tϕ Te Tm
where the mode number m and charges qe,m in the near-horizon region are equal to the original mode number and charges ˆm,qˆ e,m. This completes the argument that the Frolov–Thorne vacuum is non-trivially populated in the extremal limit.

Now, as noted in [4Jump To The Next Citation Point], 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 (34View Equation). This condition is violated for the extremal Kerr black hole or for any extremal Kerr–Newman black hole with √ -- a ≥ Q ∕ 3, 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 √ -- a < Q ∕ 3 and the Kerr–Newman–AdS black holes do as well when 4a2∕(Δ0r2+) < 1, which is true for large black holes with r+ ≫ l. 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 (25View Equation). More precisely, these quantities should only depend on the metric and matter fields and not on their equations of motion. Indeed, from the derivation (60View Equation) – (61View Equation), 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 (1View Equation), including possible higher-order corrections, one can derive [83Jump To The Next Citation Point, 20Jump To The Next Citation Point] the very simple formula

-1-- Tϕ = 2πk. (67 )
From similar considerations, it should also be possible to derive a formula for Te in terms of the functions appearing in (25View Equation). We propose simply that
Te = -1-. (68 ) 2πe
While we do not have a direct proof of the equivalence between (68View Equation) and (61View Equation), 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 [83Jump To The Next Citation Point, 20Jump To The Next Citation Point] to prove the equivalence.

Similarly, one can work out the thermodynamics of five-dimensional rotating black holes. Since there are two independent angular momenta 𝒥1, 𝒥2, there are also two independent chemical potentials Tϕ1, T ϕ2 associated with the angular momenta. The same arguments lead to

--1-- -1--- Tϕ1 = 2πk1 , T ϕ2 = 2πk2, (69 )
where k1 and k2 are defined in the near-horizon solution (36View Equation).

When considering the uplift (2View Equation) of the gauge field along a compact direction of length 2πR χ, one can use the definition (69View Equation) to define the chemical potential associated with the direction ∂χ. Since the circle has a length 2πR χ, the extremal Frolov–Thorne temperature is expressed in units of R χ,

R χ Tχ ≡ TeR χ = ----, (70 ) 2 πe
where Te is defined in (68View Equation).

Temperatures and entropies of specific extremal black holes

The entropy of the extremal Kerr black hole is 𝒮ext = 2 πJ. Integrating (57View Equation) or using the explicit near-horizon geometry and using (67View Equation), we find

1 T ϕ = ---, (71 ) 2π
and Te is not defined.

The entropy of the extremal Reissner–Nordström black hole is 𝒮ext = πQ2. Integrating (57View Equation), we obtain

1 Te = ----, (72 ) 2πQ
while Tϕ is not defined.

For the electrically-charged Kerr–Newman black hole, the extremal entropy reads as 2 2 𝒮ext = π(a + r+ ). Expressing the entropy in terms of the physical charges ∘ -------- Q = r2+ − a2 and J = ar+, we obtain

( ) 2 ∘ --------- 𝒮ext = π- ∘------4J--------+ Q4 + 4J 2 − Q2 . (73 ) 2 Q4 + 4J2 − Q2
Using (57View Equation) and re-expressing in terms of the parameters (a,r+) we find
a2-+-r2+- ---a2-+-r2+---- Tϕ = 4πar , Te = 2π(r2 − a2)3∕2. (74 ) + +
We can also derive T ϕ from (67View Equation) and the explicit near-horizon geometry (39View Equation). Te is consistent with (68View Equation).

For the extremal Kerr–Newman–AdS black hole, the simplest way to obtain the thermodynamics at extremality is to compute (60View Equation) – (61View Equation). Using the extremality constraint (46View Equation), we obtain

(a2 + r2 )Δ0 (a2 + r2 )Δ0 Tϕ = ------+-----, Te = -------+2----2-, (75 ) 4πar+ Ξ 2πQe (r+ − a )
where we used the definitions (48View Equation). The magnetic potential Tm can then be obtained by electromagnetic duality. The expressions coincide with (67View Equation) – (68View Equation). These quantities reduce to (74View Equation) in the limit of no cosmological constant when there is no magnetic charge, qm = 0. The extremal entropy is given by 𝒮ext = π(r2+ + a2)∕Ξ.
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