2.7 Near-extremal near-horizon geometries

An important question about near-horizon geometries is the following: how much dynamics of gravity coupled to matter fields is left in a near-horizon limit such as (23View Equation)? We will explore in the following Sections 2.8 and 2.9 several aspects of the dynamics in the near-horizon limit. In this section, we will discuss the existence of near-extremal solutions obtained from a combined near-horizon limit and zero temperature limit. We will discuss in Section 2.8 the absence of non-perturbative solutions in the near-horizon geometries, such as black holes. In Section 2.9, we will argue for the absence of local bulk degrees of freedom, and finally in Section 4.4 we will discuss non-trivial boundary dynamics generated by large diffeomorphisms.

Let us first study infinitesimal perturbations of the near-horizon geometry (25View Equation). As a consequence of the change of coordinates and the necessary shift of the gauge field (23View Equation), the near-horizon energy δ𝒬 ∂t of an infinitesimal perturbation is related to the charge associated with the generator of the horizon ξtot ≡ (ξ,Λ ) = (∂t + ΩexJt∂ϕ,Φexet) as follows,

δ𝒬 = λ-δ𝒬 , λ → 0, (76 ) ξtot r0 ∂t
as derived in Sections 2.2 and 2.3. Assuming no magnetic charges for simplicity, the conserved charge δ𝒬 ξtot is given by δℳ − Ωextδ𝒥 − Φextδ𝒬 J e e.10 Using the first law of thermodynamics valid for arbitrary (not necessarily stationary) perturbations, the left-hand side of (76View Equation) can be expressed as
T δ 𝒮 = δ𝒬 . (77 ) H ext ξtot
Any geometry that asymptotes to (25View Equation) will have finite near-horizon energy 𝒬 ∂t. Indeed, an infinite near-horizon energy would be the sign of infrared divergences in the near-horizon geometry and it would destabilize the geometry. It then follows from (76View Equation) – (77View Equation) that any infinitesimal perturbation of the near-horizon geometry (25View Equation) will correspond to an extremal black-hole solution with vanishing Hawking temperature, at least such that TH = O (λ). Common usage refers to black-hole solutions, where TH ∼ λ as near-extremal black holes. Nevertheless, it should be emphasized that after the exact limit λ → 0 is taken the Hawking temperature of such a solution is exactly zero.

We can obtain a near-extremal near-horizon geometry as follows. Starting from a stationary non-extremal black hole of mass M in Boyer–Lindquist coordinates, we perform the near-horizon scaling limit (23View Equation) together with the scaling of the temperature

T → λ-T near−ext. (78 ) H r0
While the form of the general non-extremal solution would be required to perform that limit in detail, all examples so far in the class of theories (1View Equation), such as the Kerr–Newman–AdS black hole, lead to the following metric
[ ] 2 near−ext 2 dr2 2 2 ds = Γ (𝜃) − r(r + 4πT )dt + ----------near−-ext-+ α(𝜃)d𝜃 + γ(𝜃)(dϕ + krdt) , r(r + 4πT ) χA = χA(𝜃), AI = f I(𝜃)(dϕ + krdt ) − eIdϕ. (79 ) k
The near-extremal near-horizon solution (79View Equation) is diffeomorphic to the near-horizon geometry in Poincaré coordinates (25View Equation). Denoting the finite temperature coordinates by a subscript T and the Poincaré coordinates by a subscript P, the change of coordinates reads as [207Jump To The Next Citation Point, 247, 4Jump To The Next Citation Point, 53Jump To The Next Citation Point]
tP = 1(τ+ + τ− ), (80 ) 2 ---2---- rP = τ− − τ+ , (81 ) ( + 2) ϕP = ϕF − 2πT near−extktF + k-log 1-−-(τ-)-- , (82 ) 2 1 − (τ− )2
[ 1 ( r )] τ ± = tanh πT near−exttF ± --log -------F-near−ext . (83 ) 4 rF + 4πT
Therefore, the classical geometries are equivalent. However, since the diffeomorphism is singular at the boundary rF → ∞, there is a distinction at the quantum level. Since the asymptotic time in near-extremal geometries (79View Equation) is different than in extremal geometries (25View Equation), fields will be quantized in a different manner in the two geometries.

Let us now compute the energy of these geometries. Multiplying Eq. (76View Equation) by r0∕λ and using (77View Equation) and (78View Equation), we get that the energy variation around the near-extremal geometry is given by

∕δ𝒬 ∂t = Tnear−extδ𝒮ext, (84 )
where the extremal entropy 𝒮ext can be expressed in terms of the near-horizon quantities as (50View Equation). We denote the variation by ∕δ to emphasize that the energy is not the exact variation of a quantity unless T near−ext is constant or 𝒮ext is fixed (which would then lead to zero energy). Therefore, the charge ∕δ𝒬 ∂ t is a heat term, which does not define a conserved energy. Since our derivation of the formula (84View Equation) was rather indirect, we check that it is correct for the Kerr–Newman–AdS family of black holes by computing the energy variation directly using the Lagrangian charges defined in [36Jump To The Next Citation Point, 90Jump To The Next Citation Point, 97Jump To The Next Citation Point].
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