2.5 Entropy

The classical entropy of any black hole in Einstein gravity coupled to matter fields such as (1View Equation) is given by
∫ --1--- 𝒮 = 4G ℏ vol(Σ ), (49 ) N Σ
where Σ is a cross-section of the black-hole horizon and GN is the four-dimensional Newton’s constant. In the near-horizon geometry, the horizon is formally located at any value of r as a consequence of the definition (23View Equation). Nevertheless, we can move the surface Σ to any finite value of r without changing the integral, thanks to the scaling symmetry ζ 0 of (29). Evaluating the expression (49View Equation), we obtain
∫ π 𝒮 = --π--- d𝜃α(𝜃)Γ (𝜃 )γ (𝜃). (50 ) 2GN ℏ 0
In particular, the entropy of the extremal Kerr black hole is given by
𝒮 = 2π𝒥 . (51 )
In units of ℏ the angular momentum 𝒥 is a dimensionless half-integer. The main result [156Jump To The Next Citation Point, 203Jump To The Next Citation Point, 21Jump To The Next Citation Point, 159Jump To The Next Citation Point, 225Jump To The Next Citation Point, 83Jump To The Next Citation Point, 173Jump To The Next Citation Point, 22, 204Jump To The Next Citation Point, 97Jump To The Next Citation Point] of the extremal spinning black hole/CFT correspondence that we will review below is the derivation of the entropy (50View Equation) using Cardy’s formula (90View Equation).

When higher derivative corrections are considered, the entropy does not scale any more like the horizon area. The black-hole entropy at equilibrium can still be defined as the quantity that obeys the first law of black-hole mechanics, where the mass, angular momenta and other extensive quantities are defined with all higher-derivative corrections included. More precisely, the entropy is first defined for non-extremal black holes by integrating the first law, and using properties of non-extremal black holes, such as the existence of a bifurcation surface [262, 176]. The resulting entropy formula is unique and given by

2π ∫ δcovL 𝒮 = − --- ------𝜖ab𝜖cdvol(Σ ), (52 ) ℏ Σ δRabcd
where 𝜖ab is the binormal to the horizon, i.e., the volume element of the normal bundle to Σ. One can define it simply as 𝜖ab = naξb − ξanb, where ξ is the generator of the horizon and n is an outgoing null normal to the horizon defined by 2 n = 0 and a n ξa = − 1. Since the Lagrangian is diffeomorphism invariant (possibly up to a boundary term), it can be expressed in terms of the metric, the matter fields and their covariant derivatives, and the Riemann tensor and its derivatives. This operator δcov∕δRabcd acts on the Lagrangian while treating the Riemann tensor as if it were an independent field. It is defined as a covariant Euler–Lagrange derivative as
cov -δ---- ∑ i --------∂--------- δRabcd = (− 1) ∇ (e1 ...∇ei )∂∇ (e1 ...∇ei )Rabcd . (53 ) i=0
Moreover, the entropy formula is conserved away from the bifurcation surface along the future horizon as a consequence of the zeroth law of black-hole mechanics [178]. Therefore, one can take the extremal limit of the entropy formula evaluated on the future horizon in order to define entropy at extremality. Quite remarkably, the Iyer–Wald entropy (52View Equation) can also be reproduced [20Jump To The Next Citation Point] using Cardy’s formula as we will detail below.

In five-dimensional Einstein gravity coupled to matter, the entropy of extremal black holes can be expressed as

π2 ∫ π 𝒮 = ----- d𝜃α (𝜃 )Γ (𝜃)γ(𝜃), (54 ) ℏGN 0
where Γ (𝜃) and α (𝜃) have been defined in (36View Equation) and γ (𝜃)2 = det (γij(𝜃)2).

From the attractor mechanism for four-dimensional extremal spinning black holes [17], the entropy at extremality can be expressed as an extremum of the functional

∫ I A -2π-- I √--- f (Γ (𝜃),γ(𝜃),f (𝜃),χ (𝜃),k,eI) = G ℏ (k𝒥 + eI𝒬 − d𝜃dϕ − gℒ ), (55 ) N
where ℒ is the Lagrangian. The entropy then only depends on the angular momentum 𝒥 and the conserved charges 𝒬Ie,m,
𝒮 = 𝒮ext(𝒥 ,𝒬Ie,𝒬Im ), (56 )
and depend in a discontinuous fashion on the scalar moduli [240]. The result holds for any Lagrangian in the class (1View Equation), including higher-derivative corrections, and the result can be generalized straightforwardly to five dimensions.

When quantum effects are taken into account, the entropy formula also gets modified in a non-universal way, which depends on the matter present in quantum loops. In Einstein gravity, the main correction to the area law is a logarithmic correction term. The logarithmic corrections to the entropy of extremal rotating black holes can be obtained using the quantum entropy function formalism [241Jump To The Next Citation Point].

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