2.2 Near-horizon geometries of static extremal black holes

As a warm-up, let us first review the near-horizon limit of static extremal black holes. In that case, the generator of the horizon (located at r = r+) is the generator of time translations ∂t and the geometry has SO (3 ) rotational symmetry. Since the horizon generator is null at the horizon, the coordinate t diverges there. The near-horizon limit is then defined as
r0t t → λ--, r → r+ + λr0r, (17 )
with λ → 0. The scale r0 is introduced for convenience in order to factor out the overall scale of the near-horizon geometry. In the presence of electrostatic potentials, a change of gauge is required when taking the near-horizon limit (17View Equation). Indeed, in the near-horizon coordinates (17View Equation) the gauge fields take the following form,
ΦI AI = − --er0dt + AIrdr + AI𝜃d𝜃 + AIϕdϕ, (18 ) λ
where ΦIe is the static electric potential of the gauge field AI. Upon taking the near-horizon limit one should, therefore, perform a gauge transformation AI → AI + dΛI of parameter
I,ext ΛI = Φ-e---r0t, (19 ) λ
where ΦIe,ext is the static electric potential at extremality.

It is important to note that one is free to redefine the near-horizon limit parameter λ as λ → α λ for any α > 0. This transformation scales r inversely proportionally to t. Therefore, the near-horizon geometry admits the enhanced symmetry generator

ζ0 = r∂r − t∂t (20 )
in addition to ζ− 1 = ∂t and the SO (3) symmetry generators. Using the properties of static horizons, one can further derive an additional symmetry generator at the horizon ζ1, which together with ζ−1 and ζ0 forms a SL (2,ℝ ) algebra. This argument is purely kinematical and does not involve the field equations; see, e.g., [194Jump To The Next Citation Point] for a detailed derivation. The general near-horizon solution compatible with an SL (2,ℝ ) × SO (3) symmetry is then given by
2 2 2 dr2 2 2 2 ds = v1(− r dt + --2-) + v2(d𝜃 + sin 𝜃dϕ ), r I χA = χA, AI = eIrdt − -p-cos 𝜃dϕ, (21 ) ⋆ 4π
where v1,v2,χA,eI,pI ⋆ are parameters, which are constrained by the equations of motion. The geometry consists of the direct product 2 AdS2 × S.

For some supersymmetric theories, the values v1,v2,χA⋆,eI are generically completely fixed by the electric (qI) and magnetic (pI) charges of the black hole and do not depend continuously on the asymptotic value of the scalar fields in the asymptotic region – the scalar moduli. This is the attractor mechanism [141, 250, 140]. It was then realized that it still applies in the presence of certain higher-derivative corrections [199, 200, 198]. The attractor mechanism was also extended to non-supersymmetric extremal static black holes [139, 240Jump To The Next Citation Point, 150, 179]. As a consequence of this mechanism, the entropy of these extremal black hole does not depend continuously on any moduli of the theory.7 The index that captures the entropy can still have discrete jumps when crossing walls of marginal stability in the scalar moduli space [227Jump To The Next Citation Point, 118Jump To The Next Citation Point]. This allows one to account for their black-hole entropy by varying the moduli to a weakly-coupled description of the system without gravity, where states with fixed conserved charges can be counted. Therefore, the attractor mechanism led to an explanation [18, 111] of the success of previous string theory calculations of the entropy of certain nonsupersymmetric extremal black holes [181, 172, 110, 260, 131, 132Jump To The Next Citation Point].

As will turn out to be useful in the development of the Reissner–Nordström correspondence, let us discuss the near-horizon geometry (21View Equation) under the assumption that one gauge field A can be lifted as a Kaluza–Klein vector to a higher-dimensional spacetime, as discussed in Section 1.2. In the simple model (2View Equation), the change of gauge A → A + dΛ is implemented as the change of coordinates χ → χ + Λ. Using the definition of the electrostatic potential ext Φe (10View Equation) at extremality, it is straightforward to obtain that in the geometry (2View Equation) the horizon is generated by the vector field ext ξtot = ∂t + Φe ∂χ. The change of coordinates (17View Equation) combined with χ → χ + Λ with Λ defined in (19View Equation) then maps this vector to

λ- ξtot → r ∂t. (22 ) 0

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