Due to the conformal factor problem, the definition of the Euclidean gravitational path integral requires a nontrivial choice of contour. The present work examines a generalization of a recently proposed ruleofthumb [1] for selecting this contour at quadratic order about a saddle. The original proposal depended on the choice of an indefinitesignature metric on the space of perturbations, which was taken to be a DeWitt metric with parameter
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A<sc>bstract</sc> α = – 1. This choice was made to match previous results, but was otherwise admittedlyad hoc . To begin to investigate the physics associated with the choice of such a metric, we now explore contours defined using analogous prescriptions forα ≠ – 1. We study such contours for Euclidean gravity linearized about AdSSchwarzschild black holes in reflecting cavities with thermal (canonical ensemble) boundary conditions, and we compare pathintegral stability of the associated saddles with thermodynamic stability of the classical spacetimes. While the contour generally depends on the choice of DeWitt parameterα , the precise agreement between these two notions of stability found atα = – 1 continues to hold over the finite interval (– 2, – 2/d ), whered is the dimension of the bulk spacetime. This agreement manifestly fails forα > – 2/d when the DeWitt metric becomes positive definite. However, we also find dramatic failures forα < – 2 that correlate with breakdowns of the de Donderlike gauge condition defined byα , and at which the relevant fluctuation operator fails to be diagonalizable. This provides criteria that may be useful in predicting metrics on the space of perturbations that give physicallyuseful contours in more general settings. Along the way, we also identify an interesting error in [1], though we show this error to be harmless.Free, publiclyaccessible full text available May 1, 2025 
A<sc>bstract</sc> It was recently shown that (near)extremal Kerr black holes are sensitive probes of small higherderivative corrections to general relativity. In particular, these corrections produce diverging tidal forces on the horizon in the extremal limit. We show that adding a black hole charge makes this effect qualitatively stronger. Higherderivative corrections to the KerrNewman solution produce tidal forces that scale inversely in the black hole temperature. We find that, unlike the Kerr case, for realistic values of the black hole charge large tidal forces can arise before quantum corrections due to the Schwarzian mode become important, so that the nearhorizon behavior of the black hole is dictated by higherderivative terms in the effective theory.
Free, publiclyaccessible full text available May 1, 2025 
Free, publiclyaccessible full text available August 1, 2024

A bstract In holography, the IR behavior of a quantum system at nonzero density is described by the near horizon geometry of an extremal charged black hole. It is commonly believed that for systems on S 3 , this near horizon geometry is AdS 2 × S 3 . We show that this is not the case: generic static, nonspherical perturbations of AdS 2 × S 3 blow up at the horizon, showing that it is not a stable IR fixed point. We then construct a new near horizon geometry which is invariant under only SO(3) (and not SO(4)) symmetry and show that it is stable to SO(3)preserving perturbations (but not in general). We also show that an open set of nonextremal, SO(3)invariant charged black holes develop this new near horizon geometry in the limit T → 0. Our new IR geometry still has AdS 2 symmetry, but it is warped over a deformed sphere. We also construct many other near horizon geometries, including some with no rotational symmetries, but expect them all to be unstable IR fixed points.more » « less

A bstract We investigate the geometry near the horizon of a generic, fourdimensional extremal black hole. When the cosmological constant is negative, we show that (in almost all cases) tidal forces diverge as one crosses the horizon, and this singularity is stronger for larger black holes. In particular, this applies to generic nonspherical black holes, such as those satisfying inhomogeneous boundary conditions. Nevertheless, all scalar curvature invariants remain finite. Moreover, we show that nonextremal black holes have tidal forces that diverge in the extremal limit. Holographically, this singularity is reflected in anomalous scaling of the specific heat with temperature. Similar (albeit weaker) effects are present when the cosmological constant is positive, but not when it vanishes.more » « less

A bstract We revisit the stability of black hole saddles for the Euclidean path integral describing the canonical partition function Z ( β ) for gravity inside a spherical reflecting cavity. The boundary condition at the cavity wall couples the transversetraceless (TT) and puretrace modes that are traditionally used to describe fluctuations about Euclidean Schwarzschild black holes in infinitevolume asymptotically flat and asymototically AdS spacetimes. This coupling obstructs the familiar GibbonsHawkingPerry treatment of the conformal factor problem, as Wick rotation of the puretrace modes would require that the TT modes be rotated as well. The coupling also leads to complex eigenvalues for the Lichnerowicz operator. We nevertheless find that the Lichnerowicz operator can be diagonalized in the space of coupled modes. This observation allows the eigenmodes to define a natural generalization of the puretrace Wickrotation recipe used in infinite volume, with the result that a mode with eigenvalue λ is stable when Re λ > 0. In any cavity, and with any cosmological constant Λ ≤ 0, we show this recipe to reproduce the expectation from black hole thermodynamics that large Euclidean black holes define stable saddles while the saddles defined by small Euclidean black holes are unstable.more » « less

A bstract We study a family of fourdimensional, asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Surprisingly, the maximum charge for given mass is a nonsingular hairy black hole with nonzero Hawking temperature. The implications for Hawking evaporation are discussed.more » « less

A bstract We study the interior of a recently constructed family of asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Inside the horizon, these black holes resemble the interior of a holographic superconductor. There are analogs of the Josephson oscillations of the scalar field, and the final Kasner singularity depends very sensitively on the black hole parameters near the onset of the instability. In an appendix, we give a general argument that Cauchy horizons cannot exist in a large class of stationary black holes with scalar hair.more » « less

null (Ed.)A bstract The gravitational dual to the grand canonical ensemble of a large N holographic theory is a charged black hole. These spacetimes — for example Reissner NordströmAdS — can have Cauchy horizons that render the classical gravitational dynamics of the black hole interior incomplete. We show that a (spatially uniform) deformation of the CFT by a neutral scalar operator generically leads to a black hole with no inner horizon. There is instead a spacelike Kasner singularity in the interior. For relevant deformations, Cauchy horizons never form. For certain irrelevant deformations, Cauchy horizons can exist at one specific temperature. We show that the scalar field triggers a rapid collapse of the EinsteinRosen bridge at the wouldbe Cauchy horizon. Finally, we make some observations on the interior of charged dilatonic black holes where the Kasner exponent at the singularity exhibits an attractor mechanism in the low temperature limit.more » « less