Abstract
The gravitational dual to the grand canonical ensemble of a large N holographic theory is a charged black hole. These spacetimes — for example Reissner- Nordström-AdS — 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 Einstein-Rosen bridge at the would-be 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.
Highlights
Background and equationsIn the grand canonical ensemble the dual field theory is held at a chemical potential μ for some global U(1) symmetry
We show that a deformation of the CFT by a neutral scalar operator generically leads to a black hole with no inner horizon
We note that in the discrete cases with m2 > 0 where an inner horizon survives, as discussed in section 3.2, the singularity beyond the inner horizon will be that of the Reissner-Nordstrom black hole, with the scalar field becoming unimportant towards the singularity (φ ∼ 1/z) and the equations dominated by the flux terms
Summary
In the grand canonical ensemble the dual field theory is held at a chemical potential μ for some global U(1) symmetry. In the bulk we must correspondingly introduce a Maxwell field A such that At → μ at the boundary. To deform the boundary theory by a scalar operator O we must introduce a dual scalar field φ in the bulk. The radial functions should obey the following leading asymptotic behavior at the AdS boundary as z → 0 f → 1 , χ → 0 , Φ → μ , φ → φ(0)z3−∆. This behavior fixes the normalization of time on the boundary as well as the chemical potential μ and source φ(0) for the dual operator O.
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