Abstract
In the context of massive gravity theories, we study holographic flows driven by a relevant scalar operator and interpolating between a UV 3-dimensional CFT and a trans-IR Kasner universe. For a large class of scalar potentials, the Cauchy horizon never forms in presence of a non-trivial scalar hair, although, in absence of it, the black hole solution has an inner horizon due to the finite graviton mass. We show that the instability of the Cauchy horizon triggered by the scalar field is associated to a rapid collapse of the Einstein-Rosen bridge. The corresponding flows run smoothly through the event horizon and at late times end in a spacelike singularity at which the asymptotic geometry takes a general Kasner form dominated by the scalar hair kinetic term. Interestingly, we discover deviations from the simple Kasner universe whenever the potential terms become larger than the kinetic one. Finally, we study the effects of the scalar deformation and the graviton mass on the Kasner singularity exponents and show the relationship between the Kasner exponents and the entanglement and butterfly velocities probing the black hole dynamics. Differently from the holographic superconductor case, we can prove explicitly that Josephson oscillations in the interior of the BH are absent.
Highlights
More general Kasner form.1 This provides a first step beyond the non-generic and classically unstable black hole interiors which have been the main focus of previous holographic literature
In absence of scalar hair, the black hole presents an inner Cauchy due to the finite graviton mass, the deformation induced by a neutral scalar operator generically removes this Cauchy horizon such that the deformed black hole with non-trivial scalar hair approaches a spacelike singularity at late interior time
As we have proved in the last section, the black hole with non-trivial scalar hair has no inner horizon, and the black hole interior ends at a spacelike singularity at r → ∞
Summary
Where G is the Newton constant, R the Ricci scalar, Λ the cosmological constant and φ a neutral bulk scalar field whose potential is denoted as V. We have chosen the cosmological constant Λ = −3/L2 with the AdS radius L set to one These coupled differential equations do not allow analytical solutions, one has to solve them numerically. Depending on the choice of the potential K(X), the model in eq (2.1) corresponds to different boundary field theories. Type II: K(X) = a1 X + a2X with a1 and a2 two constants This form of potential corresponds to the non-linear dRGT massive gravity model [24] as proven in [26]. We stress that K (X) > 0 to avoid ghost instability [29] Both Type I and Type II corresponds to the explicit breaking of translations in the dual field theory and their physics is that described by momentum dissipation.
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