What’s inside a hairy black hole in massive gravity?

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.

Inside an asymptotically flat hairy black hole

“…In higher dimensions, there are examples of hairy black holes in Einstein-Gauss-Bonnet gravity with a minimally coupled charged scalar field with no self interactions [14]. 2 For other recent discussions of the interior of hairy black holes, see [16,17].…”

Inside anisotropic black hole with vector hair

“…While the existence of the final Kasner universe has been shown to be sensitive to the potential of the scalar field [20,21], other epochs seem to be general features closely associated with the instability of the inner horizon of charged black holes without scalar hair. See other studies of black holes for which the Cauchy horizon is removed by scalar hairs [23][24][25][26][27]. Thus far, the structure as well as dynamical epochs inside the event horizon was studied by introducing a scalar field to black holes with maximally symmetric horizon.…”

Trans-IR Flows to Black Hole Singularities

“…A natural extension of our work is to consider more types of flows, such as those sourced by combinations of interacting scalars, higher-rank fields, and axions [36][37][38][39][40]72]. One may also consider higher-curvature gravity [52], but note that we would expect corrections to the a T -function by analogy to the a-function of [9,10].…”

“…It is reasonable to ask how to generalize from free Kasner flows. One way is to construct Kasner flows sourced by more complicated matter sectors [36][37][38][39][40], via numerical methods, and plot a T -a concrete but ad hoc approach. Another is to assume a Belinskii-Khalatnikov-Lifshitz (BKL) singularity [41][42][43]-reviewed by [44]-as the trans-IR endpoint and analyze the near-singularity geometry.…”

“…One may ask about Kasner flows sourced by more complicated matter sectors, such as the self-interacting φ 4scalar theory of [36] or φ coupled to Maxwellian and/or axionic fields [37][38][39]. One could even consider dimensional reductions of supergravity theories [40] or modify the gravitational theory to include higher-curvature terms [52] (upon appropriate changes to a T (7) [9,10]).…”