Inside Anisotropic Black Hole with Vector Hair

Abstract

We study the internal structure of anisotropic black holes with charged vector hairs. Taking advantage of the scaling symmetries of the system, some radially conserved charges are found via the extension of the Noether theorem. Then, a general proof of no inner horizon of these black holes is presented and the geometry ends at a spacelike singularity. Before reaching the singularity, we find several intermediate regimes both analytically and numerically. In addition to the Einstein-Rosen bridge contracting towards the singularity, the instability triggered by the vector hair results in the the oscillations of vector condensate and the anisotropy of spatial geometry. Moreover, the latter oscillates at twice the frequency of the condensate. Then, the geometry enters into Kasner epochs with spatial anisotropy. Due to the effects from vector condensate and U(1) gauge potential, there is generically a never-ending alternation of Kasner epochs towards the singularity. The character of evolution on approaching the singularity is found to be described by the Kasner epoch alternation with flipping of powers of the Belinskii-Khalatnikov-Lifshitz type.

The final Kasner regime inside black holes with scalar or vector hair

“…The number of independent relevant walls is d (d − 1 symmetry walls and 1 electric wall), which is insufficient to bound a finitevolume region in hyperbolic space. Hence, there are "escape directions" to infinity forming an open set, in which the final Kasner regime can settle 6 .…”

“…In the absence of the charged matter fields (φ, ρ µ ), the solution is the Reissner-Nördstrom AdS black hole, with an inner Cauchy horizon and a timelike singularity. However, as shown in [1][2][3][4][5][6], the matter fields generically destroy the inner horizon and the black hole develops matter hair. Furthermore, the black hole singularity is spacelike.…”

“…The study of the internal structure of charged black holes with scalar or vector hair has attracted recently a considerable attention [1][2][3][4][5][6]. These solutions are relevant to the holographic realization of superconductors [7][8][9][10][11] and are described by the Einstein action with negative cosmological constant coupled to the sum of the Maxwell action and the action for massive charged scalar or vector fields.…”

“…As one falls into the black hole, a wealth of interesting phenomena with a remarkable dual interpretation occur (see again [1][2][3][4][5][6]). The purpose of this note is to describe the final, close-to-the-spacelike-singularity, stage of the dynamical evolution under generic initial condition (e.g., no assumed isometry or simplification leading to "accidental" behaviors).…”

Trans-IR Flows to Black Hole Singularities

“…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]). Either scalar hair [53,54] or vector hair [55] generally prevents the formation of inner horizons in black holes, so trans-IR flows even in these theories will still end at a spacelike singularity, just as in the free scalar theory. Assuming a Kasner singularity, we may conjecture that a T still vanishes at the trans-IR endpoint.…”