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 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.
Highlights
The Kerr-Newman black hole has at most two horizons
We study the internal structure of anisotropic black holes with charged vector hairs
We show the nonlinear dynamics that occur near the would-be inner horizon due to the instability of the Cauchy horizon triggered by vector hairs
Summary
We introduce a charged vector field ρμ into the (d+2) dimensional Einstein-Maxwell theory with a negative cosmological constant Λ. The gauge symmetry and equation of motion (2.3) ensure that the phase factor of ρx is constant, so we can take ρx to be real everywhere. Since the resulting non-linear coupled equations do not have analytical solutions, one may try to solve the system numerically and check whether the inner horizon exists or not. This is complicated in numerics and is only able to check some particular cases. Note that the conserved charge depends on the anisotropic factor u of the metric. Before proceeding to the proof of no inner horizon, we find that the system allows the following scaling symmetry:. We can obtain another radially conserved quantity f e−χ/2 √
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