Clarifying Kasner dynamics inside anisotropic black hole with vector hair

Abstract: We investigate the internal dynamics of hairy black holes in the Einstein-Maxwell-vector theory. The development of the charged vector hair necessarily removes the inner Cauchy horizon and results in an anisotropic black hole ending at a spacelike singularity. The far interior evolution is characterized by the chaotic alternations among different Kasner epochs. We show that those late interior time dynamics of bounces and epochs can be captured analytically. In particular, we obtain three distinct types of transformation laws for the alternation of Kasner epochs analytically, including the Kasner inversion, Kasner transition, and Kasner reflection. Our analytical approach is corroborated by numerical solutions to the full equations of motion. Moreover, we provide a clear explanation of the alternation of Kasner epochs found numerically in the literature.