On the Kalb-Ramond modified Lorentz violating hairy black holes and Thorne's hoop conjecture

Recently, a class of static spherically symmetric power law corrected Lorentz violating (LV) Schwarzschild black holes in the Kalb-Ramond model have been derived and studied in the specific range of LV parameters ($0<\lambda \leq 2,\Upsilon \geq 0$) that correspond to energy condition preserving ($\rho >0$) source. On the other hand, there exist well known black holes that do not preserve the energy conditions. In this paper, we shall therefore relax energy conditions and numerically explore the horizon patterns of the enlarged class of LSMA black holes. Four generic types of LV corrected black holes emerge, which interestingly include the analogue of the \textit{braneworld} black hole ($\rho <0$) lending to $\Upsilon$ a new interpretation of "tidal charge" known as an imprint from the $5d$ bulk in the Randall-Sundrum scenario. We shall then show that Thorne's hoop conjecture, $\mathcal{H} \leq 1$, where $\mathcal{H}$ is the Hod function, consistently holds for three types and their generalizations. However, intriguingly, it turns out that, for the remaining type (viz., Schwarzschild-de Sitter and its generalizations), the hoop conjecture does \textit{not} hold. It is also shown that braneworld tidal charge black holes increases the LV correction to planetary perihelion advance in contrast to the decrease due to ordinary black holes thereby providing a qualitative distinction between them.