On the inverse hoop conjecture of Hod

Recently, Hod has shown that Thorne’s hoop conjecture ($$\frac{C(R)}{4\pi M(r\le R)} \le 1\Rightarrow $$ C ( R ) 4 π M ( r ≤ R ) ≤ 1 ⇒ horizon) is violated by stationary black holes and so he proposed a new inverse hoop conjecture characterizing such black holes (that is, horizon $$\Rightarrow \mathcal {H =} \frac{\pi \mathcal {A} }{\mathcal {C}_{{eq} }^{2}} \le 1$$ ⇒ H = π A C eq 2 ≤ 1 ). In this paper, it is exemplified that stationary hairy black holes, endowed with Lorentz symmetry violating Bumblebee vector field related to quantum gravity and dilaton field of string theory, also respect the inverse conjecture. It is shown that stationary hairy singularity, recently derived by Bogush and Galt’sov, does not respect the conjecture thereby protecting it. However, curiously, there are two horizonless stationary wormholes that can also respect the conjecture. Thus one may also state that throat $$\Rightarrow \mathcal {H \le }1$$ ⇒ H ≤ 1 , suggesting that the inverse conjecture may be a necessary but not sufficient proposition.