Inter--disks inversion surfaces

We consider a counter–rotating torus orbiting a central Kerr black hole ($\mathbf{\text{BH}}$) with dimensionless spin $a$, and its accretion flow into the $\mathbf{\text{BH}}$, in an agglomerate of an outer counter–rotating torus and an inner co–rotating torus. This work focus is the analysis of the inter–disks inversion surfaces. Inversion surfaces are spacetime surfaces, defined by the condition $u^{\phi }=0$ on the flow torodial velocity, located out of the $\mathbf{\text{BH}}$ ergoregion, and totally embedding the $\mathbf{\text{BH}}$. They emerge as a necessary condition, related to the spacetime frame–dragging, for the counter–rotating flows into the central Kerr $\mathbf{\text{BH}}$. In our analysis we study the inversion surfaces of the Kerr spacetimes for the counter–rotating flow from the outer torus, impacting on the inner co–rotating disk. Being totally or partially embedded in (internal to) the inversion surfaces, the inner co–rotating torus (or jet) could be totally or in part ``shielded", respectively, from the impact with flow with $a{u}^{\phi }<0$. We prove that, in general, in the spacetimes with $a<0.551$ the co–rotating toroids are always external to the accretion flows inversion surfaces. For $0.551<a<0.886$, co–rotating toroids could be partially internal (with the disk inner region, including the inner edge) in the flow inversion surface. For $\mathbf{\text{BHs}}$ with $a>0.886$, a co–rotating torus could be entirely embedded in the inversion surface and, for larger spins, it is internal to the inversion surfaces. Tori orbiting in the $\mathbf{\text{BH}}$ outer ergoregion are a particular case. Further constraints on the $\mathbf{\text{BHs}}$ spins are discussed in the article.