Inter-disks inversion surfaces

Abstract We consider a counter-rotating torus orbiting a central Kerr black hole ( BH ) with dimensionless spin a , and its accretion flow into the 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 $$u^{\phi }=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> on the flow torodial velocity, located out of the BH ergoregion, and totally embedding the BH . They emerge as a necessary condition, related to the spacetime frame-dragging, for the counter-rotating flows into the central Kerr 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 $$a u^{\phi }&lt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msup> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We prove that, in general, in the spacetimes with $$a&lt;0.551$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.551</mml:mn> </mml:mrow> </mml:math> the co-rotating toroids are always external to the accretion flows inversion surfaces. For $$0.551&lt;a&lt;0.886$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.551</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>a</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.886</mml:mn> </mml:mrow> </mml:math> , co-rotating toroids could be partially internal (with the disk inner region, including the inner edge) in the flow inversion surface. For BHs with $$a&gt;0.886$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0.886</mml:mn> </mml:mrow> </mml:math> , 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 BH outer ergoregion are a particular case. Further constraints on the BHs spins are discussed in the article.