Schwarzschild binary supported by an Appell ring

Abstract: We continue to study black holes subjected to strong sources of gravity, again paying special attention to the behaviour of geometry in the black-hole interior. After examining, in previous two papers, the deformation arising in the Majumdar-Papapetrou binary of extremally charged black holes and that of a Schwarzschild black hole due to a surrounding (Bach-Weyl) ring, we consider here the system of two Schwarzschild-type black holes held apart by the Appell ring. After verifying that such a configuration can be in a strut-free equilibrium along certain lines in a parameter space, we compute several basic geometric characteristics of the equilibrium configurations. Then, like in previous papers, we calculate and visualize simple invariants determined by the metric (lapse or, equivalently, potential), by its first derivatives (gravitational acceleration) and by its second derivatives (Kretschmann scalar). Extension into the black-hole interior is achieved along particular null geodesics starting tangentially to the horizon. In contrast to the case involving the Bach-Weyl ring, here each single black hole is placed asymmetrically with respect to the equatorial plane (given by the Appell ring) and the interior geometry is really deformed in a non-symmetrical way. Inside the black holes, we again found regions of negative Kretschmann scalar in some cases.