On geometry of deformed black holes: I. Majumdar-Papapetrou binary

Although black holes are eminent manifestations of very strong gravity, the geometry of space-time around and even inside them can be significantly affected by additional bodies present in their surroundings. We study such an influence within static and axially symmetric (electro-)vacuum space-times described by exact solutions of Einstein's equations, considering astrophysically motivated configurations (such as black holes surrounded by rings) as well as those of pure academic interest (such as specifically "tuned" systems of multiple black holes). The geometry is represented by the simplest invariants determined by the metric (the lapse function) and its gradient (gravitational acceleration), with special emphasis given to curvature (the Kretschmann and Ricci-square scalars). These quantities are analyzed and their level surfaces plotted both above and below the black-hole horizons, in particular near the central singularities. Estimating that the black hole could be most strongly affected by the other black hole, we focus, in this first paper, on the Majumdar--Papapetrou solution for a binary black hole and compare the deformation caused by "the other" hole (and the electrostatic field) with that induced by rotational dragging in the well-known Kerr and Kerr--Newman solutions.