A black hole is not just a dark spot in the sky. It is a complex object whose properties depend on the geometry of spacetime around it. In five dimensions, such objects can take on whimsical shapes: not only spheres, but also rings, and even 'lenses'—as if a drop of ink in water could freeze not just as a sphere, but as a donut or a biconvex disk.
The authors of the work have figured out how to quickly compute the 'action'—a quantity similar to the total energy of the system over its entire lifetime — for such multidimensional black holes. They represented the complex geometry of the solution as a mosaic of simple triangular funnels (called simplicial cones). In each funnel, the contribution to the action is collected only at its tip — where the singularity is located. This is similar to how you could determine the weight of an entire iceberg by measuring the pressure only at its peak.
This approach allowed not only to recheck already known formulas, but also to predict new ones. For example, it provides a way to compute the Wald entropy, a measure of the disorder hidden behind the horizon, even for extremal black holes, whose temperature is absolute zero. These results apply both to worlds with a negative cosmological constant (as in the anti-de Sitter model) and to the familiar flat infinity.
as written by the authors arXiv:2606.16986
We consider five-dimensional supergravity coupled to vector multiplets, gauged or ungauged, and propose an efficient method to evaluate the on-shell action of supersymmetric black saddle solutions with toric $${\rm U}(1)^3$$ symmetry and general topology, explicitly known or just assumed to exist, including higher-derivative corrections. This is equivalent to equivariant integration of the anomaly polynomial six-form over simplicial cones obtained by decomposing the toric diagram of the solution. The on-shell action is then expressed as a sum over contributions localized at the tip of each cone. We obtain a simple derivation of recently calculated expressions as well as new predictions, both for the on-shell action of the black saddles and the Wald entropy of the related extremal solutions. These may be asymptotically AdS$$_5$$ or asymptotically flat. As examples, we discuss black holes, black rings and black lenses, including a black hole in the background of a topological soliton.
In five dimensions, a black hole can take the shape of a ring — a 'black ring'—and its event horizon is topologically similar to a donut, not a sphere.
In Carl Sagan's novel 'Contact', a machine built from alien blueprints warps spacetime using extra dimensions — much in the same way that five-dimensional gravity gives rise to exotic black hole shapes in this work.