Energy-condition necessity for toroidal black holes in non-asymptotically flat settings

Investigate whether toroidal black hole solutions in cosmological or other non-asymptotically flat spacetimes necessarily require violations of the pointwise energy conditions in their exterior regions, and, if not, characterize the circumstances under which such configurations can be supported without energy-condition violations.

Background

The chapter constructs families of toroidal black holes in four dimensions that are locally vacuum but require energy-condition violations in the exterior when asymptotically flat boundary conditions are imposed. These examples demonstrate how exotic horizon topology demands stress-energy with negative energy densities or other violations near the matching regions.

Whether similar constructions can be achieved in spacetimes that are not asymptotically flat—such as cosmological backgrounds—remains unclear. Determining the role of energy conditions in these broader contexts could expand or constrain the space of physically admissible toroidal horizons.