Uniqueness of spherically symmetric pre-geometric black-hole solutions

Determine whether the exact pre-geometric black-hole solutions obtained in the Wilczek and MacDowell–Mansouri theories are unique under static and spherical symmetry, or whether additional classes of spherically symmetric pre-geometric configurations exist.

Background

The paper constructs exact vacuum solutions of the Wilczek and MacDowell–Mansouri pre-geometric theories whose spontaneously broken phase reproduces the (anti-)de Sitter–Schwarzschild geometry. These solutions are selected through the correspondence principle, which requires the pre-geometric configurations to reduce to known spherically symmetric metric solutions after spontaneous symmetry breaking.

The authors explicitly state that they do not establish uniqueness because of the mathematical complexity of the pre-geometric vacuum field equations. Consequently, it remains unresolved whether other solution classes satisfying the same symmetry conditions exist, and whether any such additional configurations have physically meaningful geometric interpretations in the metric phase. Resolving this issue is also identified as the remaining step in fully establishing the pre-geometric analogue of Birkhoff’s theorem.

References

Due to the mathematical complexity of the pre-geometric field equations in vacuum, we did not address the uniqueness of the exact solutions found for pre-geometric black holes. This means that it is possible that other classes of solutions satisfying the same symmetry conditions may exist. The solutions computed in this article are those highlighted by the correspondence principle, i.e. those that are relevant for recovering known metric spacetimes in the spontaneously broken phase. It is conceivable, at least from a mathematical point of view, that additional pre-geometric configurations with spherical symmetry could be predicted by the same pre-geometric theories under consideration -- though this possibility would then beg the question about the physical character of the corresponding spacetime geometries in the metric phase.

— Pre-geometric Black Holes with Spherical Symmetry  (2610.11357 - Meluccio et al., 8 Oct 2026) in Section Discussion and conclusions