Construct exact horizon-to-Kasner interior solutions beyond asymptotically flat black holes

Construct exact cosmological interior solutions with a horizon that are not smoothly integrated to asymptotic Minkowski infinity, in order to realize nontrivial independent Kasner scalar-charge variations in the interior first law.

Background

The paper formulates an interior first law involving the Kasner potential, horizon entropy, electric charge, and normalized Kasner scalar charge. For ordinary asymptotically flat black holes, the no-scalar-hair theorem makes the Kasner scalar charge dependent on the other parameters, so its variation does not contribute nontrivially to the first law.

The authors therefore consider more general black interiors whose horizons need not connect smoothly to asymptotic Minkowski spacetime. They construct numerical horizon-to-Kasner solutions with independently variable scalar hair, but explicitly state that exact solutions of this type are not known. Finding such analytic examples would provide an exact realization and calibration of the nontrivial scalar-charge sector of the proposed first law.