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We define a smooth extension of the Schwarzschild exterior geometry,
with the special property that the curvature two-form field is finite
everywhere. This spacetime is a
solution to the first order field equations in vacuum by construction.
Unlike the Kruskal-Szekeres construction based on invertible metrics
in the Einsteinian theory, it exhibits a vanishing metric determinant
phase over an extended region. Such solutions could be
particularly relevant in the study of singularities in general
relativity as well as in the
context of information loss problem.
We also demonstrate that it is not possible to
define a similar extension for the case of a negative mass
Schwarzschild solution. This is consistent with the general
expectation
that degenerate metric spacetimes constructed within the
Hilbert-Palatini Lagrangian
framework should satisfy the energy conditions. |