Relational Wigner-Smith Duration in Wheeler-DeWitt Scattering

Abstract

A closed Friedmann-Lemaitre-Robertson-Walker Wheeler-DeWitt model is formulated as an exact reflection problem. The derivative of its reflection phase defines a relational crossing duration whose first two clock moments follow from a covariant scalar-clock observable. An analytic expression is obtained for this duration, its classical recollapse limit, and the leading quantum correction. A finite spectral packet also gives an operational, equal-prior minimum error probability for distinguishing the two orientations of recollapse with a geometric reading. The construction is extended to a bounded finite quantum detector. Its multichannel reflection matrix yields probe transitions, spectral-probe correlations, and a matrix duration. A direct weak-coupling calculation verifies the predicted transition and duration scalings. Scalar-clock conditioning and oriented geometric sections are shown to be local representations of the same positive-frequency Dirac sector. These results define duration and transition observables entirely through correlations and scattering records of a stationary constrained state, without introducing a background clock.