Geodesic incompleteness from retrodiction friendliness and non-global hyperbolicity

Determine whether retrodiction friendliness together with non-global hyperbolicity implies geodesic incompleteness, and thereby whether all non-globally hyperbolic retrodiction-friendly spacetimes require singularity resolution.

Background

The paper introduces retrodiction friendliness as a weakening of global hyperbolicity designed to distinguish failures of retrodictability associated with evaporating black holes from failures caused by asymptotic structure or other pathologies. Non-global hyperbolicity alone does not imply geodesic incompleteness, since anti-de Sitter spacetime is non-globally hyperbolic but non-singular.

The authors observe that the additional regularity conditions built into retrodiction friendliness exclude anti-de Sitter spacetime and suggest that combining retrodiction friendliness with non-global hyperbolicity might force geodesic incompleteness. Establishing this implication would clarify whether singularity resolution is required for every non-globally hyperbolic retrodiction-friendly spacetime.

References

An interesting open question is whether retrodiction friendliness and non-global hyperbolicity imply geodesic incompleteness. Non-global hyperbolicity alone is insufficient. Anti-de Sitter spacetime is non-globally hyperbolic and non-singular, so is a counterexample, but the niceness conditions (i) and (ii) of the definition of retrodiction friendliness mean this spacetime is not retrodiction friendly. It is plausible that the conjunction of non-global hyperbolicity with retrodiction friendliness is sufficient. If this is the case, then all non-globally hyperbolic retrodiction friendly spacetimes require singularity resolution.

— Is Black Hole Evaporation Prediction Friendly?  (2609.09927 - Ryder, 9 Sep 2026) in Footnote to Section 3.1, “Resuscitating the Paradox” (following the Kodama-Wald-Lesourd theorem)