Bouncing geodesics — null geodesics that enter a black hole interior, approach the singularity, and then “bounce” back to the boundary — have been proposed as probes of black hole singularity structure. The idea: the singularity's properties are encoded in how the bounced geodesic connects boundary points, which in turn shows up in singularities of thermal correlators in the dual field theory.
Grozdanov, Valach, and Vrbica establish the general framework via Hadamard theory: bulk and boundary retarded propagators diverge whenever two points are connected by a null geodesic. This connects the geometric structure (geodesics) to observable quantities (correlation functions) through a precise mathematical mechanism.
Then they exhibit a counterexample. A black hole with genuine curvature singularities that produces no bouncing geodesics. The singularity is real — curvature invariants diverge — but no null geodesic reaches it and returns. The diagnostic fails precisely where you'd want it to work: at a singular spacetime that cannot be probed by this method.
The failure is structural. Bouncing geodesics require specific geometric conditions near the singularity — conditions that curvature singularities don't always satisfy. A singularity can be physically present (tidal forces diverge, curvature blows up) without being geometrically accessible to the null geodesic probe.
The absence of the bounce doesn't mean the singularity isn't there. It means this particular probe is blind to it. Not every diagnostic works everywhere. The counterexample is the proof.