Scattering amplitudes in flat spacetime obey crossing symmetry: the same mathematical object describes a particle going in or out by swapping its momentum's sign. On curved backgrounds, the standard S-matrix doesn't exist — the vacuum is ambiguous, particle number is not conserved, and the asymptotic structure that defines scattering is modified or absent.
Aoude et al. (arXiv:2603.17903) resolve this by recasting Bogoliubov coefficients as generalized amplitudes. Bogoliubov coefficients describe how one observer's vacuum looks from another observer's perspective — they quantify particle creation by the curved background. The authors show that these coefficients satisfy crossing relations, analyticity properties, and causality constraints that parallel the flat-space amplitude program.
When the background is a coherent state — a classical field configuration — the generalized framework reduces exactly to standard flat-space scattering amplitudes. The flat-space S-matrix is a special case, not a separate structure. The curved-space generalization extends the same principles to backgrounds where the vacuum itself is dynamical.
The applications include dynamical black holes and gravitational radiation. A black hole that forms and evaporates has no static asymptotic region — the spacetime is time-dependent everywhere — and the standard scattering formalism fails. But Bogoliubov coefficients exist whenever two sets of modes can be compared, and the crossing/analyticity/causality relations constrain them regardless of the background's complexity.
The conceptual advance is that amplitudes are not about particles colliding in empty space. They are about mode transformations constrained by symmetry. Flat space privileges one set of modes; curved space doesn't. The physics is in the relations between amplitudes, not in the amplitudes themselves — and those relations survive the transition to curved backgrounds.