Beyond the Homogeneous Approximation: Exact Linear Stability, a Structural Second-Order Source, and a Quantum-Arrival-Time Obstruction at the Robin-Boundary Signature-Change Surface

Abstract

In two companion papers [1, 2] we computed the ensemble statistics of the shift in theEuclidean–Lorentzian signature-change surface of Robin-boundary quantum cosmology induced by a light stochastic spectator field, first at the homogeneous (zero-mode) level andthen, using the separate-universe approximation, at leading order in spatial gradients. PaperII explicitly identified the promotion of that treatment to genuine, non-perturbative spatialinhomogeneity as an open problem requiring “a full cosmological perturbation theory aroundthe join, including an inhomogeneous generalization of the Israel junction condition.” Thepresent paper carries out that extension. We prove three results and precisely characterizea fourth, previously unidentified, obstruction. (i) Using the non-gradient-expanded ADMconstraints together with the momentum constraint of standard cosmological perturbationtheory, we show that the linear-order corrugation of the K = 0 signature-change surface vanishes identically, for any spectator-field configuration consistent with a reflection-symmetricbackground history, as a direct consequence of the vanishing Hubble rate at the transition; we show explicitly how this conclusion is modified if that symmetry is not realized.(ii) We show that the separate-universe/gradient-expansion parameter used in Paper II,ϵ = (k/aH)2, diverges as the transition is approached, independently of gauge, so that theleading-order treatment of Paper II cannot be analytically continued through K = 0; wefurther exhibit the residual gauge ambiguity of the extrinsic curvature itself and show thatthe gauge transformation connecting standard slicings is likewise singular at this point. (iii)Working with the Hamiltonian and momentum constraints without gradient expansion, wederive the leading surviving effect in closed form: it is second order in the spectator-fieldperturbation, sourced by a self-coupling of the field’s own linear velocity and gradient viaan explicit Poisson equation, structurally paralleling the quadratic (∝ ϕ2) dependence already found for the homogeneous shift in Paper I. (iv) We show that assigning a numericalvalue to this closed-form result requires specifying the quantum state of the perturbationat the dynamically defined, internal locus K = 0– as opposed to at either of the two pathintegral boundary surfaces (Robin, t = 0; Dirichlet, t = 1) for which the existing Lorentzianquantum-cosmology toolkit is built– and we examine the candidate prescriptions availablein the literature– the Euclidean-regularity (Hartle–Hawking/Halliwell–Hawking) condition,the Dirichlet-type vacuum used for perturbations on Robin/complex-saddle backgrounds byMondal and Chakraborty, and relational/conditional-probability (Page–Wootters/Rovelli)constructions– showing that none, as currently formulated, resolves this without additionalinput; we are not aware of any other construction in the literature that does. We showthis difficulty is formally the cosmological analogue of the quantum time-of-arrival problem,and identify the specific obstacle– a non-ideal, matter-backreacted clock– that a resolutionwould need to overcome. We conclude that this specific extension of the Robin-boundary research program requires, as a distinct and precisely delimited prerequisite, the constructionof a genuine time-of-arrival observable for Lorentzian quantum cosmology.