Tidal Deformation Bounds and Perturbation Transfer in Bounded Curvature Spacetimes

TL;DR

This paper establishes model-independent bounds on tidal deformation and perturbation transfer in spacetimes with bounded curvature, providing operational scales and criteria for adiabaticity and non-adiabaticity in gravitational contexts.

Contribution

It introduces rigorous bounds on geodesic deviation and mode transfer in bounded curvature spacetimes, independent of specific metric details.

Findings

Upper bound on geodesic deviation controlled by curvature bound and timescale.

Existence of a critical wavenumber separating adiabatic and non-adiabatic regimes.

Validation of the framework in the extremal Hayward geometry.

Abstract

We derive two model-independent results for spacetimes with globally bounded tidal fields. These are operational resolution scales of the local-inertial approximation and tidal dynamics; no spacetime discreteness is implied. Given an invariant bound $\lambda_{\max}\le\lambda_{\rm bound}$ on the electric Riemann eigenvalues $E_{ij}\equiv R_{\hat{0}i\hat{0}j}$ along freely falling worldlines, we prove (i)~a rigorous upper bound on accumulated geodesic deviation through any bounded curvature interior, controlled by $\tau_*\equiv\lambda_{\max}^{-1/2}$, and (ii)~the existence of a critical wavenumber $k_*\sim\tau_*^{-1}$ separating adiabatic from non-adiabatic perturbation transfer through high-curvature epochs, with Bogoliubov coefficients exponentially suppressed for $k\,\tau_*\gg 1$. Both results depend only on the tidal bound (and, for mode transfer, on a mild timescale assumption for the curvature-driven effective potential) and are otherwise insensitive to metric details. For preparation, we collect the standard operational consequences of bounded curvature, including the accuracy-dependent local-inertial domain $L_{\rm LI}(\varepsilon)\sim\sqrt{\varepsilon}\, \lambda_{\max}^{-1/2}$ and, for conformally flat cores in four dimensions, the benchmark ratio $\tau_*/L_*=24^{1/4}$ with $L_*\equiv K_{\max}^{-1/4}$. We quantify the robustness of this coefficient under departures from maximal symmetry via the Weyl-to-Kretschmann ratio $\epsilon_C$. The general framework is validated numerically in the extremal Hayward geometry.