Tidal Deformation Bounds and Perturbation Transfer in Bounded Curvature Spacetimes

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 ${\lambda }_{\max }\le {\lambda }_{\mathrm{bound}}$ 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 ${\tau }_{\ast }\equiv {\lambda }_{\max }^{-1/2}$, and (ii)~the existence of a critical wavenumber $k_{\ast }\sim {\tau }_{\ast }^{-1}$ separating adiabatic from non-adiabatic perturbation transfer through high-curvature epochs, with Bogoliubov coefficients exponentially suppressed for $k{\tau }_{\ast }\gg 1$. 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 $L_{\mathrm{LI}}(\epsilon )\sim \sqrt{\epsilon }{\lambda }_{\max }^{-1/2}$ and, for conformally flat cores in four dimensions, the benchmark ratio ${\tau }_{\ast }/L_{\ast }=24^{1/4}$ with $L_{\ast }\equiv K_{\max }^{-1/4}$. 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.

Revised after peer review; submitted to GRG, 17 pages, 4 figures

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• Relativistic theory of tidal Love numbers
• Notes on non-singular models of black holes
• Spherically symmetric black holes in the limiting curvature theory of gravity
• A density-responsive scalar-field framework for singularity regularization and dynamical dark energy