Tidal Deformation Bounds and Perturbation Transfer in Bounded Curvature Spacetimes

TL;DR

This work analyzes spacetimes with globally bounded tidal fields and shows that a bound $\\lambda_{\\max}$ on the electric Riemann eigenvalues implies two robust dynamical consequences independent of the metric details: (i) a Jacobi-type deformation bound that caps geodesic deviations over bounded interiors with a characteristic time $\\tau_* = \\lambda_{\\max}^{-1/2}$, and (ii) a perturbation-transfer bound that defines a critical scale $k_* \\sim \\tau_*^{-1}$ separating adiabatic from non-adiabatic evolution, with Bogoliubov mixing exponentially suppressed for $k\\tau_* \\gg 1$. The results are expressed using standard curvature invariants, e.g., $L_* = K_{\\max}^{-1/4}$ and the benchmark $\\tau_*/L_* = 24^{1/4}$ in four dimensions, with corrections controlled by the Weyl-to-Kretschmann ratio $\\epsilon_C$, and are validated numerically in the extremal Hayward interior. The framework applies across theories enforcing curvature bounds, including action-level limits and density-responsive gravity, and has potential implications for regular black-hole interiors, bounce cosmologies, gravitational-wave phenomenology (echoes, Love numbers), and the interpretation of Planck-scale physics as a transition to a maximal tidal regime rather than discretization.

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 $λ_{\max}\leλ_{\rm bound}$ 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 $τ_*\equivλ_{\max}^{-1/2}$, and (ii)~the existence of a critical wavenumber $k_*\simτ_*^{-1}$ separating adiabatic from non-adiabatic perturbation transfer through high-curvature epochs, with Bogoliubov coefficients exponentially suppressed for $k\,τ_*\gg 1$. 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 $L_{\rm LI}(\varepsilon)\sim\sqrt{\varepsilon}\, λ_{\max}^{-1/2}$ and, for conformally flat cores in four dimensions, the benchmark ratio $τ_*/L_*=24^{1/4}$ 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 $ε_C$. The general framework is validated numerically in the extremal Hayward geometry.

Figures (4)

• Figure 1: Local-inertial domain size $L_{\rm LI}(\varepsilon)/\ell$ for a de Sitter core with $\lambda_{\max}=\ell^{-2}$. Dashed lines mark $\varepsilon=1$ and $\varepsilon=1/3$.
• Figure 2: Geodesic deviation through the extremal Hayward interior. Proper time in units of $\tau_*=\ell$. (a) Tidal eigenvalues; grey line marks $-\ell^{-2}$. (b) Radial separation (left axis, black): compression followed by re-expansion; angular separation (right axis, blue, log scale): monotonic growth to $|\xi^{\rm ang}|/\xi_0\approx 50$ at the cutoff.
• Figure 3: Adiabaticity analysis for a Pöschl--Teller curvature pulse with $V_0\tau_*^2=1$ ($\nu\approx 0.62$). (a) Maximum WKB adiabaticity parameter $\mathcal{A}_k$; the transition at $k\tau_*=\mathcal{O}(1)$ confirms the critical scale. (b) Exact Bogoliubov coefficient (\ref{['eq:bogoliubov_exact']}) showing exponential suppression for $k\,\tau_*\gg 1$.
• Figure 4: Radial profiles of the Weyl anisotropy parameter $\epsilon_C(y)$ (top) and the operational ratio $\tau_*/L_*(y)$ (bottom) for the Hayward metric, with $y\equiv r^3/(M\ell^2)$. In the top panel, $\epsilon_C=0$ at the de Sitter core ($y=0$) and at $y=4$ (where $f'(r)=0$), and $\epsilon_C\to 1$ in the Schwarzschild limit ($y\to\infty$) where $K=C^2$. The gray dash-dotted line marks the sign change of $\lambda_1$ at $y\approx 0.29$. The dashed horizontal line in the lower panel marks the maximally symmetric benchmark $24^{1/4}$, and the shaded band indicates ${\pm}10\%$. The blue dashed vertical line marks $y\approx 2.12$, the point of maximal $\tau_*/L_*$ deviation; the extremal horizon lies at $y=4$ (where $\epsilon_C=0$).