arXiv 2602.162852026-02gr-qc

潮汐变形限制与有限曲率时空中的扰动转移

Tidal Deformation Bounds and Perturbation Transfer in Bounded Curvature Spacetimes

Institute of Space Systems, German Aerospace Center (DLR)（德国航空航天中心空间系统研究所）

Martin Drobczyk

备注：: Revised after peer review; submitted to GRG, 17 pages, 4 figures
DOI：: 10.1007/s10714-026-03546-x

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AI总结：

本文研究了有限曲率时空中的潮汐变形限制及扰动转移，推导了局部惯性近似和潮汐动力学的严格上界，并给出了临界波数的分离条件，通过数值验证了框架的鲁棒性。

摘要：

我们为具有全局有限潮汐场的时空推导了两个与模型无关的结果。这些是局部惯性近似的操作解析尺度和潮汐动力学；不暗示时空离散性。给定一个关于电Riemann特征值E_{ij}≡R_{\hat{0}i\hat{0}j}的不变界λ_{max}≤λ_{bound}沿自由下落世界线，我们证明(i)通过任何有限曲率内部的累积测地偏差的严格上界，受τ_*≡λ_{max}^{-1/2}控制，以及(ii)存在一个临界波数k_*∼τ_*^{-1}，将其分为绝热和非绝热扰动转移通过高曲率时期，对于kτ_*>>1，Bogoliubov系数呈指数抑制。这两个结果仅依赖于潮汐界（以及，对于模式转移，对曲率驱动的有效势的轻微时间尺度假设），并且对度量细节不敏感。为准备，我们收集了有限曲率的标准操作后果，包括依赖于精度的局部惯性域L_{LI}(ε)∼√ελ_{max}^{-1/2}，以及在四维共形平坦核心中的基准比τ*/L*=24^{1/4}，其中L*≡K_{max}^{-1/4}。我们通过Weyl到Kretschmann比ε_C量化了该系数在偏离最大对称性时的鲁棒性。一般框架在极端Hayward几何中通过数值验证。

英文摘要：

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.

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