爱因斯坦-高斯-邦纳标量坍缩中的传播与视界形成 I. 特征条件、界限与形成
Propagation and horizon formation in Einstein--Gauss--Bonnet scalar collapse. I. Characteristic conditions, bounds and formation
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中文总结 AI 辅助
研究五维和六维爱因斯坦-高斯-邦纳引力中标量坍缩的特征条件与视界形成，推导出边际球面半径和质量界限，并揭示非球对称情形下经典存在性与唯一性的局限。
中文摘要 AI 辅助
我们研究了具有正耦合的五维和六维爱因斯坦-高斯-邦纳引力中的标量坍缩，将其视为精确的经典理论。在球对称标量背景上，张量、矢量和标量引力扇区的特征条件归结为两个局部不变量的不等式：$K=1+\tilde\alpha F/R^2$，它比较了封闭质量与面积半径 $R$，以及一个标量梯度不变量 $j$。在五维中，这些条件蕴含了GR分支上规则边际球面的面积半径 $R_h$ 满足 $R_h^2>\tilde\alpha$，以及对非负质量有尖锐界限 $\tilde\alpha^2\mathcal R_{abcd}\mathcal R^{abcd}<28$。对于具有规则中心的初始未俘获五维球对称数据，仅对 $K$ 的界限就能确保在紧致依赖域上光滑延拓，因此任何首次特征损失都发生在光滑背景上，且在损失前形成的边际球面必然导致后续损失。完整光滑数据在每个允许半径处形成边际球面，其质量任意接近其尖锐下界；在六维中，半径和质量都可以任意小。在没有球对称性的情况下，具有弱曲率的完整初始数据在足够大的尺度上包含最外层的严格稳定边际曲面，其具有满足谱条件的指定 $S^3$ 几何；其中Berger曲面在固定质量下具有任意小的面积。它们从未俘获数据的形成问题仍未解决。在收缩均匀核心附近，光滑约束数据的精确非球对称发展（其所有导数收敛）具有有界曲率，但在核心特征损失后，几何潮汐记录中存在固定差异。经典存在性、局部几何唯一性以及该逐点记录的连续性不能都在共同观测区域上得到保证。配套论文处理窄入射脉冲。
英文摘要
We study scalar collapse in five- and six-dimensional Einstein--Gauss--Bonnet gravity with positive coupling, treated as an exact classical theory. On spherical scalar backgrounds, the characteristic conditions of the tensor, vector and scalar gravitational sectors reduce to inequalities in two local invariants: $K=1+\tildeαF/R^2$, which compares the enclosed mass with the areal radius $R$, and a scalar-gradient invariant $j$. In five dimensions they imply $R_h^2>\tildeα$ for the areal radius $R_h$ of a regular marginal sphere on the GR branch and the sharp bound $\tildeα^2\mathcal R_{abcd}\mathcal R^{abcd}<28$ for nonnegative mass. For initially untrapped five-dimensional spherical data with a regular center, a bound on $K$ alone ensures smooth continuation on compact domains of dependence, so any first characteristic loss occurs on a smooth background, and a marginal sphere formed before any loss forces a later loss. Complete smooth data form marginal spheres at every allowed radius with mass arbitrarily close to its sharp lower bound; in six dimensions radius and mass can both be arbitrarily small. Without spherical symmetry, complete initial data with weak curvature contain, at sufficiently large scale, outermost strictly stable marginal surfaces with prescribed $S^3$ geometries satisfying a spectral condition; Berger surfaces among them have arbitrarily small area at fixed mass. Their formation from untrapped data is left open. Near a contracting homogeneous core, exact nonspherical developments of smooth constrained data converging with every derivative have bounded curvature but, after the core's characteristic loss, a fixed discrepancy in a geometric tidal record. Classical existence, local geometric uniqueness and continuity of this pointwise record cannot all be guaranteed on a common observation region. A companion paper treats narrow incoming pulses.
发表机构
- University of Waterloo（滑铁卢大学）
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