McVittie事件视界处的最优延拓正则性
Optimal Extension Regularity at the McVittie Event Horizon
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中文总结 AI 辅助
该研究确定了McVittie解中未来黑洞事件视界的最优局部延拓正则性，由参数 $p$ 的取值决定，不同 $p$ 对应不同正则性，整数 $p\geq2$ 时存在实解析延拓。
中文摘要 AI 辅助
我们确定了由正宇宙学常数和具有常数状态方程参数 $w>-1$ 的正压流体源产生的精确空间平坦McVittie解中未来黑洞事件视界的最优局部延拓正则性。设 $H_\infty$ 为渐近哈勃常数，$\kappa$ 为极限黑洞根的表面引力，$p=3(1+w)H_\infty/\kappa$。内向径向类光测地线在有限仿射长度内到达视界，平行传播的角曲率分量渐近于 $C s^{p-2}$（其中 $C\neq0$，$s$ 为剩余仿射距离），这排除了 $0<p<2$ 时所有带锚定的 $C^2$ 延拓。对于 $p\geq2$，我们构造了参数一致的高斯-零紧致化和显式双侧洛伦兹领。若 $p=N+\vartheta$ 为非整数（$N\geq2$ 且 $0<\vartheta<1$），则最优正则性为标准大赫尔德类 $C^{N,\vartheta}$：该类延拓存在，而对于 $\vartheta'>\vartheta$，不存在 $C^{N,\vartheta'}$ 延拓。每个整数 $p\geq2$ 则属于解析岛，且容许实解析局部延拓。在临界值 $p=2$ 处，边界爱因斯坦自同态具有非零秩-1幂零部分。因此，宇宙学衰减与视界红移的比值决定了几何正则性的清晰算术层级。
英文摘要
We determine the optimal local extension regularity of the future black-hole event horizon in the exact spatially flat McVittie solutions sourced by a positive cosmological constant and a barotropic fluid with constant equation-of-state parameter $w>-1$. Let $H_\infty$ be the asymptotic Hubble constant, $κ$ the surface gravity of the limiting black-hole root, and $p=3(1+w)H_\infty/κ$. Ingoing radial null geodesics reach the horizon in finite affine length. A parallelly propagated angular curvature component is asymptotic to $C s^{p-2}$, with $C\ne0$ and $s$ the remaining affine distance, which excludes every anchored $C^2$ extension for $0<p<2$. For $p\ge2$ we construct a parameter-uniform Gaussian-null compactification and an explicit two-sided Lorentzian collar. If $p=N+\vartheta$ is nonintegral, with $N\ge2$ and $0<\vartheta<1$, the optimal regularity is the standard big Hölder class $C^{N,\vartheta}$: extensions of this class exist, whereas no $C^{N,\vartheta'}$ extension exists for $\vartheta'>\vartheta$. Every integer $p\ge2$ instead belongs to an analytic island and admits a real-analytic local extension. At the critical value $p=2$ the boundary Einstein endomorphism has a nonzero rank-one nilpotent part. The ratio of cosmological decay to horizon redshift therefore determines a sharp, arithmetic hierarchy of geometric regularity.
