发表机构
Universidad de La Frontera; Universidad Católica del Norte; Shanghai Institute for Advanced Studies; University of Science and Technology of China; Herzen State Pedagogical University of Russia; Khazar University; SAO RAS; Pontificia Universidad Católica de Valparaíso（弗龙特拉大学; 北天主教大学; 上海高等研究院; 中国科学技术大学; 俄罗斯赫尔岑国立教育大学; 哈扎尔大学; 俄罗斯科学院特别天体物理观测台; 瓦爾帕萊索天主教大學）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究引力解耦中最小几何形变产生的黑洞与虫洞分支，证明临界耦合两侧分别对应黑洞和虫洞完备化，并以第二同调群作为区分不变量。
AI 中文摘要
最小几何形变（MGD）应用于静态史瓦西黑洞种子时，会产生一个单一的解耦函数$h(r)$，该函数通过求解$\theta$-扇区场方程并结合一个状态方程得到。一旦$h(r)$固定，所得的单参数族由耦合强度$k$通过$F(r;k)=1+k\\,h(r)$控制。我们证明，每当形变产生一个穿过种子视界的简单最外层根时，同一个固定的解耦函数会导致两个相互排斥的分支，它们与不同的全局完备化相关联：在临界耦合的一侧，形变后的度量将种子视界保持为黑洞，而在另一侧，根$r_*>2M$位于外部，不能解释为黑洞几何的内部修正。我们证明该根导致区间$(2M,r_*)$上洛伦兹号差的丧失，因此一旦$r_*$位于种子视界$r=2M$之外，就不存在通过该视界的光滑外部度量延拓。在我们考虑的静态、球对称类中，相应的光滑洛伦兹完备化是一个两端虫洞，通过切除$(2M,r_*)$并将区域$r\geq r_*$沿最小球面$\mathcal T=\{r=r_*\}$加倍而获得。我们不声称也不要求任何单一时空的拓扑改变：$k>k_c$和$k<k_c$仅对应于两个不同的、非微分同胚的流形，下面的引理1表明，度量本身决定了对于给定的$k$，哪一个是允许的完备化。我们显式计算了两个完备化的第二同调群：黑洞外部相对于其视界的$H_2(\Sigma_{\rm BH},\mathcal H)=0$，以及完备虫洞流形的$H_2(\Sigma_{\rm WH})\cong\mathbb Z$，给出了区分这两个分支的离散不变量。
英文摘要
Minimal Geometric Deformation (MGD) applied to a static Schwarzschild black hole seed generates a single decoupler function $h(r)$, obtained by solving the $θ$-sector field equations together with an equation of state. Once $h(r)$ is fixed, the resulting one-parameter family is controlled by the coupling strength $k$ through $F(r;k)=1+k\,h(r)$. We show that, whenever the deformation develops a simple outermost root that crosses the seed horizon, the same fixed decoupler leads to two mutually exclusive branches associated with different global completions: on one side of the critical coupling the deformed metric preserves the seed horizon as a black hole, whereas on the other side the root $r_*>2M$ lies in the exterior and cannot be interpreted as an interior modification of the black hole geometry. We prove that this root forces a loss of Lorentzian signature on the interval $(2M,r_*)$, so that no smooth extension of the exterior metric through the seed horizon $r=2M$ exists once $r_*$ lies outside it. Within the static, spherically symmetric class considered here, the corresponding smooth Lorentzian completion is a two-ended wormhole obtained by excising $(2M,r_*)$ and doubling the region $r\geq r_*$ across the minimal sphere $\mathcal T=\{r=r_*\}$. No topology change of any single spacetime is claimed or required: $k>k_c$ and $k<k_c$ simply correspond to two different, non-diffeomorphic manifolds, and Lemma~1 below shows that the metric itself dictates which of the two is the admissible completion for a given $k$. We compute the second homology group of both completions explicitly, $H_2(Σ_{\rm BH},\mathcal H)=0$ for the black hole exterior relative to its horizon and $H_2(Σ_{\rm WH})\cong\mathbb Z$ for the completed wormhole manifold, giving a discrete invariant that distinguishes the two branches.
Comments12 pages, accepted to PRD
