宇宙学Oppenheimer-Snyder模型：McVittie时空与FLRW时空的匹配
On a cosmological Oppenheimer-Snyder model: matching McVittie and FLRW spacetimes
AI总结：
该研究探讨McVittie时空与FLRW时空沿任意超曲面平滑匹配的条件，证明匹配系统的解在未来整体存在唯一，得出McVittie时空全局无各向同性源的结论。
AI中文摘要：
我们研究膨胀的McVittie时空与空间平坦的Friedmann-Lemaître-Robertson-Walker（FLRW）时空沿任意超曲面（即具有任意因果特征，且该特征可能逐点变化）平滑匹配的必要与充分条件。我们确定了若干特殊情形，并得出一些不可行性结论。在一般情形下，我们证明全部匹配条件等价于一个受单个初值约束的三维一阶非线性常微分方程组。以McVittie区域已确定为视角，我们证明该方程组的半整体存在性结果，并从物理设定角度解释这些解；解在未来整体存在且唯一，对应所需FLRW时空存在且唯一，匹配超曲面几乎处处存在且唯一。在过去方向上，匹配超曲面延伸至与McVittie区域的过去奇点交汇。我们确定了这些极限下匹配超曲面的因果特征，并探讨其对McVittie宇宙学黑洞时空是否存在各向同性源的意义。研究发现，一般而言匹配超曲面同时具有类空和类时部分，因此不能将各向同性区域视为McVittie外部的内部区域，这与Oppenheimer-Snyder模型不同——在后者中，各向同性尘埃球沿处处类时边界与Schwarzschild外部匹配。基于超曲面的有时类空性质，我们得出结论：全局意义上McVittie时空不存在各向同性源。
英文摘要：
We consider the necessary and sufficient conditions for the smooth matching of an expanding McVittie spacetime and a spatially flat Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) spacetime across a general hypersurface - that is, a hypersurface of arbitrary causal character, where the character possibly changes from point to point. We identify a number of special cases, and determine some no-go results. In the general case, we show that the full set of matching conditions is equivalent to a 3-dimensional non-linear system of first order ODEs, subject to a single initial value constraint. Taking the perspective that the McVittie region is specified, we prove a semi-global existence result for this system and interpret these solutions in terms of the physical set-up. Solutions exist and are unique globally to the future. In this direction, the required FLRW spacetime exists and is unique and the matching hypersurface exists almost everywhere and is unique. To the past, the matching hypersurface extends to meet the past singularity of the McVittie region. We determine the causal character of the matching hypersurface in these limits, and consider the implications for the existence (or otherwise) of an isotropic source of the McVittie cosmological black hole spacetime. We find that generically, the matching hypersurface has spacelike as well as timelike portions, and so the isotropic region cannot be considered to be an interior for the McVittie exterior. This is in distinction to the Oppenheimer-Snyder model, in which an isotropic dust sphere matches across an everywhere-timelike boundary to the Schwarzschild exterior. On the basis of the sometimes-spacelike nature of the hypersurface, we conclude that an isotropic source of the McVittie spacetime does not exist in a global sense.
