Stable homology as an indicator of manifoldlikeness in causal set theory
Stable homology as an indicator of manifoldlikeness in causal set theory
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DOI：
10.1088/0264-9381/26/17/175008
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发表时间：
2009-02
影响因子：
3.5
通讯作者：
S. Major;D. Rideout;S. Surya
中科院分区：
文献类型：
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作者：
S. Major;D. Rideout;S. Surya
我们提出了一个计算工具，可用于获得的因果集的“空间”同源群。在因果集中的本地化是由一个不可扩展的反链，这是一个类空超曲面的模拟种子，和一个单参数的家庭神经单纯形复合物是由“加厚”这个反链。然后可以使用现有的同源性软件计算相关的同源群，并研究它们的行为作为稠化参数的函数。早期的分析工作表明，对于一个不可扩展的反链，在一个因果集中，可以近似为一个全球双曲时空区域，有一个单参数的子家庭，这些单纯复形是同源的连续，提供反链满足一定的条件。使用因果集近似的一组二维时空，我们的数值分析表明，这些条件一般满足不可扩展的反链。在2D和3D模拟中，随着增厚参数的增加，连续统同调组倾向于出现在第一个区域，在该区域中，同调是恒定的，或“稳定的”，高于离散尺度。在这个尺度之下，同源基团作为增厚参数的函数迅速波动。这提供了一个必要的，但不是充分的标准来测试的manifoldlikeness因果集。
We present a computational tool that can be used to obtain the ‘spatial’ homology groups of a causal set. Localization in the causal set is seeded by an inextendible antichain, which is the analogue of a spacelike hypersurface, and a one-parameter family of nerve simplicial complexes is constructed by ‘thickening’ this antichain. The associated homology groups can then be calculated using existing homology software, and their behaviour studied as a function of the thickening parameter. Earlier analytical work showed that for an inextendible antichain in a causal set which can be approximated by a globally hyperbolic spacetime region, there is a one-parameter sub-family of these simplicial complexes which are homological to the continuum, provided the antichain satisfies certain conditions. Using causal sets that are approximated by a set of 2D spacetimes, our numerical analysis suggests that these conditions are generically satisfied by inextendible antichains. In both 2D and 3D simulations, as the thickening parameter is increased, the continuum homology groups tend to appear as the first region in which the homology is constant, or ‘stable’, above the discreteness scale. Below this scale, the homology groups fluctuate rapidly as a function of the thickening parameter. This provides a necessary though not sufficient criterion to test for manifoldlikeness of a causal set.