arXiv math-ph/01100192001-10-17math-phmath.MP

时空的共形紧化与保圈对称性

Conformal compactification and cycle-preserving symmetries of spacetimes

Francisco J. Herranz, Mariano Santander

备注：: 43 pages, LaTeX
发表信息：: J. Phys. A 35 (2002) 6601-6618; ibid 6619-6629
DOI：: 10.1088/0305-4470/35/31/306

摘要：

在Cayley-Klein框架下，得到了9个二维常曲率实空间的保圈对称性.这种方法提供了一个统一的和全球性的研究共形结构的三个经典黎曼空间，以及六个相对论性和非相对论性时空（闵可夫斯基，德西特，反德西特，牛顿胡克和伽利略），并产生了一般表达式举行同时为所有这些。它们的度量结构和周期（包括测地线和圆的具有常测地曲率的线）被明确地表征。相应的循环（类高斯）李群及其代数的微分实现被推导出来;这个推导是新的，比通常的推导简单得多，并且适用于凯莱-克莱因家族中的任何齐性空间，无论是平坦的还是弯曲的，并且具有任何签名。构造了具有共形代数对称性的Laplace型和波动型微分方程。此外，共形群被实现为矩阵群作为全球定义的线性变换在一个四维的“共形环境空间”，这反过来又导致了一个明确的描述“共形完成”或紧化的九个空间。

英文摘要：

The cycle-preserving symmetries for the nine two-dimensional real spaces of constant curvature are collectively obtained within a Cayley-Klein framework. This approach affords a unified and global study of the conformal structure of the three classical Riemannian spaces as well as of the six relativistic and non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both Newton-Hooke and Galilean), and gives rise to general expressions holding simultaneously for all of them. Their metric structure and cycles (lines with constant geodesic curvature that include geodesics and circles) are explicitly characterized. The corresponding cyclic (Mobius-like) Lie groups together with the differential realizations of their algebras are then deduced; this derivation is new and much simpler than the usual ones and applies to any homogeneous space in the Cayley-Klein family, whether flat or curved and with any signature. Laplace and wave-type differential equations with conformal algebra symmetry are constructed. Furthermore, the conformal groups are realized as matrix groups acting as globally defined linear transformations in a four-dimensional "conformal ambient space", which in turn leads to an explicit description of the "conformal completion" or compactification of the nine spaces.

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