Type III and N Einstein spacetimes in higher dimensions: general properties
Type III and N Einstein spacetimes in higher dimensions: general properties
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高维中的 III 型和 N 型爱因斯坦时空：一般性质
DOI：
10.1103/physrevd.82.064043
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发表时间：
2010
影响因子：
5
通讯作者：
A. Pravdová
中科院分区：
文献类型：
--
作者：
M. Ortaggio;V. Pravda;A. Pravdová
在潜在应用中感兴趣的几种情况下，控制大地测量 WAND（Weyl 对齐零方向）光学矩阵演化的萨克斯方程在 n 维中明确求解。然后将其用于研究高维纽曼-彭罗斯形式主义中的 III 型和 N 型爱因斯坦时空，同时考虑 Kundt 和扩展（可能扭曲）解。特别地，度量和Weyl张量对仿射参数r的一般依赖性以封闭形式获得。这使我们能够表征大 r 值下 Weyl“物理”组件的剥离行为，从而讨论扭曲的存在如何影响偏振模式，以及四个维度和更高维度之间的质量差异。此外，扩展时空的某些非零标量曲率不变量的 r 依赖性被用来证明曲率奇点通常可能存在。作为说明，最终提出了在更高维度上求解爱因斯坦真空方程（具有可能的宇宙学常数）的几个显式 N/III 型时空。
The Sachs equations governing the evolution of the optical matrix of geodetic WANDs (Weyl aligned null directions) are explicitly solved in n dimensions in several cases which are of interest in potential applications. This is then used to study Einstein spacetimes of type III and N in the higher dimensional Newman-Penrose formalism, considering both Kundt and expanding (possibly twisting) solutions. In particular, the general dependence of the metric and of the Weyl tensor on an affine parameter r is obtained in a closed form. This allows us to characterize the peeling behavior of the Weyl 'physical' components for large values of r, and thus to discuss, e.g., how the presence of twist affects polarization modes, and qualitative differences between four and higher dimensions. Further, the r dependence of certain nonzero scalar curvature invariants of expanding spacetimes is used to demonstrate that curvature singularities may generically be present. As an illustration, several explicit type N/III spacetimes that solve Einstein's vacuum equations (with a possible cosmological constant) in higher dimensions are finally presented.