时空块对角化的推广与挑战
Generalizations and challenges for the spacetime block-diagonalization
AI总结：
本文重新审视了具有等距的块对角度规的几何背景，分析了多种物质源满足等距相容块对角形式的充分条件，并将静态时空中不存在零电磁场的定理推广至任意维度及非线性电磁场。
摘要：
引力场方程可能迫使具有等距同构的时空度规达到与这些等距相容的块对角形式，这一发现是构建黑洞唯一性定理体系的瑰宝之一。我们重新审视了具有等距的块对角度规的几何背景、由Killing向量场定义的叶状结构以及相应的Godbillon-Vey特征类。此外，我们分析了包括标量场、非线性电磁场和Proca场在内的各种物质源满足的充分条件，这些条件意味着度规具有等距相容的块对角形式。最后，我们将关于静态时空中不存在零电磁场的定理推广到任意数量的时空维度、广泛的引力场方程类和非线性电磁场。
英文摘要：
Discovery that gravitational field equations may coerce the spacetime metric with isometries to attain a block-diagonal form compatible with these isometries, was one of the gems built into the corpus of black hole uniqueness theorems. We revisit the geometric background of a block-diagonal metric with isometries, foliation defined by Killing vector fields and the corresponding Godbillon-Vey characteristic class. Furthermore, we analyse sufficient conditions for various matter sources, including scalar, nonlinear electromagnetic and Proca fields, that imply the isometry-compatible block-diagonal form of the metric. Finally, we generalize the theorem on the absence of null electromagnetic fields in static spacetimes to an arbitrary number of spacetime dimensions, wide class of gravitational field equations and nonlinear electromagnetic fields.
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