Generalizations and challenges for the spacetime block-diagonalization

Abstract

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.

22 pages (ver. 3: equation (51) corrected to m-dimensional case)

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• On the `Stationary Implies Axisymmetric' Theorem for Extremal Black Holes in Higher Dimensions
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Cited by in corpus (6)

• Lagrangian reverse engineering for regular black holes
• Conundrum of regular black holes with nonlinear electromagnetic fields
• Properties of general stationary axisymmetric spacetimes: circularity and beyond
• Weighing the curvature invariants
• Hexadecapole at the heart of nonlinear electromagnetic fields
• From rotating attractors to extremal black holes with axionic hair