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arXiv · 0806.2423

Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction

Alena Pravdova·Vojtech Pravda
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Abstract

Vacuum spacetimes admitting a non-twisting geodetic multiple Weyl aligned null direction (WAND) are analyzed in arbitrary dimension using recently developed higher-dimensional Newman-Penrose (NP) formalism. We determine dependence of the metric and of the Weyl tensor on the affine parameter r along null geodesics generated by the WAND for type III and N spacetimes and for a special class of type II and D spacetimes, containing e.g. Schwarzschild-Tangherlini black holes and black strings and branes. For types III and N, all metric components are at most quadratic polynomials in r while for types II and D the r-dependence of the metric as well as of the Weyl tensor is determined by an integer m corresponding to the rank of the expansion matrix S_{ij}. It is shown that for non-vanishing expansion, all these spacetimes contain a curvature singularity. As an illustrative example, a shearing expanding type N five-dimensional vacuum solution is also re-derived using higher-dimensional NP formalism. This solution can be, however, identified with a direct product of a known four-dimensional type N metric with an extra dimension.

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Alena Pravdova, Vojtech Pravda. 2008-11-03. Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction. https://doi.org/10.1088/0264-9381%2F25%2F23%2F235008

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