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Curvature invariants in type N spacetimes

J. Bicak·V. Pravda
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Abstract

Scalar curvature invariants are studied in type N solutions of vacuum Einstein's equations with in general non-vanishing cosmological constant Lambda. Zero-order invariants which include only the metric and Weyl (Riemann) tensor either vanish, or are constants depending on Lambda. Even all higher-order invariants containing covariant derivatives of the Weyl (Riemann) tensor are shown to be trivial if a type N spacetime admits a non-expanding and non-twisting null geodesic congruence. However, in the case of expanding type N spacetimes we discover a non-vanishing scalar invariant which is quartic in the second derivatives of the Riemann tensor. We use this invariant to demonstrate that both linearized and the third order type N twisting solutions recently discussed in literature contain singularities at large distances and thus cannot describe radiation fields outside bounded sources.

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J. Bicak, V. Pravda. 1998-04-14. Curvature invariants in type N spacetimes. https://doi.org/10.1088/0264-9381%2F15%2F6%2F011

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