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A Metric-Based, Coordinate-Dependent Condition for Petrov Type D Spacetimes

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Preprints.org
DOI
10.20944/preprints202511.1694.v1

To the best of the author's knowledge, this work is among the first to present a coordinate-dependent, metric-component condition that must be satisfied by all Type D spacetimes, each admitting two geodesic, shear-free null congruences. While such conditions are well known, this work provides a clear and practical formulation that enables direct verification of Type D spacetimes and the computation of metric components without assuming an ansatz. Derived from the vanishing of the Newman–Penrose spin coefficient lambda in Kerr spacetime, the condition is expressed entirely in terms of metric components and their radial derivatives, eliminating the need for tetrad construction. The condition has been validated numerically for Kerr, Kerr–Newman, Schwarzschild, and static de Sitter spacetimes. It may also guide the construction of new exact solutions or the testing of proposed ansatz with desirable geometric properties. Similar relations may exist for other spin coefficients and algebraic types of spacetimes.

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