Warped Product Metrics with Rank-1 Ricci Tensor

¹ Mathematics Engineering Department, Istanbul Technical University, Faculty of Science and Letters, Maslak, 34469 Istanbul, Turkey
² Department of Mathematics, Faculty of Sciences and Letters, Nigde University, Nigde, Turkey

Abstract

Locally 2+2 warped product metrics with a Lorentzian base and Riemannian fiber are described in Newman-Penrose formalism. It is shown that solutions with rank-1 Ricci tensor are characterized by Φ₁₁ + 3Λ = 0 and Φ₀₀Φ₂₂ = (Φ₁₁-3Λ)². Integrability conditions for the Bianchi identities are studied and it is proved that the sources are null Einstein-Maxwell fields, massless scalar fields or stiff matter fields corresponding respectively to Λ = 0, Λ > 0 or Λ < 0.