Warped Product Metrics with Rank-1 Ricci Tensor

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 Φ 11 + 3Λ = 0 and Φ 00 Φ 22 = (Φ 11 -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.