Double warped space–times

Abstract

An invariant characterization of double warped space-times is given in terms of Newman-Penrose formalism and a classification scheme is proposed. A detailed study of the conformal algebra of these space-times is also carried out and some remarks are made on certain classes of exact solutions.

Conformal Vector Fields on Doubly Warped Product Manifolds and Applications

“…But the conformal factor is given by 2div (ζ) = ρn which completes the proof. [31]. Among many other results, they obtained necessary and sufficient conditions for a (locally) double warped spacetime to be conformally related to a 1 + 3 or 2 + 2 decomposable spacetime.…”

“…In particular, if for example f 2 = 1, then M = M 1 × f1 M 2 is called a (singly) warped product manifold. A singly warped product manifold M 1 × f1 M 2 is said to be trivial if the warping function f 1 is also constant [1,18,21,29,31,36]. It is clear that the submanifolds M 1 × {q} and {p} × M 2 are homothetic to M 1 and M 2 respectively for each p ∈ M 1 and q ∈ M 2 .…”

“…Let An investigation of 4−dimensional spacetimes that are conformally related to 1+ 3 reducible spacetimes was carried out with many examples in the aforementioned references [13,31]. A classification of these spacetimes according to their conformal algebra is considered in the first reference whereas a special attention is paid to gradient conformal vector fields in the second reference.…”

Conformally reducible 1+3 spacetimes

“…We will not repeat the details of that investigation here, but we will repeat the resulting theorem (Theorem 10 of [8]) which gives results that are crucial to the present investigation. However, the theorem presented here differs from that in [8] in that in parts (2) and (3) we have added comments pertaining to the conformally flat spaces and in part (4) we have a more general result than the corresponding part in [8]. (iii) Two or more GCKV are admitted by (V, h) if and only if (V, h) is of constant curvature (hence conformally flat) and it is flat if one of the GCKV admitted is a GHV.…”

“…(a) If it is a GKV ξ then, it must be null else (M,ĝ) degenerates into a 1+1+2 reducible spacetime [8]. Thus ξ is a covariantly constant null KV and (M,ĝ) is a pp-wave spacetime specialised to a 1+3 spacetime with metric (see section 35.1 of [6]) ds 2 = dη 2 − 2dudv − 2H(u, x)du 2 + dx 2 (7) and the null KV ξ = ∂ v .…”