Invariants of the Riemann tensor for Class B Warped Product Spacetimes

Abstract

We use the computer algebra system \textit{GRTensorII} to examine invariants polynomial in the Riemann tensor for class \(B\) warped product spacetimes - those which can be decomposed into the coupled product of two 2-dimensional spaces, one Lorentzian and one Riemannian, subject to the separability of the coupling: \(ds^2 = ds_{\Sigma_1}^2 (u,v) + C(x^\gamma)^2 ds_{\Sigma_2}^2 (\theta,\phi)\) with \(C(x^\gamma)^2=r(u,v)^2 w(\theta,\phi)^2\) and \(sig(\Sigma_1)=0, sig(\Sigma_2)=2\epsilon (\epsilon=\pm 1)\) for class \(B_{1}\) spacetimes and \(sig(\Sigma_1)=2\epsilon, sig(\Sigma_2)=0\) for class \(B_{2}\). Although very special, these spaces include many of interest, for example, all spherical, plane, and hyperbolic spacetimes. The first two Ricci invariants along with the Ricci scalar and the real component of the second Weyl invariant \(J\) alone are shown to constitute the largest independent set of invariants to degree five for this class. Explicit syzygies are given for other invariants up to this degree. It is argued that this set constitutes the largest functionally independent set to any degree for this class, and some physical consequences of the syzygies are explored.