Invariants of the Riemann tensor for class B warped product space-times Original Research Article

We use the computer algebra system GRTensorII to examine invariants polynomial in the Riemann tensor for class B warped product space-times — 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 ds2 = dsϵ12 (u,v) + C(xγ)2dsϵ22 (θ, φ). with C(xγ)2 = r(u,v)2w(θ, φ)2 and sig(ϵ1) = 0, sig(ϵ2) = 2ϵ ( ϵ = ± 1) for class B1 space-times and sig(ϵ1) = 2ϵ, sig(ϵ2) = 0 for class B2. Although very special, these spaces include many of interest, for example, all spherical, plane, and hyperbolic space-times. 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.