Invariants of the Riemann tensor for Class B Warped Product Spacetimes

TL;DR

This paper identifies the fundamental polynomial invariants of the Riemann tensor for class B warped product spacetimes, including many physically relevant models, using computer algebra to analyze their algebraic structure and syzygies.

Contribution

It determines the largest independent set of polynomial invariants for class B warped spacetimes and provides explicit relations among invariants up to degree five.

Findings

The first two Ricci invariants, Ricci scalar, and real part of the second Weyl invariant form a maximal independent set.

Explicit syzygies among invariants are derived up to degree five.

The results have implications for understanding the algebraic and physical properties of these 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.