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: $d{s}^2=d{s}_{{\Sigma }_1}^2(u,v)+C(x^{\gamma }{)}^{2}d{s}_{{\Sigma }_2}^2(\theta ,\varphi )$ with $C(x^{\gamma }{)}^2=r(u,v{)}^{2}w(\theta ,\varphi {)}^2$ 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.

19 pages. To appear in Computer Physics Communications Thematic Issue on "Computer Algebra in Physics Research". Uses Maple2e.sty

References in corpus (1)

• Geometric Interpretation of the Mixed Invariants of the Riemann Spinor

Cited by in corpus (25)

• Anisotropic Stars in General Relativity
• Spherically symmetric dissipative anisotropic fluids: A general study
• All static spherically symmetric perfect fluid solutions of Einstein's Equations
• Can the galactic rotation curves be explained in brane world models?
• Vacuum solutions of the gravitational field equations in the brane world model
• Remarks on the formulation of the cosmological constant/dark energy problems
• Generating static spherically symmetric anisotropic solutions of Einstein's equations from isotropic Newtonian solutions
• Regular black holes inspired by quasi-topological gravity
• Spherically symmetric black holes in the limiting curvature theory of gravity
• Curvature Invariants for the Alcubierre and Natário Warp Drives
• Curvature Invariants for Lorentzian Traversable Wormholes
• Visualizing Spacetime Curvature via Gradient Flows I: Introduction
• An inverse approach to Einstein's equations for non-conducting fluids
• Curvature Invariants for the Accelerating Natario Warp Drive
• On perfect fluid models in non-comoving observational spherical coordinates
• Hydrodynamics in Class B Warped Spacetimes
• Generalized Swiss-Cheese Cosmologies II: Spherical Dust
• An Invariant Characterization of the Levi-Civita Spacetimes
• Spacetimes with a vanishing second Ricci invariant
• A Newman-Penrose Calculator for Instanton Metrics
• Constructing Maximal Extensions of the Vaidya Metric in Israel Coordinates: I. Integration of the Field Equations
• Non conducting spherically symmetric fluids
• Visualization and analysis of the curvature invariants in the Alcubierre warp-drive spacetime
• A Closer Look at Natário's Zero-Expansion Warp Drive
• Inverse approach to Einstein's equations for fluids with vanishing anisotropic stress tensor