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Register a new userBasis for scalar curvature invariants in three dimensions
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$mathcal{I}$-non-degenerate spaces are spacetimes that can be characterized uniquely by their scalar curvature invariants. The ultimate goal of the current work is to construct a basis for the scalar polynomial curvature invariants in three dimensional Lorentzian spacetimes. In particular, we seek a minimal set of algebraically independent scalar curvature invariants formed by the contraction of the Riemann tensor and its covariant derivatives up to fifth order of differentiation. We use the computer software emph{Invar} to calculate an overdetermined basis of scalar curvature invariants in three dimensions. We also discuss the equivalence method and the Karlhede algorithm for computing Cartan invariants in three dimensions.
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We wish to construct a minimal set of algebraically independent scalar curvature invariants formed by the contraction of the Riemann (Ricci) tensor and its covariant derivatives up to some order of differentiation in three dimensional (3D) Lorentzian spacetimes. In order to do this we utilize the Cartan-Karlhede equivalence algorithm since, in general, all Cartan invariants are related to scalar polynomial curvature invariants. As an example we apply the algorithm to the class of 3D Szekeres cosmological spacetimes with comoving dust and cosmological constant $Lambda$. In this case, we find that there are at most twelve algebraically independent Cartan invariants, including $Lambda$. We present these Cartan invariants, and we relate them to twelve independent scalar polynomial curvature invariants (two, four and six, respectively, zeroth, first, and second order scalar polynomial curvature invariants).
For a homotopically energy-minimizing map $u: N^3to S^1$ on a compact, oriented $3$-manifold $N$ with boundary, we establish an identity relating the average Euler characteristic of the level sets $u^{-1}{theta}$ to the scalar curvature of $N$ and the mean curvature of the boundary $partial N$. As an application, we obtain some natural geometric estimates for the Thurston norm on $3$-manifolds with boundary, generalizing results of Kronheimer-Mrowka and the second named author from the closed setting. By combining these techniques with results from minimal surface theory, we obtain moreover a characterization of the Thurston norm via scalar curvature and the harmonic norm for general closed, oriented three-manifolds, extending Kronheimer and Mrowkas characterization for irreducible manifolds to arbitrary topologies.
We evaluate the entanglement entropy of exactly solvable Hamiltonians corresponding to general families of three-dimensional topological models. We show that the modification to the entropic area law due to three-dimensional topological properties is richer than the two-dimensional case. In addition to the reduction of the entropy caused by non-zero vacuum expectation value of contractible loop operators a new topological invariant appears that increases the entropy if the model consists of non-trivially braiding anyons. As a result the three-dimensional topological entanglement entropy provides only partial information about the two entropic topological invariants.
We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)le 1. Let g be another metric with scalar curvature k, such that gge g on 2-vectors. We show that kge k everywhere on M implies k=k. Under an additional condition on the Ricci curvature of g, kge k even implies g=g. We also study area-non-increasing spin maps onto such Riemannian manifolds.
A short review of scalar curvature invariants in gravity theories is presented. We introduce how these invariants are constructed and discuss the minimal number of invariants required for a given spacetime. We then discuss applications of these invariants and focus on three topics that are of particular interest in modern gravity theories.
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