Nonvanishing Local Scalar Invariants even in VSI Spacetimes with all Polynomial Curvature Scalar Invariants Vanishing

TL;DR

This paper explores nonpolynomial local scalar invariants derived from the Riemann tensor that can remain nonzero in VSI spacetimes, contrasting with polynomial invariants that always vanish, and examines their behavior in various spacetimes.

Contribution

It introduces and evaluates nonpolynomial local scalar invariants in VSI spacetimes, showing they can be nonzero even when polynomial invariants vanish, expanding the understanding of scalar invariants in general relativity.

Findings

Nonpolynomial invariants can be nonzero in VSI spacetimes.

Examples include invariants related to gravitational waves.

Invariants are also analyzed for Schwarzschild, Kerr, and near Earth's surface.

Abstract

VSI (`vanishing scalar invariant') spacetimes have zero values for all total scalar contractions of all polynomials in the Riemann tensor and its covariant derivatives. However, there are other ways of concocting local scalar invariants (nonpolynomial) from the Riemann tensor that need not vanish even in VSI spacetimes, such as Cartan invariants. Simple examples are given that reduce to the squared amplitude for a linearized monochromatic plane gravitational wave. These nonpolynomial local scalar invariants are also evaluated for non-VSI spacetimes such as Schwarzschild and Kerr and are estimated near the surface of the earth. Similar invariants are defined for null fluids and for electromagnetic fields.