Nonvanishing local scalar invariants even in VSI spacetimes with all polynomial curvature scalar invariants vanishing

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.