Curvature Diagnostics in Lorentzian Geometry with Applications to General Relativity

Abstract

Scalar curvature invariants are coordinate independent, but their ability to detect curvature and identify geometric structure depends on the class of Lorentzian metrics under consideration. This focused survey compares scalar and observer-dependent tidal diagnostics in four-dimensional general relativity. After fixing the curvature convention, we give explicit proofs for three model classes. For Schwarzschild space-time, the Kretschmann scalar and the squared norm of its gradient recover the areal radius and positive mass parameter within the specified solution family. The gradient invariant vanishes at the horizon, while a regular coordinate extension establishes that the horizon is not a curvature singularity. For homogeneous and isotropic metrics, the Kretschmann scalar separates temporal and spatial curvature contributions and provides a class-specific flatness criterion. An exact vacuum plane wave supplies a counterexample to any unrestricted inference from vanishing scalar polynomial curvature invariants to flatness: its algebraic curvature contractions vanish while its tidal operator remains nonzero. The examples are related to established work on invariant characterization, stationary-horizon diagnostics and low-regularity singularity theorems. They show why a diagnostic must be accompanied by metric-class assumptions, an observer specification where appropriate, and a distinction between local curvature and global causal structure. The contribution is a comparative mathematical exposition with explicit derivations; no new classification theorem or observational result is claimed.