Tensorial versus Scalar Characterizations of Spacetime Singularities and Their Resolution in Loop Quantum Gravity

Abstract

Chur Chin

We propose a coordinate-independent notion of tensorial singularity for Lorentzian manifolds, defined via blow-up or non-extendability of the Riemann curvature tensor as a smooth section of the curvature bundle. We analyze its relation to scalar curvature singularities (e.g., divergence of the Kretschmann scalar) and geodesic incompleteness [1,2]. We prove that scalar invariant divergence implies tensorial blow-up, while the converse need not hold without additional regularity assumptions [3]. We then examine effective models arising from Loop Quantum Gravity [4,5], showing that quantum geometry corrections impose upper bounds on curvature scalars and thereby exclude tensorial singularities in homogeneous cosmological settings [6,7]. This suggests that classical curvature singularities may be artifacts of smooth manifold geometry. Our results provide a rigorous mathematical framework linking the geometric structure of spacetime singularities to their potential resolution through quantum corrections [8,9].

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