Geodesic completeness, curvature singularities and infinite tidal forces

TL;DR

This work investigates the nuanced relationship between geodesic completeness, curvature singularities, and tidal forces in general relativity. By analyzing extended bodies in static, spherically symmetric spacetimes and in a Palatini-wormhole geometry, the authors show that infinite tidal forces can occur near curvature singularities even when the metric is well-defined there, and that geodesic completeness does not guarantee the absence of tidal divergences. They demonstrate that radial tidal forces diverge with curvature when a central singularity is present, while nonradial (or ultrarelativistic) motion can amplify tidal effects by a Lorentz boost factor $\\gamma^2$. The wormhole example reveals that a claimed geodesically complete throat can be physically incomplete due to a genuine singularity at the throat, with tidal stresses diverging as the singularity is approached. Overall, the paper emphasizes the universality of divergent tidal forces near singularities and discusses potential observational implications for ultra-high-energy cosmic rays and strong-field environments.

Abstract

We report some new findings regarding the subtle relations among geodesic completeness, curvature singularities and tidal forces. It is well known that any particle may be torn up near a singularity at the center of a black hole due to the divergent tidal force. However, we find that singularity is not the only cause of tidal force divergence. Even on the surface of the Earth, the tidal force experienced by a particle could be arbitrarily large if the particle moves arbitrarily close to the speed of light in a nonradial direction. Moreover, we find that the maximum tidal force always occurs along the radial direction, regardless of the particle's motion. Usually, a curvature singularity implies geodesic incompleteness since in many cases the metric cannot be defined at the location of the singularity. Counterexamples have been found in recent years, suggesting that geodesics could pass through curvature singularities. By taking into account the fact that any real particle is an extended body, we calculate the tidal force acting on the particle in a static and spherically symmetric spacetime. We explicitly show that an infinite tidal force always occurs near such a singularity. Therefore, no particle can actually reach the curvature singularity even if the metric is well defined at that point. We also demonstrate that the tidal acceleration along a null geodesic at the coordinate origin is divergent. Finally, we examine a wormhole solution which possesses a curvature singularity at its throat and was previously asserted to be geodesically complete in the literature. However, we prove that no metric can be defined at the throat and thus the spacetime is geodesically incomplete. we also show that the tidal forces experienced by any particle near the singularity are divergent.

Figures (1)

• Figure 1: For nonradial motion, the potential $V(r)$ goes to infinity at $r=0$.