Geodesic completeness, curvature singularities and infinite tidal forces*

Abstract We report new findings on the subtle relationships among geodesic completeness, curvature singularities, and tidal forces. It is well known that particles may encounter infinite tidal forces near a black hole singularity. However, we find that singularities are not the only source of tidal-force divergence. Even on the surface of the Earth, the tidal force experienced by a particle can become arbitrarily large if the particle moves arbitrarily close to the speed of light in a nonradial direction. For fixed particle energy, the maximum tidal acceleration occurs for motion parallel to the surface with separation vectors oriented radially. Recent discoveries of spacetimes in which the metric remains well defined at curvature singularities have suggested that geodesics might extend through such points. Taking into account the fact that any real particle is an extended body, we calculate the tidal force acting on a 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 that 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 that the spacetime is therefore geodesically incomplete.