![]() ![]() Geodesics are incomplete there basically because there's no way to say which way the geodesic should go once it hits the tip. Another example of a curvature singularity is the Big Bang singularity.Ī conical singularity is like the one at the tip of a cone. There are two types of singularities, curvature singularities and conical singularities.Ī black hole singularity is an example of a curvature singularity as you approach the singularity, the curvature of spacetime diverges to infinity, as measured by a curvature invariant such as the Ricci scalar. (This also covers lightlike geodesics, which have zero metric length.) ![]() Geodesic incompleteness means that there exists a geodesic that can't be extended past a certain affine parameter. The way to get around this is to use an affine parameter, which can be defined without a metric. A careful definition of geodesic incompleteness is a little tricky, because we want to talk about geodesics that can't be extended past a certain length, but length is measured by the metric, and the metric goes crazy at a singularity so that length becomes undefined. You (and the subatomic particles you're made of) have no future world-lines. For example, if you drop yourself into a black hole, your world-line terminates at the singularity. A singularity is a condition in which geodesics are incomplete. ![]()
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