A singularity is where a quantity in an expression (e.g., the curvature of spacetime in the solutions to the equations of general relativity) goes off to infinity, usually as a result of division by 0. It’s a point where the maths breaks down and the equation can’t meaningfully describe what’s going on physically anymore.
It’s important to note that the presence of singularities in the *maths* of GR does not mean real physical reality has singularities inside black holes. Actually, the fact that *the equations* (e.g., in the Schwarzchild metric) have singularities in them is a suggestion to many that general relativity, for all its resounding successes, is still not the complete picture. Usually when an equation has division by zero, it’s a sign something is missing from your model.
Singularities are the reason that GR is in irreconcilable conflict with quantum mechanics, and either both are wrong and we need a paradigm shift (exotic stuff like string theories), or we’ll find a unified theory of quantum gravity that unifies the two.
We don’t *know* that black holes have singularities with infinite gravity or infinite density. Our *models* of black holes (the equations of GR, and the solutions to GR we derive like the Schwarzchild metric—the Kerr metric is a little more complicated b/c rotating black holes don’t necessarily have a point-like singularity) have singularities in them. But our models are incomplete, and the mere presence of a singularity in the model is highly suggestive of the common interpretation that at that point, the model breaks down and fails to describe what’s actually going on physically.
Nobody’s ever jumped inside a black hole and taken measurements of gravity or density or spacetime curvature. Rather, our models *predict* there’s a singularity, a terminus of spacetime at the center of black holes.
And in fact, some argue that we’re interpreting it wrong. The Penrose Singularity theorem has widely been interpreted to prove that the interior spacetime region of any black hole surrounded by an event horizon must be geodesically incomplete, i.e., it must contain a singularity. But Roy Kerr (the same guy after whom the Kerr metric for rotating black holes is named) argues that’s a faulty conclusion. He argues that just because the affine parameter is bounded doesn’t mean there has to be singularities.
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