Proper time is invariant but undefined at the event horizon in schwarzschild metric. How can we remove this problem with a coordinate transformation?

Question

We know that the line element gives us the proper time taken to traverse “nearby” coordinates.This is an invariant as well, so how can we remove the issue of undefined proper time at the event horizon in the schwarzschild metric by a coordinate transformation?

Answer 1

so how can we remove the issue of undefined proper time at the event horizon in the schwarschild metric by a coordinate transformation?

A coordinate chart is a smooth invertible mapping between an open subset of the manifold and an open subset of R4. The Schwarzschild chart does not cover the event horizon. There is no coordinate transform to or from the Schwarzschild chart at the horizon because the chart itself does not cover the horizon.

In other words, the issue is not that proper time is undefined at the event horizon in the Schwarzschild chart. The issue is that the Schwarzschild chart itself does not exist at the event horizon.

This is not something that can be removed nor does it need to be removed. We can simply use a chart that does cover the horizon to calculate any invariant we wish to calculate at the horizon. There is no need (nor any possibility) to transform to a chart that doesn’t cover it.