Summary: The Schwarzschild metric, so useful in describing the strange behaviors in the vicinity of a black hole, is clearly an approximation. It is valid neither in empty space (where it shows c² = c² - 1) nor at the event horizon of a black hole (where it fails with a divide by zero error as time goes to zero and space goes to infinity). This is not an opinion, it is an easily demonstrated mathematical fact.
The error arises from using the precision of calculus where reality is less precise. The potential energy gradient across a particle with a finite size is not equivalent to the tangent angle at its center. The steeper the curvature, the greater the differential between the two methods of calculating the force acting upon the blue particle.
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