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Free-falling object: When should we consider General Relativity?

My question is related to a statement:

If a pendulum is experiencing free fall, then it will not oscillate.

The statement is true in the sense that its acceleration is (approaching to) zero, then according to the period equation $T$ would be infinity.

I think this sense comes from General Relativity. However, I'm not sure exactly where this result is coming from, and why we should consider General Relativity for this pendulum, rather than considering it to be a pure Newtonian scenario.

ZR-
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