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If you put the Schwarzschild spacetime into a coordinate system in which the time coordinate is identified with the proper time of an observer falling radially from infinity, but keep the other coordinates the same, you get Gullstrand–Painlevé coordinates.

Amazingly, it is quickly seen that a constant-time slice in this system is flat! I thought this was a rather spectacular result, but I'm having trouble thinking of what it means physically. Can anyone help me out in understanding this beyond just the mathematical result?

Thanks!