Gullstrand–Painlevé coordinates.
coordinate system for the Schwarzschild metric such that the time coordinate is the proper time of a free-falling observer who starts from far away at zero velocity, and the spatial slices are flat
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About Gullstrand–Painlevé coordinates
Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes the gravitational field in the vicinity of a spherical mass with no electric charge or angular momentum. The ingoing coordinates are such that the time coordinate follows the proper time of a free-falling observer who starts from far away at zero velocity, and the spatial slices are flat. There is no coordinate singularity at the Schwarzschild radius. The outgoing ones are simply the time reverse of ingoing coordinates.
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