Flat time slices in Gullstrand–Painlevé coordinates?

[ericbrown86]

Hi there,

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!

 

[PeterDonis]

I'm having trouble thinking of what it means physically

The simplest way to think of it is that the spacelike slices of constant Painleve coordinate time are cut "at an angle" relative to the spacelike slices of constant Schwarzschild coordinate time, and the difference in the angle of cut is just enough to make the distance between two 2-spheres with surface areas $4 \pi r_1^2$ and $4 \pi r_2^2$ equal to the Euclidean distance $r_2 - r_1$ in the Painleve slices, instead of the larger distance that it is in the Schwarzschild slices.