Physical meaning of the time coordinate in Gullstrand-Painleve coordinates

Question

The Gullstrand-Painleve coordinates are a set of coordinates used to define Schwarzschild spacetime. The time coordinate $t_{GP}$ of this coordinate system is related to the time coordinate $t$ of the Schwarzschild coordinate system as follows:

$$d{t_{GP}}=dt + \frac{\sqrt{2GM/r}}{1-2GM/r}dr$$

where $r$ is the radial coordinate.

My question is, what is the physical meaning of the ${t_{GP}}$ coordinate in the Gullstrand-Painleve coordinates?

For example the time coordinates $t$ in the Schwarzschild coordinate system are understood as follows. We imagine "t-meters" at every point in space which receive light signals from clocks radially far away at infinity. These light signals are emitted in 1 second intervals as measured by the clocks at infinity and the t-meter advances 1 second after receiving each light signal. These t-meters determine the Schwarzschild $t$ coordinate of any event.

I have read that the Gullstrand-Painleve ${t_{GP}}$ coordinate can be physically interpreted in a similar way by clocks falling to the origin from infinity but I don't understand how. Can someone please help me with this?

Answer 1

In Schwarzschild coordinates, using the geodesic equation, the inverse velocity of an object free-falling radially from infinity is $$\left(\frac{\mathrm{d}r}{\mathrm{d}t}\right)^{-1} = \frac{\text{d}t}{\text{d}r} = \frac{\sqrt{r/r_s}}{1-r/r_s} = \frac{\sqrt{r_s/r}}{1-r_s/r}$$ where $r_s = 2GM$. This means by defining the Gullstrand–Painlevé time coordinate as $$\text{d}t_{GP} = \text{d}t + \frac{\text{d}t}{\text{d}r} \text{d}r = \text{d}t + \frac{\sqrt{r_s/r}}{1-r_s/r}\text{d}r$$ we can make $\text{d}t_{GP}$ behave like the proper time of a radially free-falling clock, at the cost of adding a cross term $\text{d}t_{GP} \text{d}r$ to the metric.