Gullstrand-Painlevé coordinates

[stevebd1]

While I understand Doran coordinates and Doran form (Gullstrand-Painlevé form at a=0), I'm not entirely convinced with Gullstrand-Painlevé coordinates.

While the Doran time coordinate $(\bar{t})$ is expressed-

$$d\bar{t}=dt+\frac{\beta}{1-\beta^2}dr$$

where

$$\beta=\frac{\sqrt{2Mr}}{R}$$

$$R=\sqrt{r^2+a^2}$$

and M=Gm/c² and a=J/mc

the G-P time coordinate $(t_r)$ is expressed-

$$t_r=t-\int_r^\infty \frac{\beta\,dr}{1-\beta^2}$$

and sometimes, β is expressed in negative form (see wiki entry on G-P coords).

Are Doran and G-P coordinates suppose to be different or is there some process involving the cancelling out of signs that makes them the same? Calculating the G-P time coordinate using a positive β appears to give convincing results. Even though t_r becomes negative outside the event horizon, proper time is 1 at infinity, zero at the ergosphere and divergent at the event horizon, but there appears to be a 'spike' in proper time at about 5M where it drops to zero (possibly becoming negative) and then rising before dropping down to zero again at the ergosphere. I'm assuming this has something to do with the time coordinate becoming negative. Apart from this, everything else works fine.source-
Doran coordinates
http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.2369v1.pdf page 3

G-P coordinates
http://en.wikipedia.org/wiki/Gullstrand-Painlevé_coordinates#Rain_coordinates
http://arxiv.org/PS_cache/gr-qc/pdf/0411/0411060v2.pdf page 3, 6

 

[PeterDonis]

Are Doran and G-P coordinates suppose to be different or is there some process involving the cancelling out of signs that makes them the same?

I'm not sure what you mean by "different" vs. "the same". The two charts are intended for two different spacetimes. The definition of the time coordinate is formally similar in the two cases (note that your expression for the GP case can be easily converted into an expression that is formally the same as the Doran one you give, just take the differential of both sides). But the definition of $\beta$ is different (although the Doran definition reduces to the GP definition if $a = 0$).

Calculating the G-P time coordinate using a positive β appears to give convincing results.

I'm not sure what you mean by a positive $\beta$.

There are two versions of GP coordinates, ingoing and outgoing. Ingoing is the kind that is usually discussed; in this version, the time coordinate is the proper time of an observer who is free-falling inward radially from rest at infinity. This version gives a coordinate patch that covers the exterior region and the black hole interior (regions I and II on a Kruskal diagram such as the one in my Insight series on the Schwarzschild geometry). In this version, the time coordinate going to minus infinity corresponds to approaching the past horizon (the white hole horizon); the coordinates become singular at that horizon.

In outgoing GP coordinates, the time coordinate is the proper time of an observer who is free-falling outward at exactly escape velocity, i.e., the observer will just come to rest at infinity. This version gives a coordinate patch that covers the exterior region and the white hole interior (regions I and IV on the Kruskal diagram in my Insight series). In this version, the time coordinate going to plus infinity corresponds to approaching the future horizon (the black hole horizon); the coordinates become singular at that horizon.

It is easy to get confused about what GP coordinates are telling you if you don't take the above points into account. I'm not sure exactly what you are trying to calculate with them, but the above might help to clarify things.

The usual form of Doran coordinates that I have seen would correspond to ingoing GP coordinates as I described them above. I would expect there to also be an outgoing form for Doran coordinates, but I have not seen that form discussed.