Gullstrand-Painlevé coordinates

1. Dec 19, 2008

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 tr 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 [Broken] 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

Last edited by a moderator: May 3, 2017