Time coordinate in the rain frame

[stevebd1]

I'm currently looking at Metric for the Rain Frame in 'Exploring Black Holes' by Taylor & Wheeler (page B-13) and while it's straightforward understanding dr_rain (which basically equals dr), I'm having a problem getting my head around dt_rain. The following is a step-by-step approach but for some reason, the results I get from the metric are negative. (Attached is an extract from the book which shows the metric in full, hopefully this is acceptable).

from the book-

$$dt_{rain}=-v_{rel}\gamma dr_{shell}+\gamma dt_{shell}$$

where-

$$dr_{shell}=dr\left(1-\frac{2Gm}{rc^2}\right)^{-1/2}$$

$$dt_{shell}=dt\left(1-\frac{2Gm}{rc^2}\right)^{1/2}$$

$$v_{rel}=-\left(\frac{2Gm}{rc^2}\right)^{1/2}$$

$$\gamma \equiv \left(1-\frac{v^2}{c^2}\right)^{-1/2}=\left(1-\frac{2Gm}{rc^2}\right)^{-1/2}$$


Incorporating the above equations into the dt_rain equation, I get-

$$dt_{rain}=\left(\frac{2Gm}{rc^2}\right)^{1/2} dr\left(1-\frac{2Gm}{rc^2}\right)^{-1} + dt$$

which tallies with equation 14 on page B-13.


Based on a 3 sol mass black hole (R_s=8861.1 m), r=11,000, dr=1 and dt=1, I get the following-

$$dt_{rain}=(0.897527 \times 5.142830)+1$$

$$dt_{rain}=5.615829$$


The metric for the rain frame (on a radial line only) is-

$$d\tau^2=\left(1-\frac{2Gm}{rc^2}\right)dt_{rain}^2 - 2\left(\frac{2Gm}{rc^2}\right)^{1/2}dt_{rain}dr -dr^2$$

The results I get here where dr=1 and dt_rain=5.615829 are-

$$d\tau^2=(0.194445 \times 5.615829^2)-(1.795054 \times 5.615829)-1$$

$$d\tau^2=-4.948399$$


Obviously something is amiss here as dτ² shouldn't be negative outside the event horizon.

I have a hunch I'm about 95% there but there's a mistake I'm making somewhere. I'd appreciate any feedback regarding getting this right.

Steve

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UPDATE
I've applied the same black hole parameters and radius to standard Schwarzschild metric and I get (more or less) the same answer, dτ²=-4.948385. This is corrected by introducing c² to dτ² and dt² providing an answer for dτ of 0.440959 (in standard Schwarzschild metric) which makes more sense. This tells me I need to incorporate c² to the rain metric but when applied to both quantities of dt_rain, the results are still negative.
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[stevebd1]

Having looked at a number of rain metrics, I thought the Gullstrand-Painlevé coordinates were the most accessible (r_s=2Gm/c² and θ=φ=0).

rain coordinates-

$$dt_r=dt+\sqrt{\frac{r_s}{r}}\left(1+\frac{r_s}{r}\right)^{-1}dr$$

$$dr_r=\left(1+\frac{r_s}{r}\right)^{-1}dr+\sqrt{\frac{r_s}{r}}\ dt$$


free-fall rain frame-

$$c^2d\tau^2=c^2dt_r^2 - dr_r^2$$


Global rain frame-

$$c^2d\tau^2=\left(1-\frac{r_s}{r}\right)c^2dt_r^2 - 2\sqrt{\frac{r_s}{r}}\ cdt_rdr - dr^2$$


The above 2 metrics appear to produce reasonable quantities for dτ. The free-fall rain frame metric remains time-like all the way to the singularity, representing the local frame of the infalling object, while the global rain frame metric becomes space-like at r<r_s* with no geometric singularity at the event horizon. When introducing c² to dτ and dt_r in the global rain frame, it seemed reasonable to introduce c to $-2\sqrt{\frac{r_s}{r}}\ cdt_rdr$ as this appears to take into account relativistic velocity.

*Looking at increments very close to the event horizon (1-|r_s/r|=1x10^-9) in the global rain frame metric, it appears that the change over from time-like to space-like geodesics doesn't occur exactly on the event horizon but somewhere just outside it which doesn't seem right, even when removing c from the second term of dt_r, space-like geodesics technically occur outside the Schwarzschild radius.

Source- http://en.wikipedia.org/wiki/Gullstrand-Painlev%C3%A9_coordinates"

 

[stevebd1]

Having looked at other sources, it's pretty certain that c is introduced to the second term of dt_r. For anyone who might be interested, I found this regarding Gullstrand-Painlevé coordinates and the Kerr solution-

http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.0206v1.pdf

I'd still be interested to hear anyones opinion regarding the fact that it appears that space-like geodesics occur just outside the Schwarzschild radius for the global rain frame metric (while it's a fraction of a mm for a small black hole, it balloons to about 50 metres for a SM BH of about 3.7 million sol mass).