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(adsbygoogle = window.adsbygoogle || []).push({}); _{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-

[tex]dt_{rain}=-v_{rel}\gamma dr_{shell}+\gamma dt_{shell}[/tex]

where-

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

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

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

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

Incorporating the above equations into the dt_{rain}equation, I get-

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

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-

[tex]dt_{rain}=(0.897527 \times 5.142830)+1[/tex]

[tex]dt_{rain}=5.615829[/tex]

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

[tex]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[/tex]

The results I get here where dr=1 and dt_{rain}=5.615829 are-

[tex]d\tau^2=(0.194445 \times 5.615829^2)-(1.795054 \times 5.615829)-1[/tex]

[tex]d\tau^2=-4.948399[/tex]

Obviously something is amiss here as dτ^{2}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, dτ^{2}=-4.948385. This is corrected by introducing c^{2}to dτ^{2}and dt^{2}providing an answer for dτ of 0.440959 (in standard Schwarzschild metric) which makes more sense. This tells me I need to incorporate c^{2}to the rain metric but when applied to both quantities of dt_{rain}, the results are still negative.

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# Time coordinate in the rain frame

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