Lorrentz Transformation in Rain Metric Derivation In order to derive the rain coordinate system for a falling body in a black hole, we use Lorrentz Transformation of time instead of time dilation formula to go from shell time to rain time (Exploring Black Holes, Chapter 7). Why is that?
We are calculating time difference between 2 events, we should have used the time dilation formula that gets us time difference (amount of time passes in a rain clock between two ticks in a shell clock). But here we are using Lorrentz Time Transformation that gets us the exact time in a rain clock when the shell clock reads dt_shell, and because of using LT we get an unnecessary time offset, i.e, when dt_shell =0 dt_rain may not be zero which makes no sense.
 A: I'm not familiar with this book, but I looked at the draft second edition which is free at Taylor's web site here.
Section 7.3, equation (15) is
$$Δt_\text{rain} = γ_\text{rel}(Δt_\text{shell} - v_\text{rel} Δy_\text{shell})$$
I think your question is why the right hand side isn't just $γ_\text{rel} Δt_\text{shell}$. The reason is that the usual time dilation formula $Δt' = γΔt$ is, in effect, the Lorentz transformation equation with $Δx=0$. But here, the stone is stationary in the primed (rain) frame and moving in the unprimed (shell) frame, not the other way around. Also, $Δt, Δx, Δy$ here describe the motion of an arbitrary object, not necessarily the stone. ($v_\text{rel}$ is always the speed of the stone.)
If $Δt, Δx, Δy$ do describe the motion of the stone, then it would be correct to write $Δt_\text{shell} = γ_\text{rel} Δt_\text{rain}$ (note the reversal of the shell and rain labels), and you could derive the same result from that (using the fact that $v_\text{rel} = Δy_\text{shell}/Δt_\text{shell}$). You could also start with $Δt_\text{rain} = \sqrt{Δt_\text{shell}^2 - Δy_\text{shell}^2}$.
When $Δt=0$ then $Δy=0$ also (the object can't have moved, unless it's tachyonic), so their equation does give $Δt_\text{rain}=0$.
This derivation of Gullstrand-Painlevé coordinates is unique to this book (or at least unusual), and I find it confusing, in part because $Δt_\text{rain}$, despite its name, is not the Gullstrand-Painlevé "rain" time coordinate, which they later call $T$. The usual approach would be to just write $T=t+f(r)$, where $t$ and $r$ are the Schwarzschild coordinates, and solve for $f$ subject to the constraint that $T$ is the proper time of the stones. In this approach you never define any Lorentzian "shell" or "rain" frames.
