Lemaitre Coordinate from the E-F coordinate

[johnahn]

How can Eddington-Finkelstein coordinate be transformed into Lemaitre coordinate?

I know the transformation between the Lemaitre and Schwarzschild coordinate, and also between Eddington-Finkelstein and Schwarzschild coordinate.

So I tried to find the connection between Lemaitre and E-F coordinate from that, but it is very complicated..

Can you give me the explicit formula for that?

 

[Mentz114]

If L₁ takes LeMaitre basis to the Schwarzschild coordinate basis and L₂ takes SS to E.F., then does L₁L₂ not take LeMaitre to EF basis ?

Formally, if μ is the Schwarzschild coordinate index, a is a LeMaitre frame index and A is an EF frame index then
$$\begin{align}<br /> v^\mu &= \Lambda^\mu_a v^a\\<br /> v^A &= \Lambda^A_\mu v^\mu = \Lambda^A_\mu(\Lambda^\mu_a v^a) = (\Lambda^A_\mu\Lambda^\mu_a)v^a<br /> \end{align}$$