Hi, I am recently reading Weinberg's Cosmology, and getting subtle on Ch5, small fluctuation. One of the subtle point is on P.225-226 (same as (F.11) -> (F.13) and (F.14) in appendix F). The equations of motion (5.1.24)-(5.1.26) are decomposed into many parts. For example, (5.1.24) is decomposed into (5.1.44), (5.1.45), (5.1.50) and (5.1.53). I found this is very common in cosmological perturbation, I google many times and read the G&C by weinberg, they usually directly decouple the equations into scalar mode and vector mode according to different parts in vector/tensor decomposition. In my understanding, I don't think (5.1.24) does imply these four equations. I wonder if this is an assumption because these four equations imply (5.1.24), but it seems there should be some meaningful stuff in between (Weinberg wrote: The solutions of Eqs. (F.10)–(F.12) can be classified according to the transformation properties of the dependent variables under three-dimensional rotations). So how do the properties help in decoupling the equation?