Proof of/Reason for SVT Decomposition

[AuraCrystal]

Using the conventions of http://www.damtp.cam.ac.uk/user/db275/Cosmology/Chapter4.pdf (not mine).

For a flat FRW perturbed universe, the metric is can be written in general as:
ds^2=a^2(\tau)[(1+2A)dt^2-B_idtdx^i-(\delta_{ij}+h_{ij})dx^idx^j]
I understand intuitively that we can decompose B_i into two parts:
B_i=B_i^{\perp}+B_i^{\parallel}
with
\nabla \cdot B^{\perp}=\nabla \times B^{\parallel}=0
In Fourier space, this means that we decompose the vector into two parts: one parallel to the wavevector k, and one perpendicular.

(And of course, we can write a curl-less vector as a gradient of a scalar.)

He then writes down a similar decomposition for the tensor h_ij (eq. 4.2.35-4.2.37). What's the reason/justification for the form of that?

 

[WannabeNewton]

Baumann explains it much better in Appendix A, section A.2, of the TASI lectures on inflation than he does in his notes you linked above: http://arxiv.org/pdf/0907.5424v2.pdf