I've been thinking about compact spactimes lately. My understanding of the Yamabe problem is that one can always conformally transform a (compact) spacetime to one of constant scalar curvature, something like:
If we narrow the field a bit to conformally flat spacetimes, we can also say that we can write our metric as a conformal transformation of the Minkowski metric, something like:
Here's what I've been wondering about. Supposing the spacetime is globally conformally flat, we can write the above no problem. Now though we want to add localized distortions which preserve the 4-volume of the spacetime in question. I know that such transformations (volume preserving) are called the special affine group , which for 4 dimensions is $SL(4,R)$. In this case we have something like:
Where of course $\Lambda(x)\epsilon SL(4,R)$. Here's where I'm not sure....$SL(4,R)$ is isomorphic (I believe) to $SL(2,C)$ which I understand is the double cover of the (restricted) Poincare group (please correct me if I'm wrong).
So basically my thinking was that I could represent a conformally flat spacetime (say general FLRW universe for example) as a composition of the Minkowski metric with a conformal transformation (representing the specific FLRW shape and volume at a givent cosmological time and it's growing/contracing with time) along with an $SL(2,C)$ Poincare transformation representing all deviations from homogeneity.
Is this do-able? What I like is that all the matter and fields en-total contribute to a globalized conformal term, while localized metric distortions (via local matter/fields) are contained within the $SL(2,C)$ term. (FYI I'm using a spinor formalism, probably Newman-Penrose when all is said and done). When I go from $SL(4,R)$ to $SL(2,C)$ maybe I have to hit the metric two sided with the Lambdas? Sorry it's late and I was curious.