# Can I decompose a compact spacetime into conformal and $SL(2,C)$ transformations?

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:

$$g_{\mu\nu}=\tilde{g}_{\mu\nu}e^{\phi(x)}$$

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:

$$g_{\mu\nu}=\eta_{\mu\nu}e^{\phi(x)}$$

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:

$$g_{\mu\nu}=\Lambda(x)\eta_{\mu\nu}e^{\phi(x)}$$

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.

• FWIW, the Lie group $SL(4,\mathbb{R})$ has 15 real dimensions, while $SL(2,\mathbb{C})$ only has 6 real dimensions. – Qmechanic Mar 3 '19 at 13:05
• @Qmechanic Gotcha, so it would be an have to still be an $SL(4,R)$ transformation. I'll try and look up what geometric constaints correspond to going from one to the other to get a better grasp of such a condition – R. Rankin Mar 3 '19 at 13:22
• There are “Ashtekar variables” for gravity that have $SL(2,\mathbb{C})$ connection. But those are used for canonical formulation of general relativity. – A.V.S. Mar 3 '19 at 14:07
• @Qmechanic it occurs to me that a volume preserving transformation propagating into a space (say around a planet moving) must also preserve causality, this would then be the additional constraint I believe to reduce the degrees of freedom you mentioned. Thoughts??? Again I'll have to confirm it, but it does make sense at first glance considering local lorentze transformations do exactly that – R. Rankin Aug 3 '19 at 2:02