# Are all spacetimes locally conformally flat?

No, is the answer.

However, I am confused.

Let $M$ be a (2+1) Lorentzian manifold (for simplicity) . Then the line element is given by :

$ds^{2}=g_{\mu\nu}dx^\mu dx^\nu=−N^2 dt^2 + γ^{ij} (dx^i + β^idt)(dx^j + β^jdt)$

where $N$ is the lapse, $\beta$ is the shift and $γ^{ij}$ is the induced metric in the leaves of the foliation.

Because the spatial part is $2$ dimensional there exist isothermal coordinates. which I think led to a decomposition of the line element the form:

$ds^{2}=−N^2 dt^2 + \Omega^2 (dx^i β^jdt)+\Omega^2 (dx^j dx^j)$

What are the conditions in general that forbid locally (and globally) a choice of $N=\Omega$ and $β^j=0$?

The answer I believe is the vanishing of the Cotton tensor (in the $2+1$ case). However, I would like an intuitive physical picture of what is going on and if possible a calculation of why the Cotton tensor must vanish.

Is this a restriction on the congruence of causal geodesics and therefore free falling observers?

If $N$ and $\beta^{i}$ encode our choice of coordinates why can't they be chosen freely? If this related to the existence of suitable coordinates, how is then the existence of certain coordinates be interpreted in GR?

How is the local case different from the global? Does this have to do with simultaneity and synchronization in the whole spacetime? How?