I have two 2-dimensional space-times ($\mathbb{S}^1\times\mathbb{R}$) with signature $(-,+)$. One of them is flat the other one has non-vanishing curvature (Riemann tensor), both have vanishing Ricci tensor. But they seem to have a similar global and causal structure. Of course, because of the 2-dimensional case they are local conformally flat.
I am looking for a global relation between them that could explain the similar causal and global structure and I think that a (global) conformal transformation would be a possible approach.
********************Edit in response to comments*******************
The two metrics in question are $$ ds^{2}=Td\psi^{2}+2dTd\psi \quad\text{ defined on }\quad S^{1}\times\mathbb{R} $$ and $$ ds^{2}=-(\frac{2m}{r}-1)d\nu^{2}+2d\nu dr \quad\text{ defined on }\quad S^{1}\times(0,\infty). $$
Note that $\psi $ and $\nu $ are the according periodic variables.
I already could calculate the local conformal relation. But what about the global relation?
The two metrics in question are $$ ds^{2}=Td\psi^{2}+2dTd\psi \text{ defined on } S^{1}\times\mathbb{R} $$ and $$ ds^{2}=-(\frac{2m}{r}-1)d\nu^{2}+2d\nu dr \text{ defined on } S^{1}\times(0,\infty). $$ Note that $\psi $ and $\nu $ are the according periodic variables.
I already could calculate the local conformal relation. But what about the global relation?