Geometry of transformed spacetimes?

Question

The main question seeks to understand whether a conformal structure can be put on "transformed" Minkowski 2-space, which will be denoted as $\Bbb R^{1,1}:=\Bbb M^{1,1}.$ I will get more specific about what I mean by "transformed."

Edit 2/28/2020:

As noted by the user @guiseppe in his answer below, it looks like the notion of conformal mapping is more proper and practical for relativity. To make this post more focused, I ask not about the geometry. I'm only interested in first mapping Minkowski space under $g,$ and then, finding out whether $N$ can be made conformal after the mapping $g.$

Main Question: How does one formalize the notion of mapping $\Bbb R^{1,1}:=\Bbb M^{1,1}$ under a nonlinear operator, $g,$ to (another) pseudo-euclidean space, $N,$ via conformal map?

$$f:\Bbb R^{1,1}\to N.$$

In other words the goal is to identify a conformal structure on $N,$ using the map $g$ defined further below.

Say $\Bbb R^{1,1}:=\Bbb M^{1,1}$ is Minkowski space. Would one map via $f$ first, and then put a geometry on $N,$ or first put a geometry on $\Bbb R^{1,1},$ and then map the geometry to $N?$ Does the order matter?

I'm not certain how the geometry would change from the pre-image space to the image space.

Here's an example I cooked up. Say you have a bi-Lipschitz nonlinear map $g,$ acting on all points $p\in\Bbb R^{1,1}$ and $p\in(u,v).$ You also have that $p'\in N$ and $p'\in (u',v').$ Define a map $g:(u,v)\mapsto(u',v')$ s.t. $u'=e^u$ and $v'=e^v.$ Note: I am considering a non-standard Minkowski diagram in which lines of constant time are rectangular hyperbolas.

Claim A: We have a correspondence between $\Bbb R^{1,1}$ and $N$ since $f$ is a diffeomorphism and since $g$ is bi-Lipschitz.

Claim B: We have a correspondence between spacetime geometries in the pre-image space and the image-space under $g.$ I think some requirements should be $1)$ strongly equivalent metrics and $2)$ compatible connections, to support such a correspondence.

I know that just thinking about the map $g$ in the layout of $\Bbb R^2$ lends one to conceptually think about the map $g$ as "contracting" some regions in the plane and "expanding" other regions in the plane. Qualitatively, Quad. III maps to $(0,1)^2,$ Quad. I maps to $x>1,y>1$, Quad. II maps to $0<x<1$, $y>1$ and finally Quad. IV maps to $0<y<1,x>1.$

Answer 1

I think that the problem with your question is that "diffeomorphism" is too weak a notion to be relevant to general relativity (GR).

In GR, a "spacetime" is a Lorentzian manifold, that is, a differentiable manifold equipped with some additional structure. You are pondering what transformations you can apply to a spacetime without destroying all this structure.

Now, a diffeomorphism is not enough; it does preserve the differentiable structure, but it needs not preserve the Lorentzian one. The "right" notion you are after is that of a conformal mapping of Lorentzian manifolds.

Answer 2

It appears that you are asking how to define the notion of conformal map for semi-Riemannian manifolds (of the Lorentzian signature $(1,1)$). Here is the general definition. Suppose you have two semi-Riemannian manifolds $(M,g), (N,h)$. A diffeomorphism $f: (M,g)\to (N,h)$ is called conformal if for every $p\in M$ there exists a scalar $\lambda(p)$ such that for all tangent vectors $u, v\in T_pM$, we have $$ h(df(u), df(v))= \lambda(p) g(u,v). $$

In particular, you get the notion of a conformal map between domains $M, N\subset R^{1,1}$. The theory is more complicated than in the classical complex analysis. For instance, there is no Lorentzian analogue of the Riemann mapping theorem.