Motivation: If you take a metric space which is also an analytic manifold, embed it in Minkowski $3$-space and perform a linear map (squeeze map/lorentz boost) i.e. $(ax,y/a)$ in $2D$ for real parameter $a,$ how do you describe the continuous deformation of the metric?
Let $\zeta^3=\Bbb R_{\gt0} \times \Bbb R_{\gt0} \times \Bbb R_{\gt0},$ with metric $ds^2=\frac{du^2}{u^2}+\frac{dv^2}{v^2}+\frac{dw^2}{w^2}.$ Consider $R=\zeta^3 \big/\{X\}$ where $X$ is the region not including $(0,1)^3.$ In other words delete all elements in $\zeta^3$ except for $(0,1)^3.$ Embed $f_i:R_i \to \Bbb R^3$ and tile $\Bbb R^3$ with the $R_i,$ for $i=1,2,3,\cdot\cdot\cdot$ s.t. every tile has the same orientation. Now generalize $\Bbb R^3,$ and let $\Bbb R^{2,1}$ be a semi-Riemannian space with metric $ds^2=dxdy-dt^2.$ Take a Lorentz boost without rotation. This transformation warps the tiles (cubes) prescribed by the boost and therefore the geometry warps at different rates due to each tile being equipped with its metric.
Do the entire collection of flat metrics of $\zeta^3$ tiling $\Bbb R^{2,1}$ remain flat under this transformation? How do we measure the metric decay of the copies of $\zeta^3.$ Can we achieve this using parameter(s) $A,B,C$ in the metric such as $ds^2=A\frac{du^2}{u^2}+B\frac{dv^2}{v^2}+C\frac{dw^2}{w^2}$ where specific values of $A,B,C$ gauge the geometry at a specific time during the transformation?
My attempt/thoughts in 2D:
A boost/squeeze mapping is a linear map (continuous), on Minkowski 2-space and the metric space is stretched from the initial metric on $(0,1)^2$ to some final metric. Because this special linear map preserves areas in $\Bbb R^2$ and the embedded $\zeta^2$ is Euclidean there must be some preservation of the "total area" (even though the metric of $\zeta^2$ implies infinite area). I would think that the specific type of infinity (cardinality) should be preserved i.e the map should preserve the cardinality.