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Let's consider the equation y=x in the x-y rectangular Cartesian frame in flat space time. We use the transformations in the first quadrant: $$y=y'^2$$ $$x=x'$$ $$t=t'$$ For the first transformation we are taking the positive values of $y'$ only. The equation of y=x in the transformed condition:

$$y'^2=x'$$ If the equation of the straight line:y=x in the original frame represents the path of a light ray in the flat space-time context , the corresponding path in the transformed frame is a parabola.

Options: 1.We may treat the second frame as just a mathematical workspace. 2. We can always think of some manifold in the physical sense where the null geodesic is parabolic!

It is also important to observe that the metrics in both the spaces represent orthogonal systems.One could of course think of going from the first space to the second via some intermediate non-orthogonal transformation.The value of the line element $ds^2$ does not change in the transformation used.

In view of the above considerations can we claim that we can pass from one manifold to another of a different type where the value of where $ds^2$ is preserved?

NB: It would be better to write $x=x’$ and $y=Ay’^2$: where A has the dimension $\frac{1}{Length}$ and y’ the dimension of length. The transformed eqn: $x’=Ay’^2$,y’ maintaining the dimension of length. Transformed metric is $ds^2=dt’^2-dx’^2-(2Ay’dy’)^2=dt’^2-dx’^2-4A^2y'^2dy’^2$ . Ay’ is a dimensionless quantity here.

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Your coordinate transformation is double-valued, and the line y=0 is singular, the transformation turns around there. This is the reason that you see turning around. When we say coordinate transformations are allowed, they are restricted to be 1-1 and nonsingular Jacobian, so that they are differentiable and differentiably invertible.

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