It's like claiming i broke physics because i drew my graph in red pen instead of a black one.

Its a lot more complicated that that

My transformation is twofold. Can you please help me understand which of these 2 steps you believe breaks curvature:

- convert everything to celestial coordinates (lat, long, distance)

This transformation is used in physics all the time and just expressing a sphere in celestial coords doesn't break distances or curvature. (the distance formula does of course get transformed along with the coordinates, just like when physics switches to/from celestial coords)

- Please have a look at the two graphs below. One in "polar coordinates", the other one in "cartesian coords".

These are both visualizations of the same function, In the "polar" visualization, the X axis is radial around the origin. In the "cartesian" rendering it's a straight line. Both graphs represent the same function and mathematically the functions curvature doesn't change, when plotted differently. From here stems the red/black pen analogy. FE/RE is just a representation of the same physics/mathematics.

The question is did you transform the metric correctly? ....

I believe so. I would use the same distance metric physics normally uses when switching to celestial coords.

Or more formally: distance_in_celest(p1, p2) = distance_in_cart(celest_to_cart(p1), celest_to_cart(p2))

So basically same formula after appyling the inverse transform.

I'm not proposing to do the celestial coordinate transformation any different than physics does it today.

When you *correctly* transform the metric when transforming from a spherical to a flat coordinate system, you get distortion, as explained here.

You're now sneakily going back to an orthonormal basis. When you take a non-orthonormal basis (like celestial coords) and start measuring it with a "straight" ruler, indeed nothing will match up. If i take my curved polar-coordinate-ruler to your globe, everything will be broken too.

You broke physics by making up your own rules for transforming the metric tensor. If you don't have any distortion in your model, you didn't transform the metric correctly and have changed the geometry.

Take the cartesian point (1,1). distance to the origin is √x²+y² or √2.

Express the point in polar coords: (45°, √2). The original distance formula is now invalid. The correct formula (in general) would be:

distance_in_polar(p) = distance_in_cart(polar_to_cart(p))

So in summary i agree with all the math you've presented. However the distortion only appears once you treat the transformed coordinates as orthonormal.

In my model you do indeed lose the possibility to measure distances with a "straight" ruler.