FlyingLemons wrote:
Crazy Eddie wrote:
You wouldn't need to treat a curved space as if it was flat, you'd just need a mathematically consistent way to account for that curvature itself. Which is, like, stupefyingly difficult, but hardly impossible

The Minkowski metric is the very definition of flat.

Yes, I know all that. All I've said is that the solutions for Lorentz transformations could be modified to account for anomalies in the actual coordinate system, provided you can quantify them in a mathematically consistent way.
A better way to conceptualize it would be, say, a superposition of two different transformations that treats the curvature of space as an inherent vector and thus becomes the coordinate system for for the observer's relative motion.
"Minkowski space with curvature" isn't Minkowski space but something else.

Minkowsky space assumes the flatness of space even when it demonstrably isn't. You could still account for that curvature even if the metric doesn't explicitly reflect its presence.
"Special relativity is just an approximation"  where's your experimental evidence for this?

The fact that Special Relativity assumes a locally flat spacetime even when when evaluating the behavior of objects moving within gravitational fields that are themselves nonuniform.
It's therefore an approximation, like Kepler's Laws or Newtonian gravity. It's a very
good approximation, but it explicitly avoids dealing with gravitational effects by assuming their effect is negligible  which is true on the small scale  despite the fact that those effects are ubiquitous in the universe. It works well enough on the small scale, but that means we lack a way to apply special relativity to very large objects (galaxies and stars, for instance) whose relative velocity is very high and whose gravitational fields cannot be ignored.
here you're saying that a flat space metric is applicable to curved spaces which kind of flies in the face of geometry.

I don't see how. It's relatively simple, geometricaly, to project a curved surface onto a flat one. Why should Minkowski space be any different?
The distance measures of Minkowski space when applied to curved space will not give correct results as they would fail to take into account the properties of that space  (1,1,1,1) would give wrong answers when mapping numbers to the manifold surrounding, say, a star or other spherically symmetric body.

And yet you say it gives correct answers when calculating transformations on GPS satellites in Earth orbit...
If you can put forth a substantial body of experimental data to show that equations of motion in curved spaces can be found using the Minkowski metric, I'd accept that as a fact.

If I could do that, we wouldn't be having this conversation, because you'd be reading about it in a newspaper and I'd be filthy rich.