Crazy Eddie wrote:
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.

No, you just use a metric more suited to the property of that space.
Crazy Eddie wrote:
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.

You can create a tangent space which appears locally flat, but not globally in strong gravity. If the universe was a purely weak gravity environment then "Minkowski everywhere" works, otherwise no.
Crazy Eddie wrote:
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.

That's why it's special relativity  a special case for locally flat spacetimes.
Crazy Eddie wrote:
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.

That's when special relativity is generalized to general relativity and more complicated math becomes involved. When dealing with galaxies and cosmological scales, we use the
FRW metric. When dealing with stars and other spherically symmetric bodies, we use the
Schwarzchild metric. The cases you talk about have very wellknown relativistic solutions backed up by experiment and observation.
We've got an entire mathematical apparatus to describe motion in curved spaces  Christoffel symbols, the Ricci tensor being two examples. They fit experimental data, and to boot you can retrieve the special relativistic and Newtonian cases from them. It's not "so mathematical it can't be checked", either. Many physics students, or indeed anyone, can develop a basic understanding with a little effort.
Crazy Eddie wrote:
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 Minkowski line element is x^2 + y^2 + z^2, and that's it. Try applying that metric to a strong gravitational field and your equations of motion will be nonsense because the geometry of space won't be described by it.
Crazy Eddie wrote:
And yet you say it gives correct answers when calculating transformations on GPS satellites in Earth orbit...

That's because it's not applying the Minkowski metric but instead an approximation of the Schwarzchild metric, the one you use for a spherically symmetric body such as the Earth.
Did you actually take a class in general relativity and cosmology at college? It sounds as if you're sort of unaware of the existence of metrics which perfectly solve the problems (motion on a cosmological scale, gravity surrounding spherically symmetric objects) that Minkowski can't handle that you describe.
iguana_tonante wrote:
]I am especially intrigued when people say that I am paid good money by the CIA, the Mossad, the Illuminati, the Vatican, and the Mafia to suppress the truth. I wish. I wish.

Remember the song we sang at the last Physics Conspiracy Meeting? 'Twas glorious.