In the instance of the GPS, there's dilation due to both the effects of special and general relativity, as confirmed by the Hafele-Keating experiment among others.
Not much time to dither this week, but I'll leave with this:
I've read about a number of experiments using clock times from satellites to test for relativistic effects of time dilation. Reflections on those experiments is a big part of what got me going on this line of thought, since the conclusion -- or so it very strong appeared to me -- did not follow from the results.
Looking a the numbers myself, and looking at the general logic and mathematics behind Special Relativity itself, I concluded that the Lorentz transformation itself is sufficient to account for the invariance of the speed of light, where dilation appears to occur in space mainly because of the shortening of moving objects.
That is, when a fast-moving object is contracted longitudinally due to its high velocity, the relative distance between that object and the observer likewise appears dilated consistent with the frequency doppler shift. The net result is almost the same -- in GPS satellites, a very small increase in pseudorange -- but the time between clocks is otherwise unchanged. The main reason I believe this is because relativistic equations for time dilation have some odd and contradictory implications if you treat time as a variable between observers; from both points of view it is equally valid to speak of the other's clock as running behind, since in both reference frames each observes the OTHER clock as the one in motion. This contradiction disappears when you treat DISTANCE as the variable, wherein each one measures the same distance to the moving object but has no way of knowing that the moving object -- and his perception of distance -- is distorted.
I am less sure about the implications for General Relativity but it seems to be the case that the same effect occurs: the physical distortion of objects in lower gravitational potentials necessarily distorts the physical distances between two objects, which IMO would be more consistent with things like the Oberth Effect (space is literally smaller in lower gravitational potentials and therefore the same impulse results in a larger displacement).
The reason I figured Minkowski space would work in the presence of strong gravity is that you could still treat Minkowski space as being effectively flat and account for the curvature of space by mapping the distortions onto the various objects within that space time. Put that another way: space ALWAYS appears flat in your own reference frame, and the only reason you know it's curved is because other objects farther from the origin are behaving strangely. A Minkowski spacetime applied to strong gravity would add an additional factor that would inherently distort all objects in given coordinate system as if they were being stretched longitudinally in a particular direction. Without running the numbers I am not sure what this would change about the nature of gravity on a cosmic scale applying to galaxies or clusters of galaxies, except to say that the physical contraction of space around a gravitating object could mean that our measurements of the SIZES of those galaxies is in error.
Crazy Eddie, you remind me of every eager grad student I've met, thinking they can tackle every major problem in their field, until reality sets in. Most research is incremental not revolutionary. Trying to redefine major components of relativity strikes me as a bit much for any one person to tackle.
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