"Earth's mass even warps what is considered "space time". "Minkowskis spacetime manifold is the most generally convenient for discussing Einsteins relativity.
In my previous post, (which I then deleted) I stated that of the 4 "dimensions" time is a bit out of place. The 3 Euclidean dimensions form an orthonormal basis for 3-d space. They are each independent of the other, and they are orthogonal to each other.
Time on the other hand is not naturally orthogonal to the euclideans. It actually appears to run parallel to each of the euclidean dimensions. (does this mean it is 3-dimensional itself+?...no, but...)
Minkopwski made a 4-d orthnormal basis but to do this he had to interpret time as an imaginary component of the basis, 
and orient it negatively to the other 3 vectors (so the magnitude could be calculated correctly I suppose)
Interestingly, by multiplying the "time" t by c, the speed of light, he actually converts it to a spatial dimension, not a time dimension, though the t is the variable, not the c. hhhmmmm....
This certainly constructs a consistent spacetime geodesic, but it is certainly a bit fudged. ;^)
I am interested in why? What is the physical implication of ict? What is the implication of c defining an independent dimension. What if c is actually the variable and t is a fixed scalar ratio?
I'll say no more...for now.
cheers.
You wrote: "What if c is actually the variable and t is a fixed scalar ratio?" An interesting question.
It "can" be in quantum theory, which is one of the myriad of reasons the relavitistic universe and the quantum universe have been so hard for theoritical physcists to mathematically relate. Ultimately, it has come down to the simple question of "what makes up spacetime (i.e. the fabric of space itself?"
It is at the ultramicroscopic level, peering into extra-dimensional space...moving from 4D to 11D, that allows c to vary. (See Paul Dirac's annoted series of lecture notes plus a whole host of books out there.)
Also, as a side note, you bring up the basis for which led Einstein to believe Time Travel to be plausible.
Euclidean geometry is done on z axis format in describing where an object is...but not "when" it is...that is where the fourth dimension, time, came in right? I can say meet me at the 3rd street (x), at the corner of 44th (y), on the 11th floor of the chrysler building coffee bar (z), but if I dont give you a time...we will never meet, right? So, I have to also tell you 11:00 A.M. (t), and that is the fourth dimension, honestly I am not so sure c defines an independent dimension, as I understand it time is considered a spatial dimension (i.e. in regard to work done in particle physics) but in general as well. A thought.
Note: In this simplistic example (t) can very, but X,Y,Z are thought of as fixed points. Still, you needed the added dimension to get where you are going.
You ask some good questions.
I'll email you the VSL(Variable speed of light) equation, if you are interested...it isn't mine but it's still cool. Ha!