lalbatros said:
I also doubt about your polar coordinates example.
The time coordinate could cover the range [-Inf,+Inf] even when describing irreversible physics.
Irreversibility in time has to do with how the system may change when it evolves from t1 to t2>t1.
I don't see any analogue for a spatial coordinate.
Well ... maybe I have an idea ...
I think this might help
Consider a 2d universe, as a piece of paper. On this page there is an object, say a square with a non symmetrical pattern of polka dots along it's edges. within the universe you can plot the square's location only if you use 3 values, it's X and Y position, and a rotation around an axis perpendicular to the page (that is Z.)
You might object that the rotation's magnitude isn't necessary for observers within the page (say the triangle guy some ways off) but it is critical if you really think about it.
A 2d person would be very confused about what this third axis was exactly since cannot experience translations along it but only rotations, where as for the other two dimensions they see translations but not rotations. (note that to the triangle if you move the square along Z it disappears from the universe. If you rotate the square along X or Y, the square either becomes a point or just changes texture, depending on the exact positions of the two shapes. A texture change in this case would be like your organs suddenly becoming visible to the person across the room.)
If you were to make a book by taking snapshots of each page from a higher (or lower) Z and stacking them upon one another, the book as a whole would describe the history of the universe. This does not mean that such a book "must" exist in order for the 2d universe to exist and go about it's business. Travel (that is translation) along this axis does not have to occur for this 2d world to have the symptoms of time. But were a very smart triangle to come up with an intellectual concept that allowed him to make such books in it's mind, the resulting tool would be very useful.
By analogy, our 3 dimensional history could be recorded in a 4 dimensional book without the universe "being" such a 4d object (in fact a list of positions and times for a moving object is a projection of such a 4d book.) This may be why space and time aren't interchangeable. If time is this type of abstraction we would see exactly what we do see. This would also explain why, while we can talk about time coordinates in a way that is meaningful, we can't necessarily traverse time the way we do space. (note certain statements made about relativity make this seem weird. I'm not qualified to say whether they make it in fact weird. Maybe I should make a topic about that discussion. Could be fun.)
We make books like this all the time, calenders as a simple example.
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When you choose a point to make a measurement, or start a simulation you have to record which page you're talking about. That's the minimum that the time coord represents.
Is our universe book like? Or do things actually move along our "page?" For physics generally it doesn't matter, that's more of a cosmology question.
Many of these ideas are presented in a fun way in a book called "Flatland: A Romance of Many Dimensions" by Edwin A. Abbotte. (maybe all of them, I can't tell how much I've elaborated on it since I last read it years ago)
... note that this could be further discussed if one considered position and momentum as coordinates and momentum as a time-independent concept ...