"...if we accept that the Earth rotates and moves and such, doesn't time travel also necessarily involve space travel? I mean, "a fixed point in space" is always relative. You walk into a "gateway" on a hill in Buffalo in 2012AD. Why would you expect to exit on a hill in Buffalo in 600BC (or whenever) and not be left out screaming in the "fixed point in space" where the earth is no longer at? Or caught in solid rock or some such?"
Of course John is right. I think, though, the problem is just semantic. Where I used the word space, I should have used the more accurate term space-time. To stick with his above example, Let's instead say that the gateway connected two points, a current location and a destination; both of which exist in space-time. It isn't enough to say the location is a hill in Buffalo. A hill in Buffalo in 2012AD is a totally different location in space-time than a hill in Buffalo in 600BC. Typically one would think of the hill in Buffalo as the location and then there is the matter of when (600BC vs. 2012AD), but, as John points out, this model doesn't seem to make sense for time travel as we want it. We have to imagine the fourth dimension as being simultaneous, not sequential; A hill in Buffalo in 2012AD and a hill in Buffalo in 600BC both exist along side each other. It helps me, when thinking about such things, to reduce all the dimensions. See the below diagram:
As a footnote, we can infer by this method of thought that if time is to be bent in such a way, there must be a dimension above time for it to be bent into. The two zero-dimensional points on the one-dimensional line are brought closer to each other without affecting other theoretical points along the same line (relative to the line) by using the second dimension. So if the line is time, then the area it is bent into would be a fifth dimension. Thoughts?