Does time dilation occur when an object is moving at a constant velocity and not accerating?
yes. If fact the math of Special Relativity only deals with constant velocity.
And please remember that you are always talking about time dilation relative to another frame of reference.
Since it deals with accelerated motion can something be at "true rest"? By that I mean at rest relative to spacetime itself? I was reading about Newton's bucket in Brian Greene's book and he was talking about how the absolute benchmark for verifying whether or not something is accelerating is with respect to spacetime as a whole. Does this translate to constant velocity and/or rest?
If the "speed" at which we move through time is (c-motion) and so the more we move through space the less we move through time, doesn't that mean that there really is "true rest" at which is when we are fully moving through time? (Relative to spacetime as a whole.)
(I distinctly remember this in his book because of it's elegance, but if I am getting this wrong please feel free to correct me; I'd rather be embarassed at my lack of knowledge then confident and know nothing. Also I'm aware that the "c-motion" is surely a drastic oversimplification of the math involved.)
When Greene talks about speed through space vs. speed through time, it's not meant to be absolute, just relative to a particular inertial frame of reference. If you want to see the math behind his definitions (which are not widely used, you don't generally see relativity explained this way in textbooks), have a look at this thread. On the other hand, acceleration is absolute in SR, but this doesn't imply absolute velocity--a comparison might be to curves drawn on a piece of paper, there is an absolute truth about whether a given curve is a straight line or not (and inertial motion means a straight worldline, while acceleration means a non-straight one), but no absolute truth about the slope of a given line, since that depends how you orient your coordinate axes.
The way you phrase it, you are referring to some absolute rest frame --- you call it "spacetime as a whole". Mentioning an absolute rest frame is generally frowned upon in the context of special relativity. The main reason is that this absolute rest frame is not detectable by any means. Consequently it is a superflous construct to be slashed by Occam's Razor.
Sometimes you may hear that the absolute rest frame is mathematically incompatible with special relativity. This is not true, and a nice description of how both fit together can be found in a recent book by Guerra and deAbreu.
If you want to keep with the mainstream interpretation, you should say goodbye to the idea of an absolute rest frame. Then "true rest" is a meaningless concept. If you feel it helps to give your head a rest, imagine the absolute rest frame. In this context then, yes, there is a trade-off between moving relative to this absolute rest frame and getting older --- the faster you move the slower you age. But keep in mind that discussing special relativity as if there were an absolute frame of reference will usually deteriorate the discussion.
This is a common myth. SR can describe acceleration, as long as it is caused by a four-force rather than by gravity. You can describe accelerated paths in gravityless spacetime such as done here.
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