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In particular, the time coordinate $t$ can be choosen so that the mathematical expression of the physical laws reflects their inherent symmetries. Already Newton's first law then fixes the time rate upto a constant multiplier (that is, up to a unit) to be such that equal spatial increments along a free path correspond to equal time increments. (Page-42, Rindler's Relativity)

I have two main questions from the above paragraph:

  1. Does the above paragraph imply that time is a derived quantity from motion of a particle?

  2. If the answer to above is yes, Suppose we have a body which is free of any force, and we are in a frame seeing it move at zero velocity. Does this mean time has stopped according to the above? Another observer would see it moving and hence say that time is flowing, but then this would lead to breaking the absolute time idea in Newtonian Mechanics

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"Time" is not a quantity at all, it is a coordinate similar to "north", "east", or "up" (or "x", "y", "z").

What Rindler is saying in the paragraph is that we may choose the time coordinate in a way that simplifies the equations of motion by exposing certain symmetries. We do the same thing with spatial coordinates all the time, e.g. for some problems it's easiest to work in polar coordinates, for other problems cylindrical coordinates are easier, and so on... also, we typically choose one of the spatial axes to point along a direction of motion or some other interesting property of the system. In a very similar way, when doing physics we typically choose the time coordinate so it reflects the proper time (time showing on a clock) of some element of the system, which is then deemed to be "at rest".

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  • $\begingroup$ Your answer helps but I feel like it was not focused to the exact doubt I had. Could you go through my highlighted points once more, I really think you have the knowledge of it's answer. $\endgroup$ May 21 at 13:05
  • $\begingroup$ I didn't see anything highlighted in your question... the answer to your point 1 is "no, time is not a derived quantity", and so point 2 is moot. You talk in your question about "time flowing" or "time stopping", which is really the wrong way to look at things. time doesn't move, things move through time (and space), and they do so at a fixed rate. You can vary how that rate is split between space and time, so moving more through space changes how fast something moves through time. $\endgroup$
    – Eric Smith
    May 21 at 16:00
  • $\begingroup$ I'll come back to this. Thanks for the answer. $\endgroup$ May 21 at 22:02

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