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In general, when I think of movement through space, I think of this:

$$\frac{dx}{dt}$$

But in special relativity, we also have a concept of relative duration, which means that $t$ must have a rate of change, but with respect to what?

$$\frac{dt}{d?}$$

user912
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This is a near duplicate of What is the speed of time? and Terry has given a comprehensive answer there. However there is one point that wasn't made in the previous question.

In special and general relativity there is an invarient called proper time, $\tau$, which is is the time measured by a freely moving observer, and it is perfectly reasonable to ask what $dt/d\tau$ is provided you're clear what you mean by $t$. For a freely moving observer $dt/d\tau$ is always one, but this won't be so for other observers. For example if we watch someone falling into a black hole we will see their time slow as they approach the event horizon. So if by $t$ we mean our (Schwarzschild) co-ordinate time then $dt/d\tau$ is not unity.

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John Rennie
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  • Then what is the dimension through which we move when we get closer to tomorrow and farther from yesterday? – user912 Feb 11 '13 at 07:49
  • @user912: questions about the experience of moving through time tend to verge on the philosophical and therefore have no answer. For physicists time is a co-ordinate just like $x$, $y$ and $z$. How/why we move through time is probably more to do with the way the brain works than with physics. Be cautious about the point made in my answer because co-ordinate time is a specific concept in physics that probably doesn't have much meaning for non-physicists or may even be misleading. – John Rennie Feb 11 '13 at 08:06
  • Would I be right in saying that coordinate time is $\sqrt{t^2 - x^2 - y^2 - z^2}$? – user912 Feb 11 '13 at 08:13
  • No, that's proper time, $\tau$, or more precisely $d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2$ (in units where $c$ is 1). Co-ordinate time is the time measured in a particular set of co-ordinates, e.g. the Schwarzschild co-ordinates. Co-ordinate time is in general not equal to proper time, though just to confuse matters it can be if you use co-moving co-ordinates. – John Rennie Feb 11 '13 at 08:17
  • @user912: if you're interested in the difference between co-ordinate time and proper time there is an interesting question to be asked there ... – John Rennie Feb 11 '13 at 09:24