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It's a statement I've seen thrown around here and there. Essentially, under relativity, everything can be said to move at c, some part of it is motion through space, some of it through time, and when we speed up or slow down we are merely exchanging one motion for another. However, I never understood what is meant by "speed" or "motion" here.

Specifically what confuses me is the idea of "motion through time". "Speed" to me normally means the rate in which an object varies in space over time, and to talk about something "moving" we talk about it changing it's position with respect to time. But what does it mean when we say that an object "moves" through time (or space-time) or has a "speed"? Speed with respect to what? And what would it mean for something to move through spacetime at a different rate?

Right now, the only way I can make sense of the idea of a speed through spacetime is by introducing some sort of meta-time or hypertime, with which we can talk about movement through space-time, but that sounds a bit extreme. Can someone help explain it for me? Hope this all sounds clear enough for you guys. Thank you in advance.

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    $\begingroup$ Possible duplicate of Why are objects at rest in motion through spacetime at the speed of light? $\endgroup$
    – Styg
    Commented Dec 10, 2017 at 14:33
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    $\begingroup$ @Samarth, I don't think my question is the same as the one you linked as the person in that question seems to be asking why the speed through spacetime is a particular value, while I am asking what it even means to talk about a speed through spacetime. Apart from that, some of the answers do seem relevant to my question, so I'll check them out. From the looks of it, it seems like some of them state that the description of "movement through time" or having a "speed through spacetime" is inaccurate, is that right? $\endgroup$
    – Mike
    Commented Dec 10, 2017 at 15:05
  • $\begingroup$ Define speed. Otherwise one might take it to be dx/dt. $\endgroup$
    – jjack
    Commented Dec 10, 2017 at 15:11
  • $\begingroup$ @jjack, well, that's the whole the point of my question. I don't know what it even means when we talk about an object having a "speed" through space-time and I want to ask you guys about it. $\endgroup$
    – Mike
    Commented Dec 10, 2017 at 15:21
  • $\begingroup$ @Mike I'm someone who would define a distance measure on spacetime with respect to the Big Bang. But on this site I've heard that all spacetime models are defined as differentiable manifolds without origin, so there is no Big Bang event in them. I would however take the derivative with respect to that metric (relative to the Big Bang event) of position in spacetime and define this as velocity in spacetime. But then, I've recently heard, that there are physicists who question the "past" existence of a Big Bang. However I haven't read anything about those people's research yet. $\endgroup$
    – jjack
    Commented Dec 10, 2017 at 15:36

2 Answers 2

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Suppose an object moves at $\beta$ times the speed of light in your reference frame. Then in an infinitesimal time $dt$ you see the object move a distance $c\beta dt$, so $$ds^2:=c^2dt^2-d\mathbf{x}^2=c^2(1-\beta^2)dt^2=c^2d\tau^2,$$ with proper time $\tau:=\int\sqrt{1-\beta^2}dt$. The "speed through spacetime" is $ds/d\tau=c$. Equivalently, our distance-to-time measure can be $\dfrac{\sqrt{ds^2+d\mathbf{x}^2}}{dt}=c$.

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You must understand that time just another spatial coordinate in relativity. Even if two protons have zero relative velocity, and even for an external observer, they still move along the time axis with the speed of light. From this point of view, everything moves along the time axis with the same speed, c. This is why you have time multiplied by the speed of light as a coordinate in the definition of distance in general relativity. And this is why you'll always have the mass-energy formula for objects with rest mass. I can give you an example of motion along the time axis: whenever a particle suffers a change in it's state, that is an evolution along the time axis. If anything describes evolution along the time axis, that is change itself and also entropy when speaking of more complex systems. How can we have a speed along the time axis? Well, some systems undergo faster transitions, some systems do that slower or even faster, so yes, there is a speed along the time axis. It is the rate of changes happening when compared to another system. Objects do not move only in the 3D system, they also move in their internal phase space which gives them what is called the proper time.

If I can relate change to a variable, then that change is a motion along the axis of that variable. It is mathematically correct. However, we both agree that time emerges from change, but try not to remain so enclosed in these definitions. If you'll accept change as being a motion, you'll make a big step ahead. It is called abstractization. Motion is not only something happening along x,y,z. It happens in lots of kinds of spaces. And if time is seen as an ordered set of changes, then a system undergoing changes is basically evolving along that set of changes. For example let's take a completely isolated system which has only two states: A and B. And it cycles continuously discretely between A and B. It's time axis has only two points A and B. Other temporal subdivisions simply do not exist since there is no change behaving as a superposition of A and B. Hence, his evolution or change is a motion along his proper time axis AB.

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    $\begingroup$ I'm not sure if time is exactly like space, but I can grant that it functions much like space does. I can try imagining that motion through time is just like motion through a 4th spatial dimension, but like I said in my OP, in order to even make sense of such motion it seems like we need to introduce another time dimension with which this "motion" occurs. Also, your example seems to describe a change within time, rather than any motion. Sure some particles can undergo processes faster or slower than others but to even speak of such speeds requires that we talk about them with respect to time. $\endgroup$
    – Mike
    Commented Dec 10, 2017 at 15:15
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    $\begingroup$ Aaaa...yeah, time is viewed as a component of the 4D space. I can understand your problem, I had it too once. The notion of speed is seen in mathematics usually as the rate of change of any type of variable and not necessarily as a movement through space. In general relativity we must however use clocks to make observations. You can not determine your own speed along the time axis without comparing it to an external clock or reference. My example says that change is what determines motion along time, hence it determines what we call time. I just gave you the way time emerges. $\endgroup$
    – Gabriel
    Commented Dec 10, 2017 at 15:30
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    $\begingroup$ This issue becomes important in cosmology when it seems that time can not be used anymore, becoming seemingly inexistent for an external observer, so instead of the time coordinate, they use another type of axis to define time, and that axis is in fact the history of the evolution of the Universe. Also in rotating black-holes, time and space exchange places. $\endgroup$
    – Gabriel
    Commented Dec 10, 2017 at 15:40
  • $\begingroup$ Yeah, I can agree change determines or defines time, however I wouldn't say that change is a "motion" through time. To me, time is emergent from change just as space is emergent from the concept of distance. $\endgroup$
    – Mike
    Commented Dec 10, 2017 at 16:10

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