# Can someone explain to me what the phrase “Everything moves through space-time at the same speed” means? [duplicate]

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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• Possible duplicate of Why are objects at rest in motion through spacetime at the speed of light? – Styg Dec 10 '17 at 14:33
• @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? – Mike Dec 10 '17 at 15:05
• Define speed. Otherwise one might take it to be dx/dt. – jjack Dec 10 '17 at 15:11
• @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. – Mike Dec 10 '17 at 15:21
• @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. – jjack Dec 10 '17 at 15:36

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 $\dfrac{\sqrt{ds^2+d\mathbf{x}^2}}{dt}=c$.