Can someone explain to me what the phrase "Everything moves through space-time at the same speed" means? 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.
 A: 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 $\dfrac{\sqrt{ds^2+d\mathbf{x}^2}}{dt}=c$.
A: 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. 
