Special relativity, which has been verified experimentally and is 100% bona fide serious business, predicts that when an object is moving more quickly, outside observers will see time moving more slowly for that object. Therefore s = c, regardless of where you are or how quickly you are moving †. = (c 2 - v 2) × c 2(c 2 - v 2) -1, by substituting in the expression for γ. So, finally, we can get an expression for the magnitude of the derivative of the spacetime vector, denoted by s: Now, the definition of γ is (1 - (v/c) 2) -1/2 = c(c 2 - v 2) -1/2, so γ 2 = c 2(c 2 - v 2) -1, v being the speed of the object being observed. This is also a geometric invariant, and the same form of the magnitude will be applicable to the derivative of the vector. It is conventional to reverse the sign of the right hand sign, making it s 2 = (ct) 2 - x 2 - y 2 - z 2. The magnitude of the 4-vector is given by: Similarly, our spacetime 4-vector ⟨ct,x,y,z⟩ should be invariant through a change in inertial frames of reference, so that all observers will record the same magnitude of the vector, even if they see the components differently. This is a geometric invariant, which basically means that you can rotate or move this vector however you like, or change the coordinate system, and its magnitude (length in this case) will stay the same. The magnitude of a spatial vector ⟨x,y,z⟩ is found by s 2 = x 2 + y 2 + z 2. This is the spacetime "speed" in vector form, but the magnitude of this vector is a little different from the magnitude of a spatial 3-vector. We differentiate the spacetime vector with respect to proper time: So after that digression, we can find the "speed" of the spacetime vector with respect to proper time, which is denoted by U. This relationship between τ and t makes it far easier to find a meaningful derivative of R, since we can turn the derivative with respect to an external proper time into a derivative with respect to the observed time of the object. Gamma depends on person 2's speed relative to person 1. Proper time is time it takes for one tick of either person's clock from their own perspective, and if τ is person 2's proper time, then person 1 sees person 2's clock taking time t to tick, which is related to τ through γ (Greek letter gamma) as such: t = τ × γ. How slowly the two clocks appear to tick depends on how quickly the train is moving the faster it is moving, the more slowly the clocks will appear to tick. Person 2, who is on the train, will see their clock ticking normally, but will see person 1's clock ticking slowly. For some background on this, the special relativity node has some good information, but the bottom line is if person 1 is standing still and watches person 2 move past at a constant speed on a train, they see person 2's ticking more slowly than their own. If we want to look at the "speed" at which something moves through spacetime, it will be the derivative of R with respect to " proper time", denoted by τ ( Greek letter tau). This makes the spacetime vector ⟨ict,x,y,z⟩, and we denote this vector by R. It seems like a small step to just bring the time dimension inside our space vector and make a spacetime vector, ⟨t,x,y,z⟩, but when this is done, the time coordinate must be multiplied by c (the speed of light) and i (square root of -1, this comes up later) so that the elements of the vector have the same units (distance). We could say that at t=1 an object is at ⟨1,2,3⟩, and at t=2 it is at ⟨2,4,6⟩. We describe an object by its spatial coordinates at a point in time, and then it may be at another set of coordinates at another point in time. We think of space being three-dimensional (x, y, and z), and time being one-dimensional (t). We normally think of space and time being distinct things, independent and fundamentally different in a way that we can't really describe. It will take a little while to get to the point here, though. It is an implication of special relativity that I think is pretty interesting and pretty cool, and I hope you'll think so too. This is not some kind of metaphysical, philosophical or metaphorical statement, it is actually true.