What this means is that the length of a relativistic four-vector [ct, x, y, z] is sqrt((ct)^2 - (x^2 + y^2 + z^2)).

It is not the same as the Euclidian magnitude, i.e. NOT sqrt((ct)^2 + x^2 + y^2 + z^2)

The article does mention this, although with slightly confusing wording, but it should be emphasized that Lorentz invariant quantities need to be computed with the Minkowski metric.

https://www.euclideanrelativity.com/

Now, the same path is described by many different parameterizations. For example, if we pick q(r) = p(2r) that's the same path. The point varies twice as fast with r, but that has no physical meaning. The total path that is traced out in spacetime is still the same path. That is, the set {p(r) | r in [-∞,∞]} is the same as the set {q(r) | r in [-∞,∞]}, and it is only this set that has physical meaning.

In fact, we can also pick q(r) = p(r^3 + r) or some other parameter transformation.

We might look at the vector p'(r) tangent to the path. The length of this vector depends on the parameterization, but its direction doesn't. So p'(r)/|p'(r)| is a true physical property of the path, but the length |p'(r)| is just an artefact of the parameterization that we chose to describe the path.

In order to make the parameterization somewhat more canonical, we usually pick a parameterization where |p'(r)| = c. As it happens, the four velocity is defined to be p'(r) given this choice of parameterization.

So the length of the four velocity is c simply by definition. This statement has no physical meaning. The statement "we travel through time with the speed of light" is uninteresting. It tells you nothing about physics. It only tells you something about the conventions we use when we describe paths through spacetime. We could very well have picked the convention |p'(r)| = c/2, and then we'd have the statement "we travel through time with half the speed of light", or even |p'(r)| = 2c, and then we'd "travel through time with double the speed of light".

Given any path through spacetime whatsoever, we can pick a parameterization such that it travels through spacetime at the speed of light. So the condition that something travels through spacetime with the speed of light places no constraints on the path that it takes. The statement "we travel through spacetime with the speed of light" makes it sound like there are paths that do travel through spacetime with the speed of light, and there are paths that do not travel through spacetime with the speed of light, and physical particles go along paths of the first kind. This is wrong. There are no paths that do not travel through spacetime at the speed of light, because whether or not it does is a property of the parameterization that we choose to describe the path, not a property of the path itself.

Four-velocity is change of spatio-temporal position (Δx,Δt) in an inertial frame per unit of time Δτ measured by the object in motion. For a concrete example, think a race track with distance markers accompanied by (synchronized) stationary clocks, and a vehicle carrying its own clock. Three-velocity will be given by Δx/Δt (the distance along the track divided by the time on the stationary clock), whereas the spatial component of four-velocity will be given by Δx/Δτ (the distance along the track divided by the time on the vehicle's clock). The temporal component of four-velocity will be c·Δt/Δτ, ie proportional to the clock ratio.

Turns out no matter the speed of the vehicle, it will always hold that

    (c·Δt/Δτ)² - (Δx/Δτ)² = c²

However, once we've baked this relation into the geometry of spacetime, we can of course take the more abstract perspective described above and think about reparametrization-invariant dynamics, with choice of eigentime as parameter an insignificant way to fix an arbitrary gauge.

If you have a clock moving through spacetime along some path p(r), put tickmarks at regular intervals [1] of its arc length ∫|p'(r)|dr. Those tickmarks indicate when the clock ticks.

[1] e.g. choosing the length of an interval by matching it to a one second tick of a stationary reference clock.

In differential geometry, we traditionally parametrize curves by a parameter "t" and think of it as "time", so the parametrization allows us to "walk along" a curve. This is very intuitive geometrically of course.

But in relativity, we also have a "time" coordinate (or at least a timelike unit-length tangent vectors), which then completely oposes this geometric intuition about curves.

Of course now physicists decide to rename well-established concepts and start calling "arc length parametrization" by "proper time parametrization", which makes it sound like it is something special, while it, as far as I can tell, has no actual physical meaning.

Some vectors can vary in magnitude with time, yes. But rotating your coordinate system does not change the magnitude of any vector in Euclidean space. A Lorentz transformation is the spacetime analogue of rotating your coordinate system, and similarly does not change the magnitude of any vector.

The weird part isn't that two observers agree on the vector magnitude. The weird part is that one observer calculates the same value for any two objects. Frame-invariance isn't the relevant property.

That's not weird either, because the value being calculated has all object-specific information removed. See my response to andi999 downthread regarding rest mass.

But moreover, the rest mass isn't the issue here. Momentum in classical mechanics also takes mass into account but doesn't have the "weird" property that two objects of the same rest mass always have the same momentum magnitude.

If you think for stationary examples it just means that the eigenzeit has a different pace, so two clocks next to each other going at a different pace. But this is ruled out by another definition, such that clocks are references by light clocks. And now it is also clear why this invariance is c.

It's a unit vector, and physicists working with relativity habitually set C=1 to make the equations simpler. That being said, it isn't really a vector -- it's just a direction. The "equal to C" bit is a mathematical artifact of both vectors being normalized.

Because this 4-vector removes the factor of rest mass, which distinguishes objects. Putting back the rest mass converts 4-velocity to 4-momentum, which does not have the same magnitude for all objects--the magnitude is the rest mass.