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Two questions marked in bold:

What is the magnitude of a Minkowski spacetime four velocity? I'm deducing that it is c for all observers, but I'd like it confirmed.

In Euclidian space, a velocity vector integrates over time to an absolute position.

Does a four velocity integrate over time to a four dimensional displacement, and is it used in physics?

Eventually; why not?

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    $\begingroup$ There are many kinds of four-vectors, and the magnitude of each type has a different interpretation. For example, energy-momentum is a four-vector, and its magnitude is the mass. $\endgroup$ – Ben Crowell Sep 1 '14 at 23:14
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It sounds as if by four vector you are actually thinking of the four velocity. The norm of the four velocity is indeed always $c$, and it does integrate to give a displacement along a world line.

The term four vector refers to anything that transforms as a vector, and as Ben notes in his comment this could refer to lots of different physical quantities that are unrelated to velocity.

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  • $\begingroup$ Small comment: some people (notably Landau & Lifshitz, if I remember correctly) define the four velocity with norm $1$ instead of $c$. $\endgroup$ – Javier Sep 2 '14 at 10:25
  • $\begingroup$ Just a bit more on the world line. This implies that a distance can be calculated in terms of "world line distance" between two objects. Is that used for anything in physics? Why (not)? $\endgroup$ – frodeborli Sep 2 '14 at 19:29
  • $\begingroup$ @frodeborli: yes indeed, the distance calculated along the worldline this way is called the line element or the proper time. For a freely falling observer it's equal to the time shown on a clock carried by the observer. The line element is an invariant quantity in both SR and GR. $\endgroup$ – John Rennie Sep 3 '14 at 4:52
  • $\begingroup$ @frodeborli: "that a distance can be calculated in terms of "world line distance" between two objects." -- Not exactly. 1: Two objects, $A$ and $B$, are characterized by a "distance" between each other only if and while they were at rest to each other (a.k.a. both having been members of the same inertial frame). If so then 2: the value of the distance between objects $A$ and $B$ (at rest to each other) is $$\overline{AB} = \overline{BA} := \frac{c}{2} \tau_A^{\text{ping to B and back}} = \frac{c}{2} \tau_B^{\text{ping to A and back}}$$ where [... to be continued] $\endgroup$ – user12262 Sep 6 '14 at 7:29
  • $\begingroup$ [contd. ...] where $A$'s ping duration $\tau_A^{\text{ping to B and back}}$ is the (constant) magnitude of any segment of $A$'s world line from any signal event in which $A$ took part until the corresponding event at which $A$ observed that $B$ had observed this signal event, $B$'s ping duration $\tau_B^{\text{ping to A and back}}$ is the (constant) magnitude of any segment of $B$'s world line from any signal event in which $B$ took part until the event at which $B$ observed that $A$ had observed this signal event, factor $\frac{1}{2}$ is conventional, and letter $c$ is a distinctive symbol. $\endgroup$ – user12262 Sep 6 '14 at 7:31

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