# Minkowski spacetime vs Euclidian space

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?

• 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. – Ben Crowell Sep 1 '14 at 23:14

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.
• Small comment: some people (notably Landau & Lifshitz, if I remember correctly) define the four velocity with norm $1$ instead of $c$. – Javier Sep 2 '14 at 10:25
• @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] – user12262 Sep 6 '14 at 7:29
• [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. – user12262 Sep 6 '14 at 7:31