# What is the meaning of the square of the four-velocity?

The square (or dotproduct) of the four-velocity u is -1 or 1 (depending on the metric convention and with c=1, I think (in flat spacetime)). What does this physically mean? It sounds like "my speed" squares to 1 so I might think I move with "c" if I denote my velocity with v/c. Is this correct? If not, has this a physical meaning/interpretation that makes sense?

• In my view, it ($\mathbf{u}^2=1$) just is a geometrical fact coming from the invariance of the square of the proper distance. That is if we remember the definition of the 4-speed (as $u^i=dx^i/ds$) and apply it, we get $(\mathbf{u}\cdot\mathbf{u})=dx_idx^i/ds^2$. But $dx_idx^i$ just is the proper length $ds^2$. Therefore $\mathbf{u}^2=1$. Oct 31, 2015 at 20:48

You have a world line. The world line has tangent vectors. The unit tangent vectors are called the four velocity.

It represents the direction in spacetime that you are going.

Just like a unit vector in 3d space represents the direction in space you are going.

Except now you don't have to worry about it failing when you are at rest.

Physically this is because you are always at different events in spacetime so there is always a nonzero tangent. (Except note that if you had some instantaneous impulse then you wouldn't have a tangent or a four velocity, you'd have a before and an after, which makes sense.)

Another physical interpretation is that you think of velocity as momentum per mass. Similarly there is a energy-momentum 4-vector, and if you take the energy-momentum 4-vector and divide by rest mass you get ... The 4 velocity.

But that's just because the energy-momentum 4-vector points in the direction in spacetime you are going and has a length equal to the rest mass, so dividing by it gives you a unit vector.

Either way it's a unit vector pointing in the direction you are going, make of it what you will.

As Sean M. Carroll states:

"This is a reflection of the fact that the four velocity is not a velocity through space, but a velocity through space-time, trough which one always travels at the same rate"

and that means that there is no dynamics in space-time.

• No, there isn't dynamics in space-time. Nothing ever changes in space-time, nothing ever happens in there, it's just a picture of pass, present and future.
– raul
Nov 1, 2015 at 17:17
• Hahahahaha No, it changed through time, but not "trough" space-time. In the space-time this was just a curve on it, but space-time remain the same.
– raul
Nov 1, 2015 at 19:33
• Yeah!, space-time has... everything
– raul
Nov 1, 2015 at 19:58