What is the Significance of 4-Velocity in Special Relativity?

[DiracPool]

I’m a bit confused about what the 4-velocity is actually portraying? I think I understand the 4-vector pretty well, but the 4-velocity has me squinting.

It seems as though the idea is that we take the derivative of the 4-vector with respect to the proper time in order to yield the 4-velocity.

This basically gives us a 4-velocity, U = [cdt/(d-tau), dx/(d-tau)] = (gamma)(c, dx/dt) = 1/(√1-v^2/c^2)(c, v).

Ok, I get that, but now we have two velocity terms here, one outside the vector bracket in the gamma expression, and one inside the vector bracket, each of which seem to depend on the other. Basically, if an object is not moving relative to the rest, or proper time frame, all of its vector component is entirely in the time domain/coordinate, and that value is a constant, c. That’s easy to see. But what happens when an object moves relative to that rest frame? How do we read what the 4-velocity is in this instance, and what is it telling us? What does it signify?

For example, if an object is moving at .87c, then we have a gamma of 2 which basically doubles the velocity vector dx/dt inside the brackets to yield a value for the 4-velocity, U. Plus, it also looks as though you are doubling the value of the vector in the time dimension, making that 2c. Is this right? An object moving at .87c relative to a rest frame yields a 4-velocity that is 2c in the time coordinate and 2 times .87c in the velocity coordinate? What does that mean? What is that supposed to tell us or be useful for? Or am I reading this all wrong?

 

[Dale]

But what happens when an object moves relative to that rest frame? How do we read what the 4-velocity is in this instance, and what is it telling us? What does it signify?

The best way to think of what it signifies is to think geometrically. Geometrically, the 4-velocity is just the unit tangent vector to the worldline.

For example, if an object is moving at .87c, then we have a gamma of 2 which basically doubles the velocity vector dx/dt inside the brackets to yield a value for the 4-velocity, U. Plus, it also looks as though you are doubling the value of the vector in the time dimension, making that 2c. Is this right? An object moving at .87c relative to a rest frame yields a 4-velocity that is 2c in the time coordinate and 2 times .87c in the velocity coordinate? What does that mean? What is that supposed to tell us or be useful for? Or am I reading this all wrong?

Yes, that is correct. The components of the 4 velocity are not limited to <c.

The components have the meaning that you mentioned above. The timelike component is the time dilation factor and the spacelike components are the 3 velocity times the time dilation factor.

The vector as a whole has the geometric meaning I mentioned, it is the unit tangent vector.