Velocity addition ain't what it used to be

[JesseM]

I'm taking A, B and C to be moving objects at rest w.r.t. inertial frames whose origin is at A, B and C.

Well, the speed (magnitude of velocity vector) of A in C's frame given a known velocity in B's frame should be the same regardless of where you place the origins of B and C's frames. Are you specifically concerned with the actual coordinate direction of the velocity vector apart from just the speed? If you don't assume that the origins of B and C coincide at times t=0 and t'=0 in those frames, then to get the actual vector you'll need to use some appropriate form of the velocity addition formula that's derived from the Poincaré transformation rather than the Lorentz transformation...I'm not sure what the formula would be but I'm sure you could find it somewhere.

Can you state more clearly what your objection to the velocity addition formula is? There are obviously going to be different versions of the formula depending on how you define the coordinate systems (where their x-axes are parallel, whether the origins coincide at a time of zero in each frame), are you objecting that sometimes people use the wrong version in a given scenario? Or are you arguing that even when people use a version that's tailored to the pair of coordinate systems being used (like using the (u + v)/(1 + uv/c^2) version in a scenario where B and C's x-axes are parallel and their spatial origins coincided at t=t'=0, and A is moving along the x-axis) there's still some problem with it?

 

[yre]

Even if B didn't exist, there would be a boost from A to C and that would be the relative velocity.

After some thought, I now understand that I was wrong to say that u@v is "the relative velocity". But I don't agree with the quote above.

Ok, yes, I was thinking C would be moving away from A in a straight line from A's perspective in which case I'd say there would be a boost from object A to object C.

However in 3+1 dimensions C could also be moving across A's line of sight, so to speak, which wouldn't be a boost.

 

[yre]

Are you specifically concerned with the actual coordinate direction of the velocity vector apart from just the speed?

Yes. The full 3D velocty addition formula is about directed velocities.

If you don't assume that the origins of B and C coincide at times t=0 and t'=0 in those frames, then to get the actual vector you'll need to use some appropriate form of the velocity addition formula that's derived from the Poincaré transformation rather than the Lorentz transformation

This thread has gotten side-tracked into talking about translations and parallel movements, but what this discussion is about is the composition of two boosts, i.e. Lorentz transformations, i.e. the origin of B and C did coincide at some time and the origin of A and B did coincide at some time.

Can you state more clearly what your objection to the velocity addition formula is? There are obviously going to be different versions of the formula depending on how you define the coordinate systems (where their x-axes are parallel, whether the origins coincide at a time of zero in each frame), are you objecting that sometimes people use the wrong version in a given scenario? Or are you arguing that even when people use a version that's tailored to the pair of coordinate systems being used (like using the (u + v)/(1 + uv/c^2) version in a scenario where B and C's x-axes are parallel and their spatial origins coincided at t=t'=0, and A is moving along the x-axis) there's still some problem with it?

The 3D-velocity formula u@v reduces to (u + v)/(1 + uv/c^2) in the 1-dimensional case.

Even if BC is not in the same line as AB they are all still in a plane, so it reduces to 2-dimensions. The full 3-dimensions is only needed when composing three boosts which might not all be in a plane.

Anyway u@v gives the parameter of the boost composition B(u)B(v)=B(u@v)gyr[u,v].

In the 1-dimensional case B(u@v)gyr[u,v] reduces to B(u@v), and so boost-parameter and composite-velocity mean the same thing in 1-dimension.

Ungar says that for non-colinear boosts in 3-dimensions, boost-parameter and composite-velocity do not mean the same thing.

 

[yre]

This discussion has made me think I am misinterpreting what Ungar says about coaddition. It's not that the magnitude |u#v| gives the speed of C which I think is given by |u@v|=|v@u|. It's that neither u@v or v@u give the direction.

The role of u#v is that when comparing the trigonometric formulae of Newtonian dynamics with relativistic dynamics, u#v appears in expressions in an analogous way to ordinary vector addition u+v and in this sense relativistic velocities behave more like u#v than u@v. In which case u#v isn't a departure from the laws of special relativity, u#v is just a quantity which can be used to compare Newtonian dynamics with relativistic dynamics.