Conservation of four-momentum concepts, frames

[binbagsss]

I've read that they must be calculated in the same frame , and so to calculate them in the easiest frame.. *

So for a collision when I compute the momentum before and after I should do this in the easiest frame.

Considering a specific collision where we are computing the minimum energy to create some given particles after a collision that occurs between a particle with energy E, colliding into a stationary particle, both particles have mass m.

Questions:

1) Isn't the chose of frame fixed by the requirement of minimum energy and so we need a frame in which the created particles are at rest - is there only one such frame?

2) In evaluating the four-momentum before the collision, I get p =(E+mc,p,0,0).
(Assuming the collision to occur in 1-d x direction and using natural units.)
Where p is the momentum of the not stationary particle, and is unknown.

- Here I'm unsure how * applies and what frame we are evaluating in - we have to use a frame where the colliding particle has energy E as this is the only data known?

So by choosing a frame in which the final particles are at rest, are we able to evaluate the four-momentum in different frames before or after the collision? Or how would we know we have calculated four-momentum before and after in the same frame?

I think I might be confused with the space-time interval s^us<sub>u[/SUB ]which is the same in every frame.

Any help greatly appreciated, thank you !</sub>

 

[mfb]

1) Isn't the chose of frame fixed by the requirement of minimum energy and so we need a frame in which the created particles are at rest - is there only one such frame?

There is only one frame where the created particles are at rest (apart from irrelevant things like translations and rotations), but you can consider physics in all frames. With 4-vectors, finding this frame first is more work than necessary. You have a direct way to get the center-of-mass energy.

So by choosing a frame in which the final particles are at rest, are we able to evaluate the four-momentum in different frames before or after the collision?

You can do that.

Or how would we know we have calculated four-momentum before and after in the same frame?

By looking at what you calculated.

 

[binbagsss]

You can do that.
.

So momentum is frame invariant?

 

[mfb]

No. The vectors will be different in different frames. I just said you can calculate them.

 

[binbagsss]

No. The vectors will be different in different frames. I just said you can calculate them.

In terms of applying momentum conservation then, you can only do it when the vectors are calculated in the same frame (before and after collision) ?

 

[mfb]

Sure.

 

[vela]

I've read that they must be calculated in the same frame , and so to calculate them in the easiest frame.. *

So for a collision when I compute the momentum before and after I should do this in the easiest frame.

Considering a specific collision where we are computing the minimum energy to create some given particles after a collision that occurs between a particle with energy E, colliding into a stationary particle, both particles have mass m.

Questions:

1) Isn't the chose of frame fixed by the requirement of minimum energy and so we need a frame in which the created particles are at rest - is there only one such frame?

2) In evaluating the four-momentum before the collision, I get p =(E+mc,p,0,0).
(Assuming the collision to occur in 1-d x direction and using natural units.)
Where p is the momentum of the not stationary particle, and is unknown.

- Here I'm unsure how * applies and what frame we are evaluating in - we have to use a frame where the colliding particle has energy E as this is the only data known?

You said the energy is a minimum when the created particles are at rest, but this can happen in only one frame. It can't happen in the lab frame because initially you have momentum ##p## in the x-direction. If everything was at rest afterward, momentum wouldn't be conserved. So to find the minimum energy, then, you want to analyze the situation in the frame where you can have all of the particles at rest after the collision, and then once you have the result of this analysis, you need to figure out what it will look like in the lab frame.

So by choosing a frame in which the final particles are at rest, are we able to evaluate the four-momentum in different frames before or after the collision?

Yes, by using the appropriate Lorentz transformation.

Or how would we know we have calculated four-momentum before and after in the same frame?

I'm not sure what you mean here.

I think I might be confused with the space-time interval s^us_u which is the same in every frame.

This is an instance of another tool you have at your disposal to relate quantities between different frames. If you have a four-vector ##p^\mu## in one frame and a corresponding four-vector ##q^\mu## in another frame (in other words, ##p^\mu## and ##q^\mu## are related by a Lorentz transformation), then the quantity ##p^\mu p_\mu## will be equal to ##q^\mu q_\mu##. The space-time interval is the particular case where the four-vector is (t, x, y, z).