# Energy-momentum Four-vectors

1. Oct 22, 2005

### Jamesss

Does anyone know how you find the length of the energy-momentum four-vector for a system of particles?
p_mu=(E/c,p)
where length is:
length(p_mu)=-(E/c)^2+(p)^2

Do you first add the corresponding vector elements then find the length
OR
find the length of each particle first then sum the individual lengths.

Cheers,
Jimmy

2. Oct 22, 2005

### masudr

Both values will give you invariants, although the energy-momentum four-vector ($p^\mu$) of the whole system is equal to the sum of all the individual $p^\mu$, and therefore the length of $p^\mu$ for the system is the length of the sum of all the individual $p^\mu$.

3. Oct 22, 2005

### Staff: Mentor

It's exactly analogous to finding the magnitude of the total three-momentum of a system of particles. In that case, you find the total x-momentum, total y-momentum, and total z-momentum of momentum, then use them to find the magnitude of the total-momentum vector.

4. Oct 22, 2005

### pmb_phy

If the particles are interacting through when they are seperated (e.g. two charged particles) then the addition of the two 4-vectors is meaningless. Only systems of non-interacting particles anmd systems of particles which interact only through contact forces can be added in a meaningful way. To add the vectors you add components and then take find the magnitude.

This web page I created will get into great detail regarding this. See
http://www.geocities.com/physics_world/sr/invariant_mass.htm

Pete

5. Oct 23, 2005

### Jamesss

Is this method ok?

Thanks for the clarification...

What I was trying to do was find the lengths of the four-vectors of this reaction before and after.
p + p ==> p + p + Z

Where a proton with 300GeV hits a stationary proton, then producing a particle Z.

I calculated the length of the Four-vector before the reaction in the stationary proton's frame.

I then equate this to the length of the four-vector after the collision in the
center-of-mass frame to extract the rest mass of the Z particle.

length(p_mu1+p_mu2)=length(p_mu3+p_mu4+p_muZ)

Question, is there anything wrong with my method?

I have assumed that after the collision the two protons and the Z particle are at rest, since I want the maximum possible rest mass of Z. Momentum in the COM from is zero so it should be ok?

Jimmy

6. Oct 24, 2005

### Staff: Mentor

Yes, that's a reasonable way to proceed. What you end up with is the largest mass the Z can have, and still be produced under these initial conditions.