Center of Mass Motion: Velocity & Momentum

[Hiero]

If we have a system of masses in motion, will the velocity of the center of mass always be given by the net momentum divided by (1/c^2 times) the total energy of the system?

 

[Orodruin]

Center of energy, but yes. As long as you do not subject it to external forces.

 

[Hiero]

For motions restricted to one spatial dimension, it's easy enough to show. For more dimensions of space things get messier. I am working straight from the definitions though; maybe there is some 4vector approach?

 

[Hiero]

Nevermind; the way I was doing it was unnecessarily difficult... We can just use the lorentz transforms of energy E and momentum P:
If we boost out of the center-frame where P=0 then the new energy will become ϒE and the new momentum will be ϒvE/c^2 and hence the velocity of the center-frame (v) will be given by c^2(P'/E')
(The linearity of the transformation means it doesn't matter if we're talking about a single particle or the sums of many.)

 

[Orodruin]

It is actually not that simple unless your particles are non-interacting. The reason for this is relativity of simultaneity. You generally need to work with the energy momentum tensor.