Understanding a Special-Relativity Frame and Comoving Center-of-Mass

[mark57]

good morning Sirs,

in my special-relativity texbook i read that, for a generic mass particle system, with respect to a particular frame where the total system linear momentum is zero the mass-energy relation will have a simple, well known and 'satisfactory' form.

Well: what is this frame? its coordinates?

I see that is not a like center-of-mass frame CM

...by the way, how can I define a relativistic CM?

thank you
mark

 

[Dale]

Well: what is this frame? its coordinates?

Are you familiar with four-vectors? If so then simply add up all of the four-momentum vectors for each part of the system to get a total four-momentum. Then, boost that to the frame where the spacelike part is 0. This is the center of momentum frame.

 

[jtbell]

Well: what is this frame? its coordinates?

Frames don't have "locations", only relative velocities. I suspect what you really mean to ask is something like this:

Given a collection of particles with specified energies and momenta, what is the velocity of their zero-momentum frame (the frame in which their total momentum is zero)?

I think this procedure should work, although I have not yet tested it with a simple example:

1. Calculate the total energy ##E_{tot}## and the total (vector) momentum ##\vec p_{tot}## of the particles.
2. The velocity of the zero-momentum frame as a fraction of ##c## is $$\frac {\vec v} c = \frac {\vec p_{tot} c} {E_{tot}}$$ [added: I've now tested this procedure, and it does indeed work. Whew. ]

I base this on the fact that the velocity of a single particle in terms of its energy and momentum is $$\frac {\vec v} c = \frac {\vec p c} E$$ In a reference frame moving with this velocity relative to the original one, the particle is of course at rest (##\vec p = 0##).

 

[jtbell]

As far as terminology is concerned, people do often call the zero-momentum frame the "CM frame" where "CM" can be understood as either "center of mass" or "center of momentum." This is in spite of the fact that the word "center" implies a "location" (at least to me!), but frames don't have "locations", as I noted above. People often don't use natural language as precisely as one might wish.

 

[PAllen]

In Newtonian physics there is utility in defining a center of mass. In relativity, there is an analog - center of energy, based energy contributing to inertia in SR. However, it has less utility than center of mass in Newtonian physics. It is computed by the same method using total energy for each particle instead of mass. Even for limited use, I believe it only makes sense to compute it in a frame with zero total momentum.

 

[Orodruin]

Minkowski frames have a specific location just as much as any Cartesian coordinate system on a Euclidean space - its spatial origin. Of course, this origin will move in a different frame - but there is nothing particular about that that was not there already in Newtonian mechanics.

Generally, the center of momentum frame only refers to any frame where the total momentum is zero while center of mass frame would refer to center of momentum frame with the center of mass in the origin (since it is a center of momentum frame, the total 4-momentum has the invariant mass of the system as the time component).

In Newtonian physics there is utility in defining a center of mass. In relativity, there is an analog - center of energy, based energy contributing to inertia in SR. However, it has less utility than center of mass in Newtonian physics. It is computed by the same method using total energy for each particle instead of mass. Even for limited use, I believe it only makes sense to compute it in a frame with zero total momentum.

It makes perfect sense to define the center of energy also in a frame where total momentum is non-zero. In fact, you can derive some rather familiar relations for the center of energy motion just from the assumption of stress-energy conservation.

 

[PAllen]

It makes perfect sense to define the center of energy also in a frame where total momentum is non-zero. In fact, you can derive some rather familiar relations for the center of energy motion just from the assumption of stress-energy conservation.

The reason I find it less useful is that the center of energy world line for a system of particles that have mutual relative motion is frame dependent. This is obviously not true of Newtonian center of mass. That is, Lorentz transform the particle histories and the COE world line computed in one frame, and find that the COE world line computed in the new frame is not the transform of the COE world line in the other frame. I didn't say it was useless, just less useful. Of these COE world lines, the one computed in the COM frame seems more physically meaningful to me, especially in cases like a ball of gas.