# The center of mass & relativistic collisions

• I
In summary, the center-of-momentum frame is more important than the center of mass frame in special relativity.
TL;DR Summary
In special relativity (especially relativistic collisions), is the center of mass frame as useful as Newtonian mechanics?
In special relativity (especially relativistic collisions), is the center of mass frame as useful as Newtonian mechanics?

Summary:: In special relativity (especially relativistic collisions), is the center of mass frame as useful as Newtonian mechanics?

In special relativity (especially relativistic collisions), is the center of mass frame as useful as Newtonian mechanics?
Even more useful! Vital, in fact.

PS Center of momentum frame, of course.

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vanhees71
PeroK said:
Even more useful! Vital, in fact.
I'm going to disagree here, since I think defining the "center of mass frame" has some subtleties (system mass or component masses?). The zero momentum frame, which is the same as the center of mass frame in Newtonian physics and the center of system mass frame in relativistic physics, is most certainly as important as you say.

So I'm picking nits, but given the knock-down-drag-out fights we've seen on here over mass and its conservation, I think they're important nits to pick.

vanhees71
Yes, zero momentum frame was what I had in mind.

vanhees71 and Ibix
I'm still confused. For example, in 'Introduction to Elementary Particles by Griffith', for relativistic collisions, the center of momentum frame is introduced to solve problems. But isn't the center of mass frame appropriate in relativistic collisions?

I'm still confused. For example, in 'Introduction to Elementary Particles by Griffith', for relativistic collisions, the center of momentum frame is introduced to solve problems. But isn't the center of mass frame appropriate in relativistic collisions?
The key concept in SR is energy-momentum. In general, you should start thinking in terms of energy and momentum and not in terms of mass and velocity.

To take one example: a photon is modeled as a massless particle in SR. It has energy and momentum, hence the centre of momentum (or zero momentum) frame can be defined for collisons/decays involving photons. But, a centre of mass frame when one particle is massless is not very useful.

PeroK said:
The key concept in SR is energy-momentum. In general, you should start thinking in terms of energy and momentum and not in terms of mass and velocity.

To take one example: a photon is modeled as a massless particle in SR. It has energy and momentum, hence the centre of momentum (or zero momentum) frame can be defined for collisons/decays involving photons. But, a centre of mass frame when one particle is massless is not very useful.
I understand now by your good example.
Thankful.

In relativity it's indeed the center-momentum frame, not the center of mass frame. That's because today 112 years after Minkowski's crucial article about the mathematical structure of special relativity ("Minkowski space") we express everything in covariant quantities rather than in some arbitrary confusing ones, and that's why "relativistic mass" is not used anymore anywhere in current research (though there are still some new textbooks introducing the confusion, because the authors are unwilling to learn century-old math ;-)).

The invariant mass of a system (e.g., a set of point particles or a continuum mechanics description of a fluid or some fields like the electromagnetic field) is given by the total four-momentum ##P^{\mu}## of the system by
$$M^2 c^2=P_{\mu} P^{\mu} = (E/c)^2-\vec{P}^2 \geq 0.$$
For ##M>0## you can always find an inertial frame, where ##\vec{P}=0##, and that's called the center-of-momentum frame, and it's considered as the "rest frame" of the system.

The reason is that from Noether's theorem applied to Lorentz boosts it follows that for a closed system the center energy-weighted average rather than the mass-weighted average moves with constant velocity.

E.g., take two interacting particles. Their total momentum is conserved (Noether's theorem applied to translation invariance in space and time), i.e.,
$$p_1+p_2=\text{const}.$$
Written in terms of the coordinate time that reads
$$m_1 \gamma_1 \dot{x}_1 + m_2 \gamma_2 \dot{x}_2=\text{const},$$
but
$$m_1 \gamma_1=E_1/c^2, \quad m_2 \gamma_2=E_2/c^2.$$
This implies
$$E_1 \dot{\vec{x}}_1 + E_2 \dot{\vec{x}}_2=\text{const}.$$
The temporal component means
$$E=E_1+E_2=\text{const},$$
and thus the energy-weighted average of the three-velocities (rather than the mass-averaged three-velocities) is conserved,
$$\vec{V}=\frac{E_1 \dot{\vec{x}}_1 + E_2 \dot{\vec{x}}_2}{E_1+E_2}=\text{const}.$$

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## 1. What is the center of mass?

The center of mass is a point in a system where the mass of the system is evenly distributed in all directions. It is the point where the system can be balanced and treated as a single object.

## 2. How is the center of mass calculated?

The center of mass is calculated by taking the weighted average of the positions of all the particles in a system, with the weights being their respective masses. Mathematically, it is represented as:

xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)
ycm = (m1y1 + m2y2 + ... + mnyn) / (m1 + m2 + ... + mn)
zcm = (m1z1 + m2z2 + ... + mnzn) / (m1 + m2 + ... + mn)

## 3. What is a relativistic collision?

A relativistic collision is a collision between two objects where the velocities of the objects are significant enough to cause a change in their mass. This is due to the effects of relativity, where the mass of an object increases as its velocity approaches the speed of light.

## 4. How is the center of mass affected in a relativistic collision?

In a relativistic collision, the center of mass can shift due to the change in mass of the objects involved. This shift is dependent on the velocities and masses of the objects, and can be calculated using the relativistic equations for momentum and energy.

## 5. Why is the concept of center of mass important in studying collisions?

The concept of center of mass is important in studying collisions because it allows us to simplify the analysis of a system by treating it as a single object. This makes it easier to apply principles of conservation of momentum and energy, and to understand the overall motion of the system. Additionally, the center of mass remains constant in a closed system, making it a useful reference point for studying the effects of collisions.

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