How to start a proof that shows that proper mass is the same for S and S' frame

[chris_0101]

Homework Statement

As the title of this post suggests, how does one show that the proper (invariant) mass is the same for two reference frames where the S' frame is moving away from the S frame at a velocity of +v. Note that this is a two mass system of m_1 and m_2 with speeds of u_1 and u_2, respectively.

I'm not too sure how to start this problem

Homework Equations

E = E_1 + E_2 ---> energy
p = p_1 + p_2 ---> momentum

Lorentz transformation for mass:

m' = $\gamma$m

The Attempt at a Solution

I've tried multiple solutions with no avail and my attempts are too long and complicated for me to post. However, I did try to solve this:

E^2 - p^2c^2 = m_o^2c^4 = E'^2 - p'^2c^2

but I'm not sure if this is correct or not because it does not involve the speed of the two frames +v.

Any help would be great

 

[anathema]

!

Thank you for your question. To show that the proper mass is the same for two reference frames, we need to use the concept of invariant mass. Invariant mass is a property of a system that remains the same regardless of the reference frame in which it is observed. In other words, it is the same for all observers in all frames of reference.

To start, we can use the equations you have listed in your post, E = E_1 + E_2 and p = p_1 + p_2, which relate the total energy and momentum of a system to the individual energies and momenta of its components. In this case, our system consists of two masses, m_1 and m_2, with velocities u_1 and u_2, respectively.

Next, we can apply the Lorentz transformation for mass, which states that the mass of an object in one reference frame is related to its mass in another reference frame by the factor \gamma. In this case, we have two reference frames, S and S', where S' is moving away from S at a velocity of +v. Therefore, the masses of the two objects in the two frames are related by:

m'_1 = \gamma m_1
m'_2 = \gamma m_2

Now, we can substitute these expressions into our equations for energy and momentum, and use the fact that E^2 - p^2c^2 = m_o^2c^4 (where m_o is the invariant mass) to get:

m_o^2c^4 = (E_1 + E_2)^2 - (p_1 + p_2)^2c^2
= (m'_1c^2 + m'_2c^2)^2 - (m'_1u_1c + m'_2u_2c)^2
= (\gamma m_1c^2 + \gamma m_2c^2)^2 - (\gamma m_1u_1c + \gamma m_2u_2c)^2
= \gamma^2(m_1^2c^4 + m_2^2c^4 + 2m_1m_2c^4) - \gamma^2(m_1^2u_1^2c^2 + m_2^2u_2^2c^2 +