Showing the rest energy doesn't depend on velocity.

-Dragoon-

1. The problem statement, all variables and given/known data
A system of two particles, each with rest masses [itex]m_1[/itex] and [itex]m_2[/itex] have velocities [itex]\vec{u}_1[/itex] and [itex]\vec{u}_2[/itex] in the S frame. Consider another frame, the S' frame, which has a velocity v in the S frame in the direction of increasing x. Show that the rest energy [itex]E_{0}[/itex] does not depend on the velocity of the two particles.


2. Relevant equations
[itex]E_3 = E_1 + E_2 = \gamma{m_{1}}c^{2} + \gamma{m_{2}}c^{2}[/itex]
[itex]\vec{p}_3 = \vec{p}_1 + \vec{p}_2 = m_1\vec{u}_1 + m_2\vec{u}_2[/itex]
[itex]E_0 = (m_1 + m_2)^{2}c^{4}[/itex]
[itex]E_3^{2} - |\vec{p}_3|^{2}c^{2} = E_0^{2} [/itex]

3. The attempt at a solution

So I first start with by squaring the expression of the total energy and get:
[itex]E_3^{2} = \gamma^{2}m_{1}^{2}c^{4} + 2\gamma^{2}m_1m_2c^{4} + \gamma^{2}m_{2}c^{4}[/itex]
and then do the same for total momentum of the system:
[itex]\vec{p}_3^{2} = m_1^{2}\vec{u}_1^{2} + 2m_{1}m_{2}\vec{u}_1\vec{u}_2 + m_2^{2}\vec{u}_2^{2}[/itex]
Multiplying through by [itex]c^{2}[/itex]:
[itex]\vec{p}_3^{2}c^{2} = m_1^{2}\vec{u}_1^{2}c^{2} + 2m_{1}m_{2}\vec{u}_1\vec{u}_2c^{2} + m_2^{2}\vec{u}_2^{2}c^{2}[/itex]
And while the total rest energy of the two particles is equal to:
[itex] E_0 = m_1^{2}c^{4} + 2m_1m_2c^{4} + m_2^{2}c^{4}[/itex]

The problem now arises when I use equation 4:
[itex]E_3^{2} - |\vec{p}_3|^{2}c^{2} = E_0^{2} [/itex]
[itex]( \gamma^{2}m_{1}^{2}c^{4} + 2\gamma^{2}m_1m_2c^{4} + \gamma^{2}m_{2}c^{4}) - ( m_1^{2}\vec{u}_1^{2}c^{2} + 2m_{1}m_{2}\vec{u}_1\vec{u}_2c^{2} + m_2^{2}\vec{u}_2^{2}c^{2}) = m_1^{2}c^{4} + 2m_1m_2c^{4} + m_2^{2}c^{4}[/itex]

This equation is only true if either both velocities are equal to zero (and hence the particles are at rest in S frame, which is clearly not the case according to the question) or the total momentum of the system is equal to 0, and finally, the gamma factor is equal to one. Assuming I did everything correctly, then which one is it? Is it the total momentum of the system equal to 0 and how is this the case? Also, how is it that the gamma factor is equal to 1 in the S frame? That is only the case if the particles are not traveling at relativistic speeds, and the question does not tell us whether [itex]\vec{u}_1^{2}[/itex] and [itex]\vec{u}_2^{2}[/itex] are a significant fraction of the speed of light or not.

Any ideas? Thanks in advance.


I'm actually using this invariance to prove something else, so "showing the rest energy does not depend on the velocities" is really not the question, but the result of this is paramount to my proof.