= Solution
The <covariance of a selected particle momentum sum> follows because, for a fixed set of particle <four-momenta>, their sum $P_A^\mu=\sum_{i\in A}p_i^\mu$ is a <four-vector> by linearity of the <Lorentz transformation>. Therefore
$$
\boxed{\left(\sum_{i\in A}E_i\right)^2-c^2\left|\sum_{i\in A}\mathbf p_i\right|^2
=-c^2P_A^\mu P_{A\mu}}
$$
is invariant. The subset need not comprise the entire plasma, and its particles need not move together. This expression includes cross terms between different particles; it is not the sum of their individual rest-mass squares in <3/a>.
The comparison must use the same subset and the transformed <four-momenta> at the same chosen events. Selecting “the lowest-energy half” anew in each frame need not do that. For example, take four <photons> along $\pm x$, with energies $E_0,2E_0$ in each direction. The two lowest in the original frame are an opposite pair and have invariant <energy> squared $4E_0^2$. After a boost at $c/2$ along $+x$, both forward <photons> are redshifted enough to become the lowest pair; that new collinear pair has zero <invariant mass>. This does not contradict covariance, because the subset changed.
If energies are resampled on a different <simultaneity> slice in an interacting plasma, they can also refer to different events. A physical conserved subsystem total must then include exchanged field <energy> and any boundary flux as appropriate. Such operational conservation requirements are distinct from the exact algebraic covariance of a sum of fixed <four-vectors>.
Back to article page
