Let's define relativistic center of mass in this way \begin{equation} \mathbf{r}_{cm} = \frac{\sum_i \gamma_i m_i \mathbf{r_i}}{\sum_i \gamma_i m_i} \end{equation} If system is isolated the total mass is constant and the speed at which center of mass goes is \begin{equation} \mathbf{v}_{cm} = \frac{\sum_i \dot{\gamma_i} m_i \mathbf{r_i} + \sum_i \gamma_i m_i \mathbf{u_i}}{\sum_i \gamma_i m_i} \end{equation} Using uppercase for total quantities I can rearrange in this way \begin{equation} \mathbf{P} = M \mathbf{v}_{cm} - \sum_i \dot{\gamma_i} m_i \mathbf{r_i} \end{equation} This remember a classical theorem, but the sum $\sum_i \dot{\gamma_i} m_i \mathbf{r_i}$ is annoying. What does it mean? Is it constant? If energy and momentum are constant, center of mass doesn't move at constant speed?

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    $\begingroup$ You need to define your symbols with more care before you can work this problem. What are the $\gamma_i$s? What speeds do they use? What is $M$? And so on. $\endgroup$ – dmckee Jul 12 at 22:12
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    $\begingroup$ See en.wikipedia.org/wiki/Center_of_mass_(relativistic) about how the center of mass is not a well-defined concept in relativity. $\endgroup$ – G. Smith Jul 12 at 22:21
  • $\begingroup$ Note that the system momentum (and thus a ``center of momentum'' frame) is well defined which (with a proper definition of system mass) makes a velocity for the system well defined. $\endgroup$ – dmckee Jul 12 at 22:27
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    $\begingroup$ But how are you defining total mass (I know of at least three ways, two of which are defensible)? And for the Lorentz factors what are you taking the speed relative (this factors into the definition of the total mass that may (or may not) make sense in you expressions). You really do have to be careful in working problems like this or you risk tripping on your own toes. $\endgroup$ – dmckee Jul 12 at 22:38
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    $\begingroup$ That is a defensible definition of the system mass if we can ignore potential terms between the components of the system (as would be the case for a typical "rigid enough" solid object that is not experiencing any strain or a system of non-interacting particles). But in those cases, what are the $\dot\gamma_i$s? $\endgroup$ – dmckee Jul 12 at 23:19

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