I'm assuming questions about the third law being invalid in relativity have already been asked on this site, but I'm specifically asking about how momentum is still conserved inspite of it.
Consider only two particles in the universe. In some reference frame, they're moving with velocities $v_1(t)$ and $v_2(t)$ at time $t$.
The relativistic momentum is:
$$\gamma (v_1(t))m_1v_1(t)+\gamma (v_2(t))m_2v_2(t)$$
The relativistic momentum after a small time $dt$ is:
$$\gamma (v_1(t+dt))m_1v_1(t+dt)+\gamma (v_2(t+dt))m_2v_2(t+dt)$$
If both these quantities are same, we can set them equal. After setting them equal, we get:
$$\gamma (v_1(t+dt))m_1v_1(t+dt)-\gamma (v_1(t))m_1v_1(t)=-(\gamma (v_2(t+dt))m_2v_2(t+dt)-\gamma (v_2(t))m_2v_2(t))$$
Dividing both sides by $dt$ and letting $dt\rightarrow 0$, we get:
So what's wrong with the above?
Edit- That other question is about General Relativity and the answers do not address the computations I've provided here. I want to know what's wrong with this specifically
Edit- After reading the other links, what I got is that the momentum conservation, as I've stated stated it in this post, is incorrect. This is because I did not account for the momentum of the field. So that means that Newton's third law is also correct if we extend the notion of force from particle-particle interactions to particle-field interactions. Is this correct? And in what sense do fields carry momentum? What is the mass and velocity of fields?