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Newton's third law says that every action has an equal and opposite reaction. This is essentially the law of conservation of momentum. From conservation of momentum perspective, the law should be obeyed by all physical theories.

But we know that action-reaction principle does not hold in classical electromagnetism. For example in case of two charged particles approaching origin one along the x and one along y axis. Still Conservation of momentum holds.

I would like to understand if the law (or some version of it) is valid in the context of general relativity and standard model of particle physics (QFT)?

The motivation for this question is to ascertain whether the action-reaction principle should/will play a role in physics beyond standard model or in grand unified theories?

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  • $\begingroup$ What do you mean by the law not holding for two charged particles approaching the origin? $\endgroup$ Feb 3, 2021 at 15:16
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    $\begingroup$ When two charges approach origin one along x-axis and one along y-axis, the electric force at one due to other is equal and opposite but the force due to magnetic field is equal in magnitude but not opposite in direction (they are at right angles to each other). Hence, the total electromagnetic force on one due to other is not equal and opposite. I cannot provide more details in a short comment and ask you to refer to a text on electromagnetism. The text by Griffiths covers it. $\endgroup$
    – prateek
    Feb 3, 2021 at 16:52

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Local conservation of momentum is obeyed in general relativity, which is encoded in the rule that $\nabla_{a}T^{ab} = 0$*, which is a direct consequence of Einstein's equation. That said, you can get the same sorts of "violations" of Newton's third law in GR that you do in Electromagnetism -- namely, that momentum can be carried away by the field, which make action/reaction forces nonequal from a picture of "mass A exerts a gravitational force on mass B", which is an incomplete picture if the dynamics of the field (or spacetime curvature if you prefer that) are important.

*The reason for this is that the stress energy tensor is constructed in such a way that, for any timelike trajectory $u^{a}$, the 4-momentum of matter observed along that trajectory is given by $P^{a} =T^{ab}u_{b}$, and so this divergence rule is just a generalization of the standard E&M continuity equation.

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  • $\begingroup$ Does similar reasoning goes for QFT as well and any yet known unified theory? $\endgroup$
    – prateek
    Feb 3, 2021 at 17:07
  • $\begingroup$ @prateek: QFT is built on special relativity, so it has local 4-momentum conversation built into it, yes. And any unified theory that is worth investigating should reduce to qft/gr in the appropriate limits. $\endgroup$ Feb 3, 2021 at 17:54

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