# Violation of Newton's 3rd law and momentum conservation

Why and when does newtons 3rd law violate in relativistic mechanics? Check this link http://www.animations.physics.unsw.edu.au/jw/Newton.htm.

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It would help a lot if you include in this question the reason why you would expect Newton's 3rd law to be violated in relativistic mechanics. Don't rely on people being able to click the link. – David Zaslavsky♦ Nov 2 '12 at 16:48
– Qmechanic♦ Nov 2 '12 at 17:32

## 1 Answer

Newton's third law is naively violated in relativistic mechanics when there is field potential momentum. This happens in basically any magnetic field situation where there are also charged objects.

Feynman gives a simple example, two charge particles, one moving directly towards the other and the other one moving in some other random direction (not towards the first). The electric forces are nearly equal and opposite (up to relativistic corrections that are lower order) and the magnetic force (the first relativistic correction to the Newton's third law consistent Coulomb repulsion) is only nonzero on one of the particles, since there is no magnetic field along the line of motion by symmetry.

Newton's third law always holds in special relativity if it is expressed as conservation of particle momentum plus field momentum, since it is a consequence of translational invariance plus a Lagrangian formulation, or of translational invariance plus a Hamiltonian formulation, or of translational invariance plus quantum mechanics. This implies conservation of momentum, which implies that any forces between two bodies must be balanced (since the force is the flow of momentum). For direct 3-body forces, Newton's 3rd law, as stated by Newton, fails even in the nonrelativistic limit, as explained in this answer: Deriving Newton's Third Law from homogeneity of Space .

There is a stronger formulation of momentum conservation that holds in relativistic field theories. Energy momentum is not just conserved, it is locally conserved, which means that there is a stress energy tensor $T^{\mu\nu}$ which obeys

$$\partial_\mu T^{\mu\nu} = 0$$

This equality tells you that the flow of momentum density ($T^{0i}$ for $i=1,2,3$) across any surface is conservative, with a current equal to the stress (the force per unit area across an infinitesimal surface). This is the local form of Newton's third law "The force is equal and opposite". If you also add "The force between distant objects is collinear" (which is implied by conservation of momentum), the relativistic version is that there is a stress tensor choice which obeys

$$T^{\mu\nu} = T^{\nu\mu}$$

This says that the flow of the j-component of momentum in the i-th direction is equal to the flow of the i-th component of momentum in the j-th direction, which implies collinearity of force for distant objects transferring momentum through a field.

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