# What does the actual third law really look like?

Newton's third law of motion states (as quoted on wikipedia):

When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

$\textbf{F}_{12}=-\textbf{F}_{21}$

Clearly, when we consider actual systems, where information is bound to progress at a finite speed, this law becomes irrelevant- it is impossible for two bodies to simultaneously exert force on each other- they each interact with a retarded field left by the other's trail through space, to do with it's location in the past (or, using the mathematically valid yet physically dubious advanced field to do with it's location on the future).

My question is, then, what is the true form of the third law for fields exchanging information at finite speed of light (if such a relativistic form exists)?

As the third law is obviously a form of conservation of momentum, the true form of it must be (according to Noether's theorem) derived from spatial transational symmetry.

If this law uses a retarded field or potential, can the advanced field or potential be used and give the same results? Will it still be physically meaningful?

• Hint: the law is momentum conservation disguised, and your observation requires the field to have a variable momentum, so it's a three-body problem and you can accordingly write a momentum-conserving force law that honors that. – J.G. Jun 25 '17 at 23:10
• How does a body exert a force on another ? Is contact required ? – HsMjstyMstdn Jun 26 '17 at 2:31
• What about retarded potentials? – SRS Aug 7 '17 at 18:29