Relativity has always been explained to me (in books I've read, etc) as a superset of newton's laws - that is; it encapsulates all of Newton's mechanics in addition to other effects (observer effect, time dilation, space-time geometry, etc).
I can kind of imagine 2 of Newton's 3 laws of motion to be incapsulated within Einstein's Special Relativity, but the one about "for every action there is an equal and opposite reaction", I'm struggling to find a place for that within Special Relativity.
Does this sit outside Special Relativity's explanatory power or is it inferred within somehow?
Answer
Newton's third law is really a special case of the conservation of momentum. Suppose you have two rigid bodies with momenta $\mathbf{p}_1$ and $\mathbf{p}_2$. If they only interact with each other, then $\mathbf{p}_1 + \mathbf{p}_2$ is constant, since total momentum is conserved. Differentiating this gives $\frac{d\mathbf{p}_1}{dt} + \frac{d\mathbf{p}_2}{dt} = 0$. But this is just $\mathbf{F}_{21} + \mathbf{F}_{12}$ (i.e., the external force exerted on body 1 plus the external force exerted on body 2).
In relativity we usually don't use the concept of force, preferring to deal with momentum instead. Momentum is conserved in relativity, just like it is in Newtonian mechanics.
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