Newton's third law in extremis

Is Newton's third law always true?

If space and time are entangled, as suggested in the special theory of relativity, we'd expect there to be some differences, as the argument for invariance depends on Charlie, Bob and Alice recording events without taking account of the time for the information about that event to reach each of them. (In fact, the argument assumes the Galilean theory of relativity, where everyone agrees on the passage of time.)

If you change how you record the measurements of space and time then you'll also change how the forces appear.

In contact forces it seems reasonable to postulate that the forces are the same at all instants of the interaction, but for the kinds of interactions that are modelled by non-contact forces this is less clearly the case.

In fact, any non-contact force presents an interesting challenge – make a change at the emitter, and you won't know about it until the photons are absorbed by your hand (1 metre or 3 nanosecond away). These are difficult ideas and not for classrooms, but it's important that you're clear that the world as imagined by Newton is a very good mimic of some aspects of the lived-in world, but that it's not the final imagining.

So Newton's third law is the one of the three laws of motion that has been shown to be inadequate at very high speeds.

Fortunately, the idea of momentum and its conservation is so useful that it does get reused in the special theory of relativity. The idea gets developed and extended to be more general, and even more powerful.

In the imagined world you've been investigating there are things on which Alice, Bob and Charlie all agree (the acceleration and the force, and the mass), so long as they're all travelling at constant velocities.

Knowing what does not change is a very valuable guide to understanding a new situation (the whole of the SPT: Energy topic is an example of the value of that kind of thinking).

Physicists, as they explored the new world that was opened up by Einstein's rethinking of time, looked for things that did not change. They found two:

• Duration and displacements might be recorded as being different by Alice and Bob, but they both agree on a new, generalised measure: the spacetime interval.
• Energy and momentum might be recorded as being different by Alice and Bob, but they both agree on a new, generalised measure: the momenergy.

Both of these unchanging entities live in four-dimensions, so are necessarily vectors.

These arise because of two more fundamental truths:

• There is one universal speed, on which Alice, Bob and Charlie will always agree, and light travels at that speed.
• The physical laws are agreed on by Alice, Bob and Charlie.

One such physical law is: d →p d t = →a.

You might remember this from episode 03.