Is determinism broken in special relativity?

Question

Under classical mechanics, in an isolated system everything is deterministic given some initial conditions. Otherwise, we would have to consider some probabilities of interactions with the outside on the border of the system.

But what if a system is very far away from other objects (and fields)? Say we have a single particle drifting through space. Determinism says it should move in a straight line with constant velocity. Would it be possible that another particle (say a photon) traveling at the speed of light comes and hits our particle, without the particle knowing this was going to happen? Then the trajectory of our particle will change due to the momentum transfer in the impact. Now, if we were to reverse time (and velocity) then the particle doesn't go back on the same line it started on; unless we know for sure that the same photon also came back and hit the particle to change its trajectory.

The photon didn't exist before in the system, it came out of nowhere by pure chance (ignorance of the outside world), but now we have to consider it when rewinding, otherwise we lose the particle's history (information). If we assumed that the whole energy and momentum of the photon gets absorbed by the particle, then in reverse it would look as if the particle spontaneously emits a photon (randomly, by itself) just so it can change course.

But let's say that colliding photon just came out of nowhere and then got deflected back to nowhere. There's no clear way of defining an isolated deterministic system. Unless of course we consider the whole Universe, then we would know through the eyes of a demon that the photon would come. But shouldn't the laws of physics be local? Shouldn't the particle "know" or "feel" that the photon is coming? But interactions only travel at the speed of light, so my logical conclusion is that you can never know in a small region of space what's coming at a certain time from very far away.

Is my logic failing and where?

Answer 1

How can I model it other than in a random fashion? If I don't know that the emitting star exists, I can't know a photon will come.

Your knowledge or ignorance of the photon is irrelevant. The state of the universe included a photon headed your way, and the universe evolved deterministically from there. In practice, there are many times when it is certainly unrealistic to expect to be able to fully model all of the relevant influences on a system, and so including some random processes may be necessary; however, this does not mean that the underlying theoretical framework itself is not deterministic.

If we assumed that the whole energy and momentum of the photon gets absorbed by the particle, then in reverse it would look as if the particle spontaneously emits a photon (randomly, by itself) just so it can change course.

Isolated charged particles cannot absorb (real) photons. They may scatter elastically (or inelastically if the particle has internal degrees of freedom), but the photon cannot be fully absorbed by a free charged particle because it is impossible to conserve energy and momentum in such an interaction. You'd need e.g. a second charged particle (see Bremsstrahlung) or some additional influence.

But shouldn't the laws of physics be local? Shouldn't the particle "know" or "feel" that the photon is coming? But interactions only travel at the speed of light, so my logical conclusion is that you can never know in a small region of space what's coming at a certain time from very far away.

You are mistaken as to what it means for interactions to be local. Indeed, you are precisely describing nonlocality - the ability to "feel" the presence of an influence which is localized to a different region of space.

If I stand in the middle of a park and get hit in the back of the head by a ball I didn't see coming, that doesn't mean that the universe is non-deterministic; it means that I was ignorant of the full state of the universe. If I had known the state of the universe, then I would have known that I was about to be hit, I would have known when and how it would have happened, etc.

That is the essence of deterministic time evolution - complete knowledge of the state of a system at one time leads to complete knowledge of the state of the system at later times. In contrast, non-deterministic time evolution means that even if I know absolutely everything about the state of a system at one time, I cannot predict with certainty the state of the system at some later time.

This is most common in (some interpretations of) quantum mechanics, wherein measurement of a system is associated with non-deterministic time-evolution. Even if I know the state of a system before a measurement, I can only predict the probabilities of various outcomes and post-measurement states. Some interpretations (e.g. Many-worlds) adopt different perspectives on this, but the "measurement problem" remains unresolved on the whole.

However, classical special relativity (as with classical mechanics) does not feature any inherent non-determinism. Of course, as you say you may choose to incorporate stochastic dynamics into a model in order to include influences which are seemingly random (see e.g. Brownian motion or diffusive processes), but this non-determinism is not "baked in" to the framework.