A Hidden Principle in Relativity

From what I understood of your question, I will try to explain my perspective. Two observers 'agreeing' on an event is a semantic detail. An $textit{event}$ is any labeled point $A(t,x,y,z)$ on the space-time diagram of a phase space $S$(say). Two different observers correspond to two different charts/coordinates of the $textbf{same}$ phase space $S$. So another observer will measure the event $A$ at $(t',x',y',z')$. The fact is that the same event (as labeled in an arbitrary frame) exists for both the observers. Now, in special relativity, you just have to carefully define what constitutes an event in a physical problem and you should not find an ambiguity in saying that two observers "agree" (measure) on a particular event.

In every thought experiment we state that observers should agree on what is happening in some way. Problem is that this is not always true! If I state that observers in different frames should agree on simultaneity, for example, I am obviously in the wrong. But if I state that both observers should see a mug breaking I am probably correct.

Some quantities are invariant*, meaning that all frames agree on their value. For logical consistency the outcome of any measurement must be invariant.

If my clock measures the time between two events to be $tau$ then all frames will agree that my clock measured $tau$ even if their clocks measured something else. Same with simultaneity, length, or any other measurement I might make.

Other frames would not agree that my measurements were valid measurements of length or time or simultaneity in their frames, but they would all agree on the values that I measured. Thus the outcome of any measurement is invariant.

This principle is necessary for logical consistency, but as far as I know it doesn’t have a special name. At a minimum, it is part of the principle of relativity. When we say “the laws of physics are the same in all frames” what we mean is precisely that we can apply the same laws of physics to any scenario described in any frame and all of the measured outcomes will be invariant.

*The most certain way to recognize an invariant quantity is to mathematically transform it to a different frame and see if it stays the same. When done for a generic transform then it definitively indicates invariance. However, usually the easiest way to recognize an invariant quantity is simply to write it as a contraction of tensor quantities. This is called "manifestly invariant" or "manifestly covariant". In practice, that is the method used most often.

If you and I both draw accurate maps of the world, our maps might disagree on things like "How is China colored?" or "Which direction (north? south? east?) does 'up' represent?". We will not disagree on things like "Does China touch India?" or (if our maps are topographical maps) "Which country has the highest mountain?".

What principle determines the ways in which our maps must agree and the ways in which our maps might disagree?

I'm not sure what you'd consider a satisfying answer to that question. But if you can form an answer that satisfies you, you can translate it into an answer to your original question via an analogy that replaces the earth with spacetime and our maps with reference frames.

First, it looks like you are interested in things that are invariant between frames, but not how we normally think about invariance. Let's look at what we usually mean though. For example, all frames will agree on the space-time interval between two space-time coordinates, even if the coordinates themselves are different. We would say that space-time intervals are invariant under Lorentz transformations in special relativity. This type of invariance has a simple rule: if the quantity does not change under any Lorentz transformation, then it is invariant.

So this covers things like space-time intervals, magnitude of four-velocity, etc. But it doesn't cover (or at least sufficiently cover for you?) what you seem to be asking about:

(From a comment) Yes. A rule that tells me which facts must be identical in every frame of reference. For example the fact that: the distance between two things is two meters, can be different in different frames; but the fact that a bombs explodes must be true in every frame. Maybe for me it explodes now and for you after a year, but it should explode for everybody. Problem is: in relativity there are different kinds of facts, some we must agree on and some not, and I want a precise way to distinguish them, and I also what to know where this rule comes from. The world "fact" here is used in a philosophical way

I suppose one could say these types of "facts" are also Lorentz invariant, but I think you want an intuitive / philosophical(?) reason why. I would say the "principle" you are looking for is that "all frames agree on the existence of events". For example, a bomb exploding is an event. The notion that my measured time interval is different than your time interval is not an event.

I might be off here, but I think you are essentially looking for is related to causality, or at least events that would produce some form of causality. For example if I say that my house burned down, but you say that it didn't, then you are also refuting all causes$^*$ leading up to my house being burned down. If all frames do not agree on the existence of events, then all frames are not following causality, which is a logical issue.

So essentially, perhaps my homemade principle for you would be "All frames agree on the existence of events, where events are things that have / produce a cause / effect".

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$^*$ Or some subset of previous causes?

I do not agree with the premise of the question, that is the structure of a though experiment. Here is how I would put it instead:

1. We present a setup as seen by an observer
2. We use SR or GR to derive how that setup is seen by another observer
3. We check that it is indeed the case that the second observer see what's predicted (and if it is then SR or GR has not been invalidated)

And the above is precisely what "both observers agree on what is happening" means.

SR or GR allows any observer to know what the other observer should see. That's the agreement.

You seem to be looking for some absolute description, that somehow defines what is really happening in the sense that it is not dependent on the perspective of any specific observer. But there is no such absolute description - this the whole point of the notion of relativity.

Let me give it a try, too. I would state the principle as follows.

We assume, that an abstract reality exists, with an abstract notion of spacetime(*), abstract notions of energy, mass, momentum, frequency, etc. This abstract reality is where the "real"/abstract physical laws live. All observers agree on this abstract physical reality.

What they don't necessarily agree upon is the numbers they apply to physical entities. This includes the coordinate system they apply onto space time, it includes the units they use and it includes all phenomena that depend on coordinate system or units. And since we merged space and time into spacetime, it includes the notion of simultaneity.

Let me take your example with the bomb that's attached on a wavemeter. The bomb explodes if the wavemeter detects a wave of a resonance frequency $nu$. An observer of a different frame will agree that the bomb exploded, but she will not agree that the wave actually had the frequency $nu$. She will neither agree, though, that $nu$ is actually the resonance frequency of that wavemeter. Instead she will say that the resonance frequency of that device was $nu'$ and that the frequency of the detected wave was $nu'$, too. Thus, they both agree on the abstract frequency of the wave as well as on the abstract resonance frequency of the measuring device, they will merely not agree on the numerical value of these frequencies.

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In mathematics this principle is realised in the notion of a manifold. There, we have an abstract point set and we have multiple charts which map these abstract points onto numbers (coordinates). With this done, one defines abstract (tangent) vectors and tensors, which can represent quantities like energy, momentum or frequency. With a given chart (also called frame), vectors and tensors obtain a numerical value.

The general covariance principle and the relativity principle now state, that it's possible (and that's what we do in practice) to describe the abstract reality from any frame (ie. apply any chart to the abstract manifold) and even though the numerical values used to describe the state of the world will differ, the physical laws used to describe the evolution look identical in every frame. The result of applying the physical laws to predict the evolution, is a local description of the evolved state, which again corresponds to an abstract state of the abstract system. No matter which chart you used upfront, after "unapplying" that chart, you'll reach the same abstract state.

In formula: Let $varphi, varphi': M to mathbb{R}^4$ be two different charts on the abstract spacetime manifold $M$. Let $Psi$ be an abstract state (a tensor) and $T$ be the time evolution of a system, in local coordinates. Then $varphi^{-1}(T(varphi(Psi))) = varphi'^{-1}(T(varphi'(Psi)))$.

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I don't think, that this principle has a name in the context of relativity theory. Though, it reminds me quite a lot of the reality principle that appears in Bell's theorem.

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(*) note: prior to SR, people would have said "abstract notion of space and time", so Einstein did modify this principle (which was obviously there before).