Thought experiments are very common in Special and General Relativity (SR,GR). Usually a thought experiment is structured as follows:
- We present a setup in a frame of reference
- We apply principles of SR or GR to derive what must happen
- We then shift perspective to another frame of reference
- We state that both observers should agree on what's happening
- We at last draw the conclusions
This structure lets us derive directly from the postulates effects like Time Dilation, Light Bending in a Gravitational Field and so on.
My problem regards the point 4.: 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.
My question is: The fact that both observers, in different frames, should agree on events is a principle? If so what is the precise wording of this principle? Does this principle have a name?
It blows my mind that apparently there isn't a precisely stated principle regarding what the frames must agree on. Dale's answer helps a bit but I can't help feeling like it's a partial exposition on what's going on.
For me the best way to put it is the following: Different frames must agree on explosions, meaning that if a frame sees an explosion then all frames must see an explosion as well; maybe they will see a fireball with a different shape or maybe they will see the fireball expanding at a different time ecc. But all frames must agree on the fact that an explosion has occurred. This is important because I can think of attaching an explosive device to a multitude of physical system; for example I can take an instrument that measures the frequency of light hitting it and attach to it a device that explodes if the instrument measures a specific frequency. So then we can surely say that all observers, in different frames of reference, must agree that the instrument measures that specific frequency.
The upper reasoning seems to solve the problem, but surely is not a precise formulation; and also remains the problem that seems to me absurd that no separate principle regarding this is clearly stated in the literature.
Edit: A lot of answers revolve around the idea that the principle of relativity (or the principle of general covariance) implies that different observers must agree on what a specific experimental setup is measuring; or to say it in another way, that all observers should agree on events like the presence of an explosion. But I cannot see why this implication holds: the principle of general covariance states that the laws of physic must be the same in every reference frame, but this does not mean that all the observers must agree on what those laws of physics predict for a specific object. This is an entirely different statement.