Is the state of a system invariant or not? In a physics textbook by Eric Mazur there is the following phrase:

“The state of any object or system cannot depend on the motion of the observer.”

Here, the state is understood as some complete set of physical parameters of the system: shape, temperature, … In the textbook this statement is used both in the Galilean relativity and special relatively to argue that some physical quantity is invariant.
I have 2 questions on this.

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*To me, the statement that “the system’s state is invariant” (which means that any attribute of a system or an object should be invariant: shape, size, inertia, mass, charge, temperature, …) intuitively makes sense, but how can one prove that?

*If this statement is true, why does it fail for inertia and length of an object, which moves at relativistic speed relative to observer (special relatively)? Inertia and length are obviously attributes of a system, thus should be invariant (like charge and temperature).

 A: The state of a system is an invariant in the sense that any well-defined experiment should yield the same result regardless of the observer. However, it is extremely common to mistake what is a well-defined experiment.
Asking what is the length of an object, or it's energy, are not well-defined experiments, because these terms have different meanings for different observers. What an observer calls "energy", another calls "a specific linear combination of the object's energy and momentum". Nevertheless, if you specify your experiment by actually giving an experimental prescription, there is no ambiguity. Different observers might interpret the result differently (one of them calls it energy, the other calls it something else), but the result is the same.
As for proving this, I'd say it is a logical necessity. Pick an arbitrary experiment. Now couple a bomb to it such that, if the result is above a certain value, the bomb goes off and kills the experimenter. Now, all observers must agree on whether the experimenter died due to the bomb or not. Hence, all observers must agree on the result of the experiment. Otherwise, changing observers would change reality itself.
Of course, different observers might give different names to the quantity being measured. One of them calls it energy, another calls it a linear combination of energy and momentum, etc. However, this just shows that our words are often poorly chosen, leading us to think an experiment is well-defined when it isn't.
As a further example, suppose two different observers try to measure the energy of some particle (each of them according to their own notion of "energy"). These are different experiments, because you have to use different apparatuses or couple them differently to the particle. Hence, asking to measure the energy of a particle is not a well-defined experiment, since each choice of observer leads to a different experiment.
