# Nature of Experiment or Measurement tied to Special Relativity concepts

Premise: In special relativity, we come across concepts like Length Contraction and Time Dilation. Pertaining to these topics, I have come across the following diagram (or its slight variations) in many books:

My Question: Is the manner in which we assign coordinates (x,y,z,t) to an event, measure length contraction and time dilation limited to experiments like these (involving setup as shown in the diagram)? How intricately is our theory tied to the nature of experiment we carry out?

Does any expression (say, in the set of equations characterizing Lorentz Transformation) change when we change the nature of the experiment?

Clarification: I do not know if it is even possible to devise a truly different experiment for measuring the same quantities. If it is absurd to even talk of doing that, kindly let me know why it is so.

Any help would be appreciated.

Usually when people study relativity in school, they learn special relativity in a formalism that gives a preference to a certain type of coordinate system, the Minkowski coordinates $$(t,x,y,z)$$. Then they learn general relativity, in which all coordinate systems are allowed. Really this is just a historical artifact or an effort to avoid introducing too much new math. Special relativity can be done in any coordinate system. For example, if $$x$$ is a Minkowski coordinate, I can choose some new coordinate $$u=x/a+(1/3)\sin (x/a)$$, where $$a$$ is a constant, and then the coordinate system $$(t,u,y,z)$$ is just fine.
What is special about Minkowski coordinates is that they're orthonormal, meaning that when you form unit vectors by doing unit coordinate displacements, those unit vectors have magnitude $$\pm 1$$ and are orthogonal to each other, e.g., $$\hat{\textbf{t}}\cdot\hat{\textbf{t}}=1$$ and $$\hat{\textbf{t}}\cdot\hat{\textbf{x}}=0$$. We generalize the dot product from 3 space dimensions to 3+1 dimensions of spacetime in order to be able to talk about these things. The underlying mathematical machinery that allows us to define these dot products is called the metric.