@Dalespam
you must remember the fact that deriving Lorentz transformation "the isotropy of space and homogeneity of time" was assumed... and now you are violating that here... now tell me one thing when space-time is curved... STR is valid? no! then how this "s" STR u expect to be valid... then...
but if in some media "s" is the ultimate speed and through this the information can travel and no more than that speed... then it is obvious that "s" will remain invariant... because we are using "s" for highest precision to measure "s" itself... as its the highest speed with which information...
I've found that the momentum expectation of a particle inside an one dimensional finite box of length 'L' is '0'... we can say the probability of going right=that of the left... so sum up to zero. Now I calculate the same(<p>) for a free particle in one dimension if I think that free particle is...