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*) Special relativity is the geometry and trigonometry of
spacetime ... nothing more and nothing less.
*) *THE* length of a ruler is its proper length, defined in
the frame comoving with the ruler.
*) *THE* timing of a clock is its proper time, defined in
the frame comoving with the clock.
*) *THE* mass is the invariant mass.
*) The physics does not care what frame (if any) is chosen
by whatever observers (if any) are involved.
*) The /projection/ onto this-or-that frame will depend on
the choice of frame, but that's just the projection; that's
not the real physics. The shadow of a ruler changes if I
rotate the ruler, but that's just the shadow; *THE* length
of the ruler is invariant. This is not entirely a new idea;
it has been understood for 2300 years that the thing that is
most easily observed is not necessarily the best representation
of the underlying reality.
https://faculty.washington.edu/smcohen/320/cave.htm
Special relativity is the geometry and trigonometry of spacetimeIt may be dangerous to represent our knowledge as an ultimate truth. First, SR is the geometry of the flat spacetime rather than any spacetime. Second, and more important, geometry of spacetime itself is, according to GR, determined by distribution of matter (its momentum-energy tensor). Quoting Wheeler & Co.: "Matter tells spacetime how to curve; spacetime tells matter how to move." So geometry, on the one hand, and dynamics (energy-momentum distribution) on the other, are the two inseparable sides of one coin. Focusing on SR: the norm of a 4-vector is invariant under the Lorentz-transformations (or spacetime rotations).For a time-like 4-displacement its norm is the proper time between the corresponding events (in units with c=1).In an arbitrary inertial RF, its temporal and spacial components are the corresponding projections of the 4-displacement. Similarly, for a time-like 4-momentum, its norm is the invariant mass of the respective object. Object's energy and momentum in an arbitrary RF are the corresponding projections of the 4-momentum vector. And one can illustrate such projections as the (appropriately cast!) shadows. No objections to this. But for a space-like 4-displacement, its norm is the proper distance (spatial separation between the events in RF K where they are simultaneous). Denoting such separation as S, we can write
... nothing more and nothing less.
by whatever observers (if any) are involved."and- contradict each other. A more detailed discussion of all these questions can be found in: Three +1 Faces of Invariance, arXiv:1001.0088 [physics.gen-ph]
"*) The /projection/ onto this-or-that frame will depend on
the choice of frame, but that's just the projection; that's
not the real physics."