[ebg] “Then, the only difference between ruler A within earth frame and ruler B within earth frame is scale.”
Unfortunately that is not true in SR. The real problem with this sort of comparison (using SR) is that from the perspective of the rocket frame each end of ruler B exists in the present. But from the perspective of the Earth frame each end of ruler B exists at different times, neither of which is the present. This makes it significantly more difficult for the two observers to agree about their observations.
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In order to measure the position interval (AKA “length”) of an object one must record the position coordinate of each end of the object and subtract. For a stationary object this is easy, but if the object is moving you have to be careful to either:
1) Take both position readings at the same time.
or
2) Record the time of each reading and correct for the time difference (using the known speed of the object).
in order to get an accurate length calculation.
But there is also that other complication mentioned above – the one where SR says that time in the moving frame is different from time in the fixed frame. This is the infamous “lack of remote simutaneity” of SR. Specifically, SR says that time in the moving frame is a function of the velocity difference between the frames AND a function of the position difference between the frames. The Lorentz transform for time is:
t_mov = gamma * ( T_fix – (v * X_fix / c^2) )
It’s that “v X over c squared” term that is the culprit.
So the moving observer and the fixed observer have to do some additional calculations related to time-of-observation for each coordinate before they can agree on the length of the rocket or the length of the rulers.
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Understanding SR is not all that hard. The math actually works, and is able to accurately describe / predict the things we see. Very useful, but not the whole story. Believing that SR is a physically accurate explanation of reality (time itself changes, rather than clocks) in addition to being a mathematically acurate description, however, is quite hard for some of us.
I lean toward Lorentzian relativity. It has a preferred frame (the local gravity field), a universal time (now everywhere) and no frame reciprocity (if I see your clock running slower, you see mine running faster).
The math of LR (see Tom’s primmer on LR) is equally able to describe and predict the things we see (so far). But physically it is much more like what you imagined where the only difference between ruler A and ruler B is their scale. None of SR’s paradoxes exist in LR because everything (right here AND way over there, moving AND stationary) happens now. Remote simultaneity does exist in LR.
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Math is not physics. Math is not even the language of physics. It is, however, one of the languages of science. It is arguably the most important one. The other three are logic, art and English (or Russian or Spanish, etc.). All of them are vital to the proper conduct of science.
Regards,
LB
