Bjarne said:
Right
I agree (and "disagree").
Notice Observer “Ex” (external) will not see any length contraction.
Seen from the perspective of "Ex" the distance of the Milkyway would be the same for both A and B.
Correct, some observer floating away from the milkyway would see these distances the same.
Bjarne said:
B is deeper inside the gravitionel field of the Sun. He will complete 1 orbit in less time as A.
If B shall have the right to claim that the orbit of the MW is shorter (length contraction), it is only possible if B’s ruler is comparable longer than A’s.
Do you understand that point? – It seems like a contradiction but it is not, but rather a mathematical necessity that B’s meter stick must be longer than A’s.
(You must also respect the mathematical reality of observer Ex, - Observer Ex must also have the possibility to understand other realities - relative to his own )
Here, you have confused several things. The length contraction I described had to do with local rulers as perceived by a distant observer. However, I never proposed a way to use local ruler measure for astronomic distances (it is possible, building a ladder of distances).
Totally independent of the issue of local rulers perceived from a distant free fall observer, I proposed a different, simple convention for astronomic distances (radar ranging, using local time, and
assumption speed of light is isotropically c. It is only using this convention (rather than local rulers) that you end up with shorter distances to distant objects, and thus the same speed measured by A and B.
Note that whatever definitions are used,
some measurements by A and B will differ (assuming each uses the same definitions). This is not unexpected or inconsistent with invariance of laws of physics.
Let's state what is really claimed by different relativity principles:
1) Galilean relativity: All laws take the same, simplest, form in any inertial frame. Note, this never meant that measurements are the same, only laws (equations) relating measurments. The main thing wrong with this was that its law for velocity transformation between inertial observers turned out to be experimentally incorrect. Between observers with relative acceleration, there is no simple relativity, and laws take more complex form.
2) Special relativity: Same principle as above, except the transformation law between different frames is different and consistent with experiment. In particular, there is no 'relativity' between observers undergoing relative acceleration.
3) General relativity gives you both less and more. The laws of special relativity only apply locally, for inertial observers, defined as those in free fall. There is no unique answer at all to such things as long distances or velocity of a distant object (whether for inertial observers or non-inertial observers). Instead there are only useful conventions you may choose, and procedures for making valid physical predictions based on whatever conventions you choose. There is a general formulation of laws such that whatever conventions are used by any observer, the laws in this form apply (but measurements are not the same). However, the same conventionality of coordinates means, in practice, you use transformation rules to convert your measurements to the most convenient coordinates for calculation.
Based on (3), your A and B observes each know they are non-inertial; they know the magnitude of their acceleration. Seeing the sun, and making measurements, they can determine the quali-local structure of spacetime. What each does, in practice, is convert their local measurements, using the predictions of GR to accomplish this, to milkyway center coordinates (each able to determine a different required clock adjustment, for example). They compute distances, speeds, etc. in this frame. Each one doing this ends up with the same predictions and values. This is all that is expected, and found to be true.
Bjarne said:
B and A’s perception of speed can also not possible be the same, simply because B’s clock is ticking slower.
You cannot make such a blanket statement. It depends on measurement conventions. I have shown that there exists a simple convention that has the property that A and B differ on distances and times such that speeds of distant objects come out essentially identical. Other equally valid measurement conventions will lead to different results. However, GR provides the precise rules allowing A and B to make the same physical predictions whatever consistent conventions they use, and compare results, as long as each knows the other's conventions. The requirement on consistency here are very broad (one-one mapping of spacetime, continuity conditions, etc.).
Bjarne said:
We should not be allowed to mix realities, hence also not to force our (A’s) perception of speed into B’s reality.
There is really one reality in GR - the spacetime manifold. There are many ways to label events in it, and many different physical processes for taking measurements, that can be used at different places, times, instrument speed etc. GR allows any of these to be used to probe the underlying reality. However, the underlying reality does not include statements such as a unique valid distance between distant objects, nor a unique valid relative speed between distant objects.
Bjarne said:
So since B’s time-rate is ticking slower, - that alone should mean that B moves FASTER than A, - but because B’s ruler (seen from a mathematical point of view) must be longer the speed is the “same” – but not comparable the same.
Notice A and B will agree to complete the orbit of the MW in the exact same period, but they can impossible agree about distance / circumstance / time / rulers.
That all depends on how they take and interpret measurements. Using raw local measurements, some of these must disagree (but not necessarily all of them, and many choices about which differ). However, if each converts their measurements to an agreed common coordinate convention,
using the predictions of GR, they will agree on everything.
Bjarne said:
I appreciate your contribution to the thread and I understand most of what you have explained, but still I wish there was a simpler way to understand and compare how B's reality really is, as well as understand how would B’s ruler would be compared to A’s.
I think there still is more to discover to make that simpler, straight and logical.