Are you assuming (or concluding) that distance and relative motion add the way they do in Newtonian physics? That is observed to not be the case, on the scales and precisions of the GPS system.
In other words, the vector difference between the velocity of object B and C varies depending on the observer, and adding the same constant vector to the velocity of B and C changes the difference between their vectors.
No, I’m saying that you can do or transform all your observations to one frame of reference, and all observers will agree on the results within that frame of reference, no matter what they’re transferring from.
Your assertion is not consistent with current observational evidence. Observers in different frames of reference will not agree on what percentage of a sample of an unstable isotope will decay in the time it takes light to travel the length of an artifact.
Not until they have transformed their observations to a common frame, no. Please re-read what I said. Then let’s see some math. Make an observation of length in one frame of reference, transform it to another, and show that it still disagrees with what observers in that frame of reference measure. Alternatively, make two observations in different frames, transform them to a third, and show that they disagree.
Observer C notices that at time 2, object C is stationary, object L is moving left at 9 Million Kilometers/minute, is 18 Million Kilometers long, and that stick L is 18 Million Kilometers long. C also notices that object R is moving right at 9 Million Kilometers/minute, is 18 Million Kilometers long, and that stick L is 18 Million Kilometers long. C notes a distance of 36 MKm between L and R when they pass the ends of their co-named sticks.
Observers L and R observe that sticks L and R are moving right at 9 Million Kilometers/minute, and that their observed length is 13.5 MKm. They observe that they are 27 MKm apart when they pass the end of their sticks.
Lights at the extreme ends of sticks L and R flashed simultaneously (at the same place and time) with lights located on objects L and R. 45 seconds later, the light from objects L and R has traveled the 13.5 MKm to C; 15 seconds after that, the light from the ends of the sticks arrives (having traveled 18 MKm).
The speed of L to R (and distance between them) is left as an exercise to the reader.
10.1 MKm (13.5MKm *(1-(1/2)^2) for the distance between L and C, but 18MKm for the length of the stick which extends between them. in the reference frame of C. (Distance only diminishes in transformation, while length increases when transformed towards a frame of reference closer to that of the object in question)
Observer L is in the rest frame of object L; consequently he measures its proper length. Observer C is not in the rest frame of object L, and therefore measures a shorter length. Your claim that L measures a shorter length than C indicates that you’re using something other than special relativity, that you’re using special relativity wrong, or that I didn’t understand your description. Please clarify which it is. A diagram may be helpful.
Additionally, this:
Distance only diminishes in transformation
is plain wrong, since you can always do the transformation in the other direction! It doesn’t diminish both ways! And this claim:
length increases when transformed towards a frame of reference closer to that of the object in question
appears to be meaningless; there’s no such thing as a frame of reference “closer to an object”. A frame of reference doesn’t have a location, it has a speed relative to an observer.
Sticks L and R are stationary in the frame of C; objects L and R just happen to be close to the other ends of the sticks at some time, and are stationary in their own reference frames.
At this point I have no idea what is supposed to be happening in your scenario. You’ve got sticks, objects, and observers, and it’s not clear whether any of them are the same or what frames they are in. Please draw a diagram.
Are you assuming (or concluding) that distance and relative motion add the way they do in Newtonian physics? That is observed to not be the case, on the scales and precisions of the GPS system.
In other words, the vector difference between the velocity of object B and C varies depending on the observer, and adding the same constant vector to the velocity of B and C changes the difference between their vectors.
No, I’m saying that you can do or transform all your observations to one frame of reference, and all observers will agree on the results within that frame of reference, no matter what they’re transferring from.
Your assertion is not consistent with current observational evidence. Observers in different frames of reference will not agree on what percentage of a sample of an unstable isotope will decay in the time it takes light to travel the length of an artifact.
Not until they have transformed their observations to a common frame, no. Please re-read what I said. Then let’s see some math. Make an observation of length in one frame of reference, transform it to another, and show that it still disagrees with what observers in that frame of reference measure. Alternatively, make two observations in different frames, transform them to a third, and show that they disagree.
Observer C notices that at time 2, object C is stationary, object L is moving left at 9 Million Kilometers/minute, is 18 Million Kilometers long, and that stick L is 18 Million Kilometers long. C also notices that object R is moving right at 9 Million Kilometers/minute, is 18 Million Kilometers long, and that stick L is 18 Million Kilometers long. C notes a distance of 36 MKm between L and R when they pass the ends of their co-named sticks.
Observers L and R observe that sticks L and R are moving right at 9 Million Kilometers/minute, and that their observed length is 13.5 MKm. They observe that they are 27 MKm apart when they pass the end of their sticks.
Lights at the extreme ends of sticks L and R flashed simultaneously (at the same place and time) with lights located on objects L and R. 45 seconds later, the light from objects L and R has traveled the 13.5 MKm to C; 15 seconds after that, the light from the ends of the sticks arrives (having traveled 18 MKm).
The speed of L to R (and distance between them) is left as an exercise to the reader.
You don’t appear to understand what I mean by “transform”. What does L say the speed (or length, or time) is in the reference frame of C?
10.1 MKm (13.5MKm *(1-(1/2)^2) for the distance between L and C, but 18MKm for the length of the stick which extends between them. in the reference frame of C. (Distance only diminishes in transformation, while length increases when transformed towards a frame of reference closer to that of the object in question)
Observer L is in the rest frame of object L; consequently he measures its proper length. Observer C is not in the rest frame of object L, and therefore measures a shorter length. Your claim that L measures a shorter length than C indicates that you’re using something other than special relativity, that you’re using special relativity wrong, or that I didn’t understand your description. Please clarify which it is. A diagram may be helpful.
Additionally, this:
is plain wrong, since you can always do the transformation in the other direction! It doesn’t diminish both ways! And this claim:
appears to be meaningless; there’s no such thing as a frame of reference “closer to an object”. A frame of reference doesn’t have a location, it has a speed relative to an observer.
Sticks L and R are stationary in the frame of C; objects L and R just happen to be close to the other ends of the sticks at some time, and are stationary in their own reference frames.
At this point I have no idea what is supposed to be happening in your scenario. You’ve got sticks, objects, and observers, and it’s not clear whether any of them are the same or what frames they are in. Please draw a diagram.