DaleSpam said:
No, even per B the coordinate system is not 1 to 1. It isn't a matter of perspective or opinion, it is an objective fact about the chosen coordinate system.
Hmmm. Well, if heavenly bodies reshift about B's spacetime coordiante system due to a rotation in B's own sense of simultaneity, why must B's own coordinate system not maintain a 1:1 relation between space and time? The way I see it, instead of forcing everything to remain where it is and contorting the B system, why not leave the B system untorted (ie remain 1:1) and allow the heavens to shift? In the end, it's the same thing viewed 2 different ways.
DaleSpam said:
Because if a coordinate system is not 1 to 1 then you cannot do well defined coordinate transforms any more. If you cannot do coordinate transforms then you cannot use coordinate transforms to determine the laws of physics in that coordinate system and to transform results to other coordinate systems. So suddenly the coordinates become physically useless.
Well, I agree they become useless, but that's only assuming twin B continues to define simultaneity per τ1 = ½( τ0+ τ2), and also assuming B runs the LTs as he would if he were purely inertial (even though he properly accelerates). However, that was defined for cases of uniform linear translation alone, which falls down during a POV undergoing an active proper acceleration.
I don't know of a valid coordinate system for a non-inertial POV. I've heard of a couple, but none were sufficient IMO. One cannot use the Einstein sync convention for non-inertial bodies, assuming spacetime coordinates are "always" to match the predictions made by inertial observers. In any valid theory, all POVs must attain the same solns using the same spacetime transformations.
DaleSpam said:
Thanks for the heads up. I fixed it.
LOL, very funny! I kinda liked it better before you tweeked it. You know, the dynamic worldline there always enters the origin at a vertical orientation at the origin, so just as in the case of a Loedel figure, the horizontal space axis and vertical time axis exist whether displayed or not.
The way I see it, B's calculations using the LTs are just less convenient than any purely inertial observer's LT calculations, because B has extra steps he must do that all inertial observers would not. The reason, when B receives light from a remote body, it does not predict how the remote body has moved since the EM left the body prior. However, if the body is purely inertial, then B should be able to determine where the body really exists by accounting for his proper accelerations from his own proper accelerations (using his own accelerometer data) relative to his initial inertial frame. It's an extra step that inertial observers need not be concermed with, but what they hey. While this is a far less convenient way to determine the location of remote bodies, it should be no less accurate assuming good enough technology is being used in the nav system ... and besides, twin B cannot do it in the usual way during periods of his own proper accelerations.
That said, while I don't really disagree with your points, I don't see the problems you see in such an approach as applied to a twin B POV. Maybe you discard it too hastily. Is there a valid method for twin B to determine the location of any inertial body during his own proper accelerations using the LTs, that you are aware of? I mean, one whose solns precisely match that the LT calculations made by any inertial POV.