Right, but SR has never revealed an inconsistency either observationally or via any consistently applied thought experiment. So any apparent dilemma just means thinking it through more carefully. That PF thread covers your scenario thoroughly enough imo. Sticking with the simple case of two rigid balls each of diameter d, a fixed center-to-center distance D apart in their co-moving frame, it's a trivial observation both are Lorentz contracted by factor 1/γ as seen in the Earth frame. Hence the number of identical such balls fitting between the original two is ((D-d)/γ)(γ/d) = (D-d)/d there (assuming an integer number fit), which is identical to that determined in the co-moving frame. Where then is there any issue?
Let's take 13 of your balls but with negligible diameter. Then, you have 12 equal distances between every 2 balls. Name them A, B, C... Then, put them on a circle with the thirteenth ball at the same point as A. So, you have 12 balls on the circle and 12 equal intervals. Make the circle to rotate at the angular speed omega, then the balls and intervals will move with velocity v = omega R, R the radius of the circle. How will the intervals contract?
The constraints are now different. Here the ball-to-ball separation r is fixed wrt lab frame, irrespective of ball speeds in lab frame. Lorentz contraction still applies for each ball as it did in the earlier scenario, hence now the (approximately) co-moving frame separation between adjacent balls expands by factor γ accordingly. Approximately because unless the ball-to-ball separation is vanishingly small, circular motion implies a non-zero relative velocity, so there is not perfect co-movement, which makes the γ factor not perfectly accurate. But good enough for our purposes. And that constrained circular motion case is simply a variant on your current loop issue(s) discussed at great length but nevr mutually settled here: http://www.sciforums.com/threads/le...etic-force-between-arbitrary-currents.159343/ Must depart.
It is not actually a paradox. It is an apparent paradox, and it does have an explanation. I am not trying to hide it. I did my research to verify my suspicions. Surely, any wise person would. That's not disingenuous. Sure, it would have been easier if I had just suggested you just post your question on PF, but I was pretty sure you would prefer help here rather than be sent somewhere else (otherwise you would have posted there). However, as it turns out, I was quite wrong - and have indeed posted complete nonsense above. I apologize for the error, and will recuse myself from this thread.
It does not have one explanation, but several. A true solution does not have many different explanations. Sorry for my joke. By the way, How do you post image here? I do not have the possibility in the box of reply.
No, it's all the same explanation - all the same physics - there are just multiple ways to look at it. For example, my muon example can be looked at either as length contraction or time dilation. 1] You can try simply copying an image (Right-click, copy image) and pasting it directly here. 2] You can use the Image tool in the enhanced text editor. My editor looks like this: Please Register or Log in to view the hidden image! You might have basic editor on by default. Look in your user settings for the option: Please Register or Log in to view the hidden image!
My point is that if the distance between balls contracts, then the interval AB is shorter. In this case, the interval BC becomes longer, see the drawing. Then, length contraction for one interval destroys length contraction for other intervals. Please Register or Log in to view the hidden image!
PengKuan; Direction of motion, in agreement with SR derivations. The spherical form becomes ellipsoidal with the field lines rotating toward the vertical.
I can't see your image, it just prompts me to sign in to Google accounts. If you have a bunch of balls moving in a circle, (perhaps constrained by a ring), all of the balls will all be length contracted according to the non-rotating inertial frame. So you can fit more balls into the circumference of the ring when the balls are moving, compared to a different case where the balls are all at rest with respect to the ring.
So, the diameter of the circle will be smaller, the circle becomes half of the original when the speed is near c
Yes, I can see that image, thank you. It depends on how they are spaced before they are accelerated up to speed, and also on how exactly they are accelerated up to speed. Assume they start off stationary in the inertial non-rotating reference frame, and equally spaced around the circle like your image shows. If they are then all accelerated up to speed simultaneously and equally (according to the inertial non-rotating reference frame) then they would remain equally spaced around the circle. Each ball would become length contracted by the same factor, (again according to the inertial non-rotating reference frame). The interesting part (to me) is that any one ball would not consider itself length contracted in its own reference frame. This assumes the balls are small enough that each one can measure itself locally, as compared with the overall circle which would relatively be much larger. This is where some people conclude that the geometry of the ring (in the ball's own rotating frame) must be non-Euclidean. This is because the ring still must have room for more balls to fit in the ring now, as compared to when they were all at rest in the inertial non-rotating frame.
Sorry but you have completely misread my #43. The constraints again: A circular track/tube of fixed diameter, stationary and non-rotating in lab frame. Equal spacing, in the lab frame, between a given number of balls or as you now want it, electrons. Then with the balls/electrons circulating at some appreciable fraction of c, in the approximate co-moving frame for any two adjacent balls/electrons, the separation distance as measured in that frame, has increased by factor γ, not contracted as you somehow read it! Your illustration in #54 is not particularly accurate as it shows the balls/electrons having zero Lorentz contraction along directions of motion. Once again, Lorentz contraction of balls seen in lab frame means more equally sized balls could fit between any two adjacent balls, than when they are stationary in lab frame. Hence, logic demands the measured separation determined in approx co-moving frame is greater, not less, than seen in lab frame. Note the instantaneous shape of constraining circle is an ellipse as seen by any given moving ball. Lorentz contraction of circle in instantaneous direction of ball motion. Transverse length unchanged.
Yes, that is a great insight, and I agree. That is why I wrote, "This is where some people conclude that the geometry of the ring (in the ball's own rotating frame) must be non-Euclidean." The geometry in one ball's own non-inertial reference frame really doesn't have to be non-Euclidean -- simply allowing the ball to substitute the approximate co-moving inertial frame coordinates keeps the geometry Euclidean. In that case, as you said, the circular track would be an ellipse, with the one ball in question at one vertex, where locally the ball would measure itself as having essentially no length contraction. But the ball in question would also say that the ball at the opposite vertex of the ellipse was very much length contracted. This would allow all of the extra balls to fit in a Euclidean way, because most of them would be concentrated toward that opposite vertex, where there is very much length contraction of the balls.