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More than 20 years ago, I plotted a chart showing two separated objects undergoing the same constant acceleration "A". (That chart still hangs on the wall above my desk, and I've never questioned it before).

The plot supposedly shows the view of things according to an inertial reference frame (the IRF) that is stationary wrt the two objects immediately before the acceleration begins. One curve starts from the origin with slope zero at the origin, but then curves upward with a curvature that monotonically decreases as time increases, and asymptotically approaching a slope of "c", the speed of light. I use units where "c" equals 1.0, so the curve approaches a slope of 1.0 on the chart.The other curve has exactly the same shape, but starts at some distance "D" above the origin. The two curves are always separated by a vertical distance of "D".

The idea, I think, was that the two curves must have exactly the same shape because of "the Principle of Relativity" ... i.e., it shouldn't matter where in space you start the curve, the curves should always have the same shape.

But here's the quandary: An observer in the inertial frame IRF is told by the chart that the two objects always have the same distance apart.

But the length contraction equation (LCE) of special relativity says that an inertial observer should conclude that a moving yardstick should get shorter and shorter as its speed wrt the inertial observer increases.

That seems to contradict what the chart says, and it seems to contradict the Principle of Relativity.

The LCE seems to require that the two curves get closer together as time increases.

Does the upper curve slowly get closer to the lower curve?

In my original post, I said:"The LCE seems to require that the two curves get closer together as time increases. Does the upper curve slowly get closer to the lower curve? Or does the lower curve approach the upper curve?"

I've realized that the bottom curve doesn't move upward, because it already has speeds that approach the speed of light "c", and so it's speeds can't be increased any. So all of the decrease in their separation has to come from a lowering of the upper curve.So I suppose that is enough information to allow the correct upper curve to be plotted ... just subtract the amount of length contraction (using the LCE) from each point of the upper curve.

Do the two observers who are doing the accelerating agree that their separation is decreasing?

(Inertial observers don't ever think the yardsticks between them contract, so maybe accelerating observers don't think the yardsticks between them contract either.)

I've scanned the chart into a jpeg. How do I post that?

More than 20 years ago, I plotted a chart showing two separated objects undergoing the same constant acceleration "A". (That chart still hangs on the wall above my desk, and I've never questioned it before). The plot supposedly shows the view of things according to an inertial reference frame (the IRF) that is stationary wrt the two objects immediately before the acceleration begins. One curve starts from the origin with slope zero at the origin, but then curves upward with a curvature that monotonically decreases as time increases, and asymptotically approaching a slope of "c", the speed of light. I use units where "c" equals 1.0, so the curve approaches a slope of 1.0 on the chart.The other curve has exactly the same shape, but starts at some distance "D" above the origin. The two curves are always separated by a vertical distance of "D".The idea, I think, was that the two curves must have exactly the same shape because of "the Principle of Relativity" ... i.e., it shouldn't matter where in space you start the curve, the curves should always have the same shape.But here's the quandary: An observer in the inertial frame IRF is told by the chart that the two objects always have the same distance apart. But the length contraction equation (LCE) of special relativity says that an inertial observer should conclude that a moving yardstick should get shorter and shorter as its speed wrt the inertial observer increases. That seems to contradict what the chart says, and it seems to contradict the Principle of Relativity. The LCE seems to require that the two curves get closer together as time increases. Does the upper curve slowly get closer to the lower curve? Or does the lower curve approach the upper curve? Or is there some combination of those two movements? Any of those movements contradicts what the chart says, and it thus seems to contradict the Principle of Relativity.Any ideas? I'm really stuck ... I don't know the answer.

Quote from: MikeFontenot on 23/04/2023 23:47:23In my original post, I said:"The LCE seems to require that the two curves get closer together as time increases. Does the upper curve slowly get closer to the lower curve? Or does the lower curve approach the upper curve?"The LCE applies to rigid objects. 'The distance between two things' is not a rigid object. Read my post above. Your chart is correct and the curves stay equally separated in that IRF.QuoteI've realized that the bottom curve doesn't move upward, because it already has speeds that approach the speed of light "c", and so it's speeds can't be increased any. So all of the decrease in their separation has to come from a lowering of the upper curve.So I suppose that is enough information to allow the correct upper curve to be plotted ... just subtract the amount of length contraction (using the LCE) from each point of the upper curve.You persist in using the mathematics of a rigid object. That's fine, but not the scenario depicted on your chart.Suppose you had a long rigid object of length D, a rocket say, stretching the distance between the two points on your chart. The rear of it accelerates per the curve shown in the chart. Now the LCE comes into play as you describe here. All of the contraction of the rocket has to come from, as you say, a lowering of the upper curve, but this also has a consequence of lower proper acceleration of the upper curve since the full proper acceleration is the not-lowered curve that your chart shows. Yes, that is enough information to allow the alternate upper curve to be plotted, albeit a somewhat complicated way to do so. The plot of the full proper acceleration curve remains unchanged as your chart correctly shows.Anyway, it means that accelerometers at either end of a rocket read different values.QuoteDo the two observers who are doing the accelerating agree that their separation is decreasing?They'd be wrong if they decided that. In the rocket (with the front guy under less proper acceleration), they'd agree that the rigid rocket remains the same proper length at all times. In the identical proper acceleration case that your chart depicts (and the Bell's scenario discusses, and you still haven't read), they'd agree that their separation is increasing as evidenced by the string between them breaking.Quote(Inertial observers don't ever think the yardsticks between them contract, so maybe accelerating observers don't think the yardsticks between them contract either.)That's right, so in the long rocket case, the rocket always remains a constant number of yardsticks in length. The marking are in fact painted along the length of the rigid rocket so it really isn't possible for them to measure a different length.Quote from: MikeFontenot on 24/04/2023 00:06:23I've scanned the chart into a jpeg. How do I post that?A bit complicated. Apologies.http://www.thenakedscientists.com/forum/index.php?topic=45718.msg397740#msg397740It all works through the 'Attachments and other options' link just below the edit window

A new question: Do the two observers who are doing the accelerating agree that their separation is decreasing?

Hi. OK, we can see the diagrams you ( @MikeFontenot ) have posted. We can also see that you have drawn some diagonal lines between the two worldlines. So I'm going to agree with previous comments from others. The situation is very much like Bells's spaceship paradox and it is probably best explained just by looking through a good explanation of that situation ("paradox" - although it isn't really a paradox, a perfectly fine explanation does exist).... Best Wishes.

...Statements to the effect of 'Einstein was wrong' will get this topic moved like all the others.

There is a problem though.The initial lab frame grid of inertial observers predicts the rockets separation increase therefore the string breaks.

Special relativity theory (and not some frame) predicts that the proper separation (which is not frame dependent) increases, and for that reason the string breaks (an objective fact, not a frame dependent one).

Quote from: Jaaanosik on 24/04/2023 20:38:16There is a problem though.The initial lab frame grid of inertial observers predicts the rockets separation increase therefore the string breaks.What exactly are the "initial lab frame grid of inertial observers"? The inertial observers who are stationary wrt the rockets immediately before the acceleration begins will say that the rockets get closer together as the acceleration progresses. So they will conclude that the string DOESN'T break.

The distance that the blue rocket measures from A to B is approximately γL (in fact, it's somewhat more than γL). But if we ask the blue rocket to re-measure the distance from itself to the red rocket at a later time marked by event P, then the line of simultaneity will have changed: it will be the upper dotted blue line. This is not parallel to the lower dotted blue line, and this line crosses the red world line at event Q. The distance PQ will be larger than distance AB, and so the blue rocket will conclude that the red rocket is actually pulling away from it.

The diagram (two identical attachments) shows worldlines of two objects with constant proper acceleration, and depicts the original IRF, thus the perspective of an inertial observer.

[...]

The important thing to understand is that the diagram, as shown, is INCORRECT. It does NOT show the correct viewpoint of that first set of inertial observers. The well-known length contraction equation (LCE) says that for ANY inertial observer (HE), a line of end-to-end yardsticks that are moving at a constant speed relative to him will be shorter than his own yardsticks, by the gamma factor 1 / sqrt( 1 - v * v ).

To obtain the correct diagram, at each instant of the given inertial observers' time, it is necessary to compute the gamma factor (where "v" is the speed of the rockets at that instant), and divide the constant separation "L" of the rockets (according to the observers on the rockets) by gamma. The result is then added to the location of the trailing rocket, to get the location of the leading rocket.That correct diagram shows that, according to the given inertial observers, the two rockets get closer together during the acceleration, and therefore the string does NOT break.

The diagram that you like, and which you contend is standard special relativity, is wrong, because it violates one of the most important laws of special relativity: the length contraction equation.The diagram that you hate, and which you contend ISN'T special relativity, is correct, because it obeys the length contraction equation of special relativity.

Now do exactly that for the lead rocket when it is 10 ly ahead of the trailing one instead of 0.5.

If you're loosing an argument, change the subject.

Hi.Quote from: MikeFontenot on 25/05/2023 03:42:06If you're loosing an argument, change the subject. That seems to be what you ( @MikeFontenot ) have done. You CAN have a situation where the two rockets get closer together in the lab frame and the string between them does not break. However, that wasn't the situation you originally described or what was shown in your original diagrams. Change the original situation and you will change the final consequences. I think we're all in agreement with that. It doesn't make the original situation an impossible situation to have, just one that you didn't really want to be examining.Best Wishes.