[MikeFontenot (OP)]

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.

[Halc]

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).

We will assume constant proper acceleration since constant coordinate acceleration isn't possible after a while. The chart sounds legit and seems to have no reason to question it.

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".

All good so far. It seems that vertical is the D axis and horizontal is T, kind of opposite of what I'm used to, but not wrong. OK, so these objects start simultaneously (relative to the frame of the chart) and thus stay at constant separation relative to that frame exactly as they should.

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.

That principle doesn't say that, but yes, relative to that IRF, those curves will be identical.

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.

Relative to the IRF, they do. The observer doesn't need to be told this if he already has the description above.

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.

Again, so far so good.

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

I don't think the chart shows contraction of moving things. Maybe it does. You didn't post an image. Principle of relativity seems unreferenced here. There's no specific contradiction specified. Yes, a ruler moving with the objects contracts in the D direction, but it doesn't sound like your chart shows moving rulers. You can fit more of them between the objects over time, as many as you want if you wait long enough.
Principle of relativity just says that physics is the same relative any inertial frame. It sounds like you have only one inertial frame specified here, so there's nothing to compare, hence no particular violation of PoR.

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

It says no such thing. It says that the moving rulers get shorter, meaning more fit between, meaning that in the accelerating frame of one of the rulers, the objects are getting further apart, not closer. The chart doesn't show that since it shows the IRF, not the frame of any of the objects.

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

Frame dependent question. Relative to the IRF in which the two objects always have identical velocity, the separation remains constant. In the accelerating frame of either object, the two get further apart, a consequence of relativity of simultaneity. All this is covered in Bell's spaceship scenario, something with which you really should familiarize yourself since so many of your topics seem to run amok on this.  https://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox

[MikeFontenot (OP)]

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.

A new question: 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.)

[MikeFontenot (OP)]

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

[Halc]

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?"

The LCE applies only to rigid inertial objects. 'The distance between two things' is not a rigid object, nor is it inertial in this case. Read my post above. Your chart is correct and the curves stay equally separated in that IRF.

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.

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.

Do 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.

(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.

I'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#msg397740

It all works through the 'Attachments and other options' link just below the edit window

[Jaaanosik]

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.

Here is a figure, is it similar to your picture above your desk?



As Halc says, Bell's paradox is important to read and analyze:
https://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox#Constant_proper_acceleration
... specifically the section: "Constant proper acceleration".

Edit: from the wiki page, these two scenarios are different, the Bell's paradox explains it.

[MikeFontenot (OP)]

Scan 2023-4-23 17.01.18.jpg (243.13 kB . 1700x2338 - viewed 2764 times)
Scan 2023-4-23 17.01.18.jpg (243.13 kB . 1700x2338 - viewed 2764 times)

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?"

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.

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.

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.

Do 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.

(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.

I'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#msg397740

It all works through the 'Attachments and other options' link just below the edit window

[Eternal Student]

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).

     I'm, not sure that the link to Wikipedia's article provides a great explanation.    Try this explanation:
https://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html
    (That's a physics  FAQ  webpage  from the University of California and should be safe enough).
I've noticed that their diagram and explanation has the same approach of drawing diagonal lines from one worldline to another (lines of constant time or of "simultaneity" in a different frames of reference).    I think it will help to remind you of what was actually done in your own diagram and answer many of your questions.   Obviously they have chosen to put time on the y-axis and not the x-axis but other than that switch it's showing exactly the same as your diagram.   

    The phrasing of these situations is always complicated and English Language is rarely the best tool to use,  diagrams really help.   The phrasing used on their  ( ucr) webpage isn't without some problems but it's a good effort and I doubt I could do much better.     Notice that they do end the discussion with a digram showing a very different pair of worldlines the objects could have traced out in the lab frame  (the one you called IRF in your posts),  where the people in the rockets now would find that the distance between the rockets remains constant - but the person in the lab frame no longer does.

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

    If you manage to follow the explanation of Bell's spaceship paradox you'll see that the answer is  "no",  not if the objects have the worldlines with the shape you've given them.   The distance also doesn't decrease as you stated, they would notice the distance has increased.   

Best Wishes.

[MikeFontenot (OP)]

 [ Invalid Attachment ]
(I couldn't get the attachment to work ... you'll just have to view it in one of the other posts.)

The above diagram (without the diagonal straight lines) shows the perspective of the two accelerating observers.

One thing that diagram DOESN'T show is how the ages of those two observers compare, as time progresses.  (The horizontal axis is the age of the TRAILING accelerating person).  The two accelerating observers do NOT age at the same rate.  Einstein (in his 1907 paper) said the leading person ages exp(D * A) times faster than the trailing person, but I showed in an earlier paper [] that that exponential equation is incorrect.

[Jaaanosik]

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.

There is a problem though.
The initial lab frame grid of inertial observers predicts the rockets separation increase therefore the string breaks.
The formation of the uniformly accelerated frame, a Rindler frame, can be done only when one origin from the initial lab frame is chosen as a preferred origin.
That means not all observers within the original lab frame grid of inertial observers are equal.

[Jaaanosik]

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

I can confirm that.

[MikeFontenot (OP)]

There 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.

[MikeFontenot (OP)]

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).

I don't believe that SR predicts that.

[Jaaanosik]

There 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.

From the link provided by Eternal Student:

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.

[MikeFontenot (OP)]

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 diagram is SUPPOSED to show the perspective of two sets of inertial observers.  The first set is the inertial observers who are stationary with respect to the rocket passengers immediately before the rockets are ignited.  For those inertial observers, a given instant of time is a vertical line on the diagram, and a given spacial location is a horizontal line on the diagram.  (The other set of inertial observers in the diagram are stationary wrt the rocket passengers at a later instant of time.  Since both sets of inertial observers must agree about whether or not the string breaks, I will focus on the first set, and ignore the second set.)

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 ).

So that means that the given inertial observers (who are stationary with the rockets immediately before the rockets are turned on) MUST conclude that the two rockets get closer together during their acceleration.  So the diagram, as drawn, is incorrect ... it does NOT show the correct conclusions of those given inertial observers.

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.

  Q.E.D.

[MikeFontenot (OP)]

[...]

Halc, you've got things exactly backwards:

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.

[Halc]

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 ).

Translation: The mathematics (the picture) and the entire physics community (Einstein included) contradict your intuitions, so the mathematics and the physicists must be wrong. Another conclusion is inconceivable.

Moving this accordingly to New Theories as this is no longer a question, but an assertion of alternate physics.

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.

Excellent!  Now do exactly that for the lead rocket when it is 10 ly ahead of the trailing one instead of 0.5.  Compute the gamma factor and compute where the trailing rocket needs to be after a year (rocket time or inertial frame time, your choice), of acceleration at 1 ly/y² (a smidge over 1g), in order for the string not to 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.

Jano posted some diagrams in post 5, which I'll call (P and QL and QR). You posted one (twice) in post 6 (R). ES put one in post 14 (S), and it's hard to tell but acceleration seems to cease after a certain amount of time in that one.
Don't remember hating any of them, but they don't all depict the same thing. QR, R & S all depict identical proper acceleration profiles. P & QL depict constant acceleration of an extended rigid object. Only those two show a ruler to be contracted.

None are inconsistent with SR. What you describe is inconsistent, which would become super apparent if you actually did this:

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

I notice you decline this. Now why is that? Could it be that your assertions can trivially be driven to contradiction?
I don't need a plot. I just need to know where the other end is after one end accelerates at about 1g for a year. What are the coordinates (relative to the initial inertial frame) of both ends?  A simple 2 digits of precision will do.
Oh wait, you can't do that. You won't do that. The length contraction which Einstein describes must be wrong as well, and relativity of simultaneity along with it. All bunk.

[MikeFontenot (OP)]

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

[Eternal Student]

Hi.

If 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.

[MikeFontenot (OP)]

Hi.

If 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.

Here is my post that you are referring to:
____________________________________________________

The diagram is SUPPOSED to show the perspective of two sets of inertial observers.  The first set is the inertial observers who are stationary with respect to the rocket passengers immediately before the rockets are ignited.  For those inertial observers, a given instant of time is a vertical line on the diagram, and a given spacial location is a horizontal line on the diagram.  (The other set of inertial observers in the diagram are stationary wrt the rocket passengers at a later instant of time.  Since both sets of inertial observers must agree about whether or not the string breaks, I will focus on the first set, and ignore the second set.)

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 ).

So that means that the given inertial observers (who are stationary with the rockets immediately before the rockets are turned on) MUST conclude that the two rockets get closer together during their acceleration.  So the diagram, as drawn, is incorrect ... it does NOT show the correct conclusions of those given inertial observers.

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.

  Q.E.D.
__________________________________________________

Show me where in the above I have changed anything.  The diagram that I am referring to is given early in this thread (the one showing the viewpoint of the inertial observers who are stationary with the rockets immediately before they are fired).  I stated why that diagram is incorrect (because those inertial observers say that the rocket separation doesn't change, and that violates the length contraction equation (LCE) of special relativity).  I then described how that diagram must be changed in order to be consistent with special relativity.