Proper accelerations and rigid bodies.

[Jorrie]

Others be the judge

Are you trying to create trouble or are you that slow in understanding?

Neither. I rest my case.

 

[Fredrik]

Boustrophedon's original post says that some books and articles are using the term "proper" incorrectly. The people who replied to that seem to think that B has either read some bad articles, or simply misunderstood them. I have a different suggestion. I think the books and articles are using the term "proper" correctly, and that B is wrong about what would be correct usage.

The "proper" quantities can always be determined by the coordinates of co-moving inertial observers.

The proper time along a curved world line is the sum of time coordinate increases of an infinite sequence of inertial frames.

(A curved world line can be approximated by N straight line segments of equal length. Each straight line segment corresponds to an inertial frame. The time coordinate of this inertial frame increases from one end of the straight line segment to the other. The sum of these increases approximates the proper time. The approximation is exact in the limit N --> infinity.)

The proper length of an accelerated object is the sum of spatial coordinate increases of an infinite sequence of inertial frames. (Imagine N points on the object. The tangent to the world line of each point is the time axis of an inertial frame. The space axis of this inertial frame assigns a spatial distance between one point and the next. The sum of all these distances approximates the proper length. The approximation is exact in the limit N --> infinity).

The proper acceleration is the second derivative of the space-time coordinates of a co-moving inertial frame with respect to proper time.

I frequently come across the statement that "the "proper acceleration" of the rear end of an accelerated rigid rod must be greater than that of the front end."
...
Since co-moving observers verify a constant "proper" length for the rod, and an observer at each end will record the same "proper" time, then by definition the "proper" acceleration must surely in fact be identical at each end of the rod, whether they are measured by inertial accelerometers or by reference to the background inertial frame.

You're making a mistake here. You're assuming that the world lines of each part of the rod is identical except for starting position. But the statement that you "frequently come across" is a statement about Born rigid motion, and that contradicts your assumption.

So although the platform clocks are perfectly synchronised to the guy on the platform, the train rider's judgement of "simultaneity" has tilted "upward" or forward in time ahead of him while dipping backward in time behind, so that "now" ahead of him is later in time than for the platform in that direction, and he therefore regards the clocks as set forward in time.

Now what happens as the train accelerates from rest ? The "lines of simultaneity" at every point of the train start parallel and smoothly rotate so as to increasingly tilt forward in time towards the front and backwards in time towards the rear.

You seem quite focused on the idea of using accelerating frames, but you also seem to underestimate the problems with accelerating frames. I believe that what you're thinking is something like this:

If an inertial observer draws the world line and the lines of simulaneity of another inertial observer, the slope of the world line is 1/v and the slope of the lines of simultaneity is 1/v. This seems to imply that the "line" of simultaneity of an accelerating observer must bend upwards, as the observer's world line bends to the right. These "lines" (actually curved space-like hypersurfaces of Minkowski space) seem to be good candidates for what the accelerating observer should call "space" at different times.

If it really makes sense to say that these space-like hypersurfaces are "space" at different times, for an accelerating observer, the length of the object as measured by him/her must be sqrt(ds^2) integrated along a curve in "space", from one end of the object to the other.

There are plenty of reasons why this doesn't make sense. The strongest one is that the length of the object would depend on how its velocity is going to change in the future.

 

[Fredrik]

Since Pervect has introduced the so-called "Bell spaceship problem" I should point out that the falsity of Bell's claim follows inevitably from...

I'm not sure if it's appropriate to answer a post about something that was the topic of a thread that's now locked, but it seems that this thread has become a discussion about the same thing, and what you say here makes me curious.

In the other thread, I solved one version of the spaceship problem explicitly. (In this version, the acceleration is arbitrary for a certain proper time, and then the engines shut off). You seemed to agree with what I said except for one detail. I made a new post after that and proved that I was right about that detail. You didn't answer that. Perhaps you didn't see it, or perhaps you didn't have time to answer before the thread got locked.

Anyway, the solution is still there if you're interested, and the result is that the string breaks. If you have any questions or comments, I'm willing to answer them, either here or in private messages if the moderators think we're getting too far off topic.

It is easy to see that the observers disagree about the length of a rod simply and solely because they disagree about whether each other recorded the ends simultaneously. This is a non-physical, purely kinematical effect and cannot possibly give rise to any "stresses", "forces", "tensions" or "string-breaking". The same causes ( differing simultaneity ) would give the same effects ( differing length estimates ) if the ends of the rod were not even connected.

Everyone who disagrees with your conclusions agree with you about these details, so there's no need to explain them to us.

 

[Doc Al]

Since Pervect has introduced the so-called "Bell spaceship problem" I should point out that the falsity of Bell's claim follows inevitably from the demonstration that both time dilation and length "contraction" effects follow in a elementary way from the divergence in simultaneity without any "extra" assumptions that a moving rod or clock is any different from a "stationary" one.
It is easy to see that the observers disagree about the length of a rod simply and solely because they disagree about whether each other recorded the ends simultaneously. This is a non-physical, purely kinematical effect and cannot possibly give rise to any "stresses", "forces", "tensions" or "string-breaking". The same causes ( differing simultaneity ) would give the same effects ( differing length estimates ) if the ends of the rod were not even connected.

Boustrophedon, it's clear that you have some basic misunderstandings of special relativity. Your interpretation of the Bell spaceship paradox is just one manifestation of this.

While the Lorentz contraction is surely a "kinematical" (or, better, a geometrical) effect, you must realize that forcing both ships to have the same acceleration (with respect to an inertial frame) does introduce stresses in the string. Of course the string breaks.

Several knowledgeable members have patiently described how proper acceleration is treated in special relativity; I suggest you study their posts and the papers they have cited.

One of the main goals of PF is to help students learn the current status of physics as practiced by the scientific community. Please reread the general posting guidelines that cover all of PF and the particular guidelines that appear in the stickies at the top of this forum. Personal interpretations and theories that contradict established, mainstream physics are not permitted in this forum.

On that note, I am going to close this thread. Please do not bring up the same topic in yet another one. If you seriously think that you have been misrepresented, you are welcome to submit your ideas to our Independent Research Forum, provided that they meet certain https://www.physicsforums.com/showthread.php?t=82301.