Is Distance/space Lorentz contracted?

[PeterDonis]

Before going into how I am misinterpreting them I would like a little clarification.
I made that statement in the general context of standard interpretation regarding inertial frames. Persuant to the apparent agreement as to the identicallity of the geometry under consideration and an inertial frame.
So is your quote here applicable to inertial frames ?
Thanks

Yes, it is. Each line of simultaneity, in a given inertial frame, is just a line parallel to the x-axis of that inertial frame. Each such line has its own time coordinate in that frame--which is just the time coordinate of the event where that line intersects the t-axis of that frame. All events that lie on a given line of simultaneity in a given frame have the time coordinate of that line. So events that lie on different lines of simultaneity have different time coordinates.

(Edit: The x-axis itself, for a given inertial frame, is just the line of simultaneity with the time coordinate t = 0 in that frame.)

 

[Austin0]

--------------------------------------------------------------------------------

Yes, it is. Each line of simultaneity, in a given inertial frame, is just a line parallel to the x-axis of that inertial frame. Each such line has its own time coordinate in that frame--which is just the time coordinate of the event where that line intersects the t-axis of that frame. All events that lie on a given line of simultaneity in a given frame have the time coordinate of that line. So events that lie on different lines of simultaneity have different time coordinates.

(Edit: The x-axis itself, for a given inertial frame, is just the line of simultaneity with the time coordinate t = 0 in that frame.)

Having thought it over I see that I was wrong in the statement as I meant it.
I was forgetting that a ICMIF was not a true inertial frame.
But I did not mean it in the sense that you seem to talking about here either.

Original post Austin0
That multiple lines of simultaneity for disparate locations in a frame are simply different segments of a single continuous line which is actually congruent with the x' axis.

This is *not* correct. Distinct lines of simultaneity indicate distinct, different clock readings.

I didn't mean that the particular lines at a given moment indicated the same clock readings which is self-evidently false. I meant that from the F frames perspective they were different segments [i.e. different time slices] but that every event [colocation of positions and clocks] that appears on one line will also, at some point in time, appear on the other line.
That over time the set of these events is singular and represents a single set of clocks ,congruent to the x' axis in the F' frame.
But I can see that this does not pertain to an accelerating system because such a system has a dynamic metric. So one line does in fact represent a unique set of clock times and locations due to a change of clock rate during the interval separating them.
If I am still incorrect in my interpretation above let me know.

 

[Austin0]

=PeterDonis;2353089]Austin0:

I do agree with you that there are plenty of complexities in how to handle accelerated objects in relativity--

I'm not unwilling to discuss all the other complexities; I'm just not ready to discuss them until we have agreement on the simple scenario I specified above, and in particular on the proposition that we currently don't seem to agree on. The simple scenario involves nothing except relativistic kinematics, and without agreement on that, the complexities will never fall into place.

Well PeterDonis, at the prospect that you might discuss all the other complexities, I would probably agree to being the real shooter of John Lennon. :-}
Aside from great knowledge, you seem to have a rare degree of objectivity. The ability to play the "devils advocate" as you put it.
Although in my case so far, there seems to be a tinge of the devils prosecutor. ;-(
So I am a little befuddled by the course of this discussion and still unclear on what it is I am supposed to agree to.
I have tried to respond to you point by point, but so far I have no response from you, aside from the few points that have been somewhat taken out of context, reinterpreted or scanned for subtext indicating doubt of SR or wild speculative alternate theories. I am not quibbling over this and understand your desire to have a firm base of understanding but it leaves me in doubt as to whether you actually read the rest.


Originally Posted by Austin0
Equal proper acceleration must result in equal distance and time. But exactly the same obvious logic is applicable from the ships frame.

I've been watching this thread for a few days now but haven't jumped in because I've been waiting for you to say something short and to the point that gets at the heart of the point of disagreement between you and JesseM. I think the above quote is it. PMFJI, but I'd like to give my take on why the above quote doesn't apply in this situation.

Austin0 Post 9
So for the moment, if you could consider this question solely within the given parameters , with no reference to how things appear in the station frame or whether the acceleration is proper or not, what does your logic predict would be the outcome of the events?

Originally Posted by Austin0
This is purely a matter of logic and known physics. They have the same acceleration ,same instantaneous thrust , as determined by the identicallity of their mechanisms of propulsion.

So if you are saying that logic and physics does not lead to productive insights within the parameters I wanted to examine the situation from, then at this point, I can't argue. It has certainly not led me to any useful conclusions and it is even more certain that it is not a productive format for discussion with those of this forum.

#1 Does the above argument convince you? I should add that this argument is purely kinematic; it doesn't make any assumptions about how objects respond to stresses under acceleration, and so forth. It's

#2 * Do you want both ships to experience equal proper acceleration?* Or do you want both ships to remain at a constant spatial separation, as seen by the ships?

You *can't* satisfy both. Which one you pick will determine what specific scenario you're talking about, and until you do that, it's hard to have a discussion because you may be talking about different scenarios.

It seems to me there is a bit of contradiction here.

#1 States there is no assumption of physics involved.

#2 Makes all kinds of physical assumptions. Born acceleration seems to be based on several physical assumptions . Non-Born acceleration, as outlined by JesseM, also makes several physical assumptions. And the assumption that there is no other possibility other than these two is also a prediction of physics , no?

SO it looks like; having, a priori , made all possible physical assumptions regarding the situation, it resolves to a purely kinematic problem.
Well it certainly leads to definitive results of a sort.
But, as the original question was an attempt to circumvent physical assumptions beyond the known physics of acceleration, it pretty much negates that pursuit.
Or at least, certainly redirects it into an examination of these physical assumptions themselves.

So if all the above assumptions are to be taken as known facts and in addition the stations frames simultaneity is to be taken as the only relevant criteria; then the transformations can, of course, only produce one set of figures. There is no logic involved there, it is only a set of functions.
So if this is the agreement that you are asking for then you have it.

I do have a question though: The Lorentz math itself makes physical predictions which are integrated into the transform. Newton and Galilleo in one.
So the Born hypothesis says that the Lorentz math is insufficient to accurately describe the physics of accelerated motion without additional assumptions. JesseM has said it is not an accurate description of the physics of non-Born accelerated motion.
So how is expecting the math [as a transform] to make accurate coordinate predictions in this context , any different than expecting Galillean transforms to produce accurate predictions regarding relativistic motion between frames where Newtonian kinematics doesn't apply?

Thanks

 

[PeterDonis]

Austin0:

I'm going to respond to things in both of your last two posts here. You said:

I have tried to respond to you point by point, but so far I have no response from you, aside from the few points that have been somewhat taken out of context, reinterpreted or scanned for subtext indicating doubt of SR or wild speculative alternate theories. I am not quibbling over this and understand your desire to have a firm base of understanding but it leaves me in doubt as to whether you actually read the rest.

I did, but I've already explained why I haven't commented on most of what you said. Once again, let me reiterate that there are at least two separate questions involved here:

(1) Given a specific worldline (or worldlines) in Minkowski spacetime, how will observers traveling along that worldline (or worldlines) "see" other events?

(2) Given a description of a physical situation, how do we translate that description into a specification for one or more worldlines to which we can then apply our knowledge with regard to question #1 above?

So far I have only been talking about question #1, because your original specification of the scenario was enough to determine a pair of specific worldlines in Minkowski spacetime. Question #1 is what I'm talking about when I say that a question is "purely kinematic": given a specific worldline in Minkowski spacetime, the question of how observers traveling along that worldline will perceive events is *entirely* determined by the specification of the worldline and the laws of SR. No "physical assumptions" other than those are involved.

In the course of this thread, you, and JesseM, and you in response to my posts, have also talked about a lot of things that *only* become involved when you're talking about question #2--in other words, when you haven't yet obtained a specification of a particular worldline or worldlines that an object or parts of an object will follow. In particular, the statements JesseM was making about whether or how acceleration would affect the length of bodies, how bodies would respond to stresses, whether length contraction would apply, etc., were talking about question #2, *not* question #1. All of those things are necessary to consider if you want to figure out, for example, what specific worldlines each end of a rod made of steel, let's say, would follow if subjected to a certain acceleration. But if you have already specified (as your original scenario did) the worldlines you want to consider, all that other stuff is irrelevant.

Right now, as I've said, I am *only* talking about question #1. So I'm simply not going to respond to things that don't involve question #1, or that mix in question #1 with other stuff. There's no point in my doing so until we have a clear understanding of question #1, by itself, with the specific worldlines specified in the scenario we're discussing.

In response to my statement that:

#1 Does the above argument convince you? I should add that this argument is purely kinematic; it doesn't make any assumptions about how objects respond to stresses under acceleration, and so forth.

...

#2 * Do you want both ships to experience equal proper acceleration?* Or do you want both ships to remain at a constant spatial separation, as seen by the ships?

You *can't* satisfy both. Which one you pick will determine what specific scenario you're talking about, and until you do that, it's hard to have a discussion because you may be talking about different scenarios.

You said:

It seems to me there is a bit of contradiction here.

#1 States there is no assumption of physics involved.

#2 Makes all kinds of physical assumptions. Born acceleration seems to be based on several physical assumptions . Non-Born acceleration, as outlined by JesseM, also makes several physical assumptions. And the assumption that there is no other possibility other than these two is also a prediction of physics , no?

There's no contradiction. My statement #1 was in reference to a specific scenario, the one in which you specified that both ships experience equal (constant) proper acceleration--which is enough to specify a worldline for each ship. In other words, #1 only involved question #1 above, and so, as I've said, is purely a matter of kinematics--the specific worldlines and the laws of SR.

My statement #2 was simply pointing out that the specification that both ships experience equal proper acceleration, and the specification that both ships will remain at constant spatial separation, as seen by the ships, are two *different* specifications; they result in two *different* pairs of worldlines. It is true that, in order to determine that, other "physical assumptions" may be involved, but that's part of question #2, which I'm not yet discussing. Even if it were to turn out (which it won't) that those two specifications, equal proper acceleration vs. constant spatial separation, *did* result in the same pair of worldlines, I would still not want to discuss that until we have agreement on question #1 by itself, dealing with the pair of worldlines that are specified by equal (constant) proper acceleration. If it helps, just suppose that I've withdrawn my statement #2 above from discussion until we are finished with question #1.

With regard to question #1, I want to first make sure that we are using the same definitions for terms. First, you said:

I was forgetting that a ICMIF was not a true inertial frame.

Yes, it is. An MCIF *is* an inertial frame, by definition; it's just that the accerating object is only at rest in that frame momentarily, because it's accelerating. But considered just as an inertial frame, an MCIF is no different from any other.

Also, you said:

I didn't mean that the particular lines at a given moment indicated the same clock readings which is self-evidently false. I meant that from the F frames perspective they were different segments [i.e. different time slices] but that every event [colocation of positions and clocks] that appears on one line will also, at some point in time, appear on the other line.

An "event", as I've been using the term (and as it's standardly used in relativity) means a single *point* in spacetime. In any given inertial frame, an event has specific coordinates t, x (and y, z too if we're using all three dimensions of space--we've just been using x in this thread to keep things simple). For a given inertial frame, no event can possibly appear on more than one line of simultaneity--the one that has the t-coordinate assigned to that event.

If you meant in your quote above the same thing by "event" that I mean, then your statement is simply false: events that appear on one line of simultaneity *cannot* appear on any other. If you were using "event" to mean something else, then please give an explicit definition.

Next, you said:

That over time the set of these events is singular and represents a single set of clocks ,congruent to the x' axis in the F' frame.
But I can see that this does not pertain to an accelerating system because such a system has a dynamic metric. So one line does in fact represent a unique set of clock times and locations due to a change of clock rate during the interval separating them.

I'm not sure exactly what you mean by this, given my uncertainty about how you're using the word "event" above. But let me state what I think you may be saying about accelerated frames in my own words:

* Given a worldline along which an observer experiences constant proper acceleration, each individual event on that worldline has a *different* MCIF. In other words: the MCIF at any given event on that worldline will be in motion relative to the MCIF at any other event on that worldline--no two MCIFs at different events on that worldline can be at rest relative to one another.

And finally, you said:

So if all the above assumptions are to be taken as known facts and in addition the stations frames simultaneity is to be taken as the only relevant criteria; then the transformations can, of course, only produce one set of figures. There is no logic involved there, it is only a set of functions.
So if this is the agreement that you are asking for then you have it.

First, I never said that "the stations frames simultaneity is to be taken as the only relevant criteria". I simply said that the station frame is the only frame in which events Rn and Fn+1 are simultaneous, for n > 1. But let me split that claim up, and take it in two stages. Let's first consider just inertial frames alone, without making any claims as to how the "accelerating frame" relates to them. Let's just consider the claim that the station frame is the only *inertial* frame in which events Rn and Fn+1 are simultaneous, for n > 1. In other words, do you agree to the following propositions? (I'm using the same names for events that I used in my last post.)

* Events Rn and Fn+1 are simultaneous in the station frame for all n > 1.

* Events Rn and Fn+1 are *not* simultaneous in R's MCIF at Rn, or in F's MCIF at Fn+1, for all n > 1.

I don't know that I'd say there's no "logic" involved, although maybe "mathematics" would be a better word--but mathematics is just logic using equations and symbols as tools. And again, please note that those two propositions don't talk about the "accelerating frame" itself--just the station frame and the two MCIFs, R's at Rn and F's at Fn+1--all inertial frames. I'm not asking here if you agree that the "accelerating frame" itself *has* to see things the same way at each event as the MCIF; I want to make sure first that you agree on how things will look in the MCIFs themselves. Since those are just inertial frames, there shouldn't be an issue, since you've already said you agree that relativistic kinematics gives correct predictions for how things look in inertial frames in relative motion. But I want to be sure before going on.

 

[PeterDonis]

So I am a little befuddled by the course of this discussion and still unclear on what it is I am supposed to agree to.

Maybe it would help if I responded to this statement as well. Here's why I came into this thread:

You (Austin0) raised the issue of an apparent logical contradiction between two propositions (to save time I'll state them using my more recent specification of the scenario):

(1) According to SR, observers R and F will *not* observe events Rn and Fn+1 to be simultaneous for all n > 1; those respective pairs of events will only be simultaneous in the station frame.

(2) However, the proposition that "equal acceleration leads to equal distances and times" would seem to imply that events Rn and Fn+1 *would* be simultaneous to R and F, for n > 1; and this appears to be confirmed by the fact that the proper time on R's clock for event Rn will be the same as the proper time on F's clock for event Fn+1, for n > 1.

(I know the above is not the only issue you raised; but as I've said, I believe that until the above issue is resolved, any discussion of the other issues you raise is premature.)

Now I believe SR to be a correct theory (within its domain of applicability--that is, as long as the effects of spacetime curvature are negligible, so that we can assume spacetime to be globally flat). That means that anything which is a logical consequence (or mathematical consequence, if you like) of the laws of SR along with whatever specific premises are assumed for a given scenario, I believe to be true. Given that, I'm quite sure proposition #1 above is not in error; the situation you posed is not a new one, and I've worked through it myself in detail in the past, and I've found plenty of references to confirm my solution. I'm also quite sure that the proper time on R's clock for event Rn *is* the same as the proper time on F's clock for event Fn+1; that's a simple consequence of the solution for proposition #1 (and anyway, you agree that it is, so it hasn't been in question).

So given the above, I believe that there has to be something wrong with the proposition that "equal acceleration leads to equal distances and times". I already know what *I* believe is wrong with that proposition: I stated it in my first post in this thread. "Equal acceleration" is an invariant (provided we mean "proper acceleration", the acceleration that would be measured by an accelerometer moving with the given observer, but we've already agreed on that), but distances and times are frame-dependent, so there can only be *one* frame in which equal accelerations can lead to equal distances and times, and we already know the station frame is that frame.

You weren't satisfied with that resolution of the issue, so I'm now trying to pinpoint exactly where our understanding diverges. Every question I've asked and you've answered has narrowed it down.

 

[Saw]

Please ignore this if it distracts you from your discussion. It’s just that I have followed it and I feel as if the discussion were incomplete without a practical background, i.e., a problem to solve. Trying to think of one, the example has reminded me of a similar discussion we’ve had.

Imagine that the ships (we keep calling them R and F) depart from the same place and fly in opposite directions. After some “whiles” with equal proper accelerations, each ship reaches its respective finish line. Both finish lines are, in the station frame, equidistant from the origin.

So a referee in the station declares that the arrivals are simultaneous and the race to be a draw.

However, in R’s MCIF, R has arrived at its finish line earlier than F at its own. In other words, when it reaches its finish line, a longer patch of the station’s frame has passed by R itself than by F, as per R's simultaneity line.

But F’s MCIF measures that F has arrived earlier! At that time, a longer patch of the station’s frame has passed by F than by R.

In spite of that, R’s and F’s MCIFs read the same proper time intervals when they reach their respective finish lines.

PeterDonis, would you agree that the solution of the problem (who wins the race) is a draw?

Austin0, if we all agreed that the solution, in accordance with orthodox SR, is a draw would you not mind that SR holds that in R’s and F’s MCIFs there is discrepancy with the station frame and between themselves as to simultaneity/distances traversed?

 

[PeterDonis]

PeterDonis, would you agree that the solution of the problem (who wins the race) is a draw?

If the definition of "who wins the race" is "who reaches their respective finish line first, as judged from the station frame", then yes; your statement of the scenario stipulates that they arrive at their finish lines simultaneously in the station frame.

However, you can't say categorically that that is "the" solution of the problem precisely because simultaneity is frame dependent. Any condition on what happens "first" when the respective events are spacelike separated has to include a specification of whose frame "first" is going to be judged from. Otherwise the condition is ambiguous.

 

[atyy]

If the definition of "who wins the race" is "who reaches their respective finish line first, as judged from the station frame", then yes; your statement of the scenario stipulates that they arrive at their finish lines simultaneously in the station frame.

However, you can't say categorically that that is "the" solution of the problem precisely because simultaneity is frame dependent. Any condition on what happens "first" when the respective events are spacelike separated has to include a specification of whose frame "first" is going to be judged from. Otherwise the condition is ambiguous.

I think this is the same scenario that Saw and I discussed a little while ago. If it is, I think Saw is right. Of the 3 frames, the station frame is the only inertial frame, because the inertial frames associated with R and F are ambiguous, since R and F both accelerated. A frame in which R is always at rest is non-inertial.

The coordinate independent way of selecting the station frame is to make R and F both carry clocks, and declare the winner the one who reaches his end point in least own proper time.

 

[PeterDonis]

The coordinate independent way of selecting the station frame is to make R and F both carry clocks, and declare the winner the one who reaches his end point in least own proper time.

Yes, this would be a good way of avoiding a frame-dependent specification, and under this definition, the race would be a draw.

Of the 3 frames, the station frame is the only inertial frame, because the inertial frames associated with R and F are ambiguous, since R and F both accelerated. A frame in which R is always at rest is non-inertial.

This is true, but I don't think it disqualifies R or F from having a legitimate "point of view" (if you don't like the word "frame") according to which they are at rest and other objects are moving. R's or F's point of view will not "work the same" in all respects as that of an inertial observer, but that's OK; as long as spacetime is flat, SR can be used to work out how R's or F's point of view will work.

 

[atyy]

This is true, but I don't think it disqualifies R or F from having a legitimate "point of view" (if you don't like the word "frame") according to which they are at rest and other objects are moving. R's or F's point of view will not "work the same" in all respects as that of an inertial observer, but that's OK; as long as spacetime is flat, SR can be used to work out how R's or F's point of view will work.

Indeed. R has the temerity to say that he won the race, even though he was at rest!

 

[Saw]

Yes, this would be a good way of avoiding a frame-dependent specification, and under this definition, the race would be a draw.

This is a way to phrase it with which I disagree. In this particular respect, at this more fundamental level, I think Atyy and I fully agree. I’ll explain myself. This is something which is at the core of SR. Someone might say it’s philosophical, but it seems to me totally physical. After all, physics is about solving problems, isn’t it?

With your statement, you seem to be implying that SR means that the “specification” of the race (the rule for determining who wins) can be arbitrary: if you choose a coordinate-independent quantity, you’re doomed to disagreement; if instead you manage to specify an invariant, then you’ll get agreement; but the choice is whimsical, there is no physical reason to prefer one over the other. If this were true, the significance of SR would be that a whole bunch of problems (those where the relevant events are non-colocal) have been shifted to the realm of problems without an objective solution.

To take an extreme example: this would amount to the same as saying that, in the classical scenario, the police has no objective reason for arresting you for driving faster than the speed limit… wrt the ground! You could always say that in your brother’s frame, who was co-moving with you, you were at rest! And you could challenge the traffic regulations on the grounds that they are arbitrary and discriminatory! That would not be right: it’s obvious that, in this context, the ground frame is the relevant reference. That does not mean that any other frame is not apt for predicting what is wrong and what is right. A spaceship flying by just has to measure the speed of the ground wrt it and your speed wrt it, apply the relativistic rule for combining velocities and thus conclude that you were creating danger on the road, which is the outcome that the rule tries to avoid, i.e., the spirit of the problem.

The same applies to this race example. You have to find the spirit of the problem, the practical objective: what do we want to reward by saying R or F won the race? Technology of the spaceship, ability in driving it? It seems to me that both proper acceleration and proper time are good indications thereof. So the choice of these indicators is not arbitrary but a must.

Of the 3 frames, the station frame is the only inertial frame, because the inertial frames associated with R and F are ambiguous, since R and F both accelerated. A frame in which R is always at rest is non-inertial.

This is true, but I don't think it disqualifies R or F from having a legitimate "point of view" (if you don't like the word "frame") according to which they are at rest and other objects are moving. R's or F's point of view will not "work the same" in all respects as that of an inertial observer, but that's OK; as long as spacetime is flat, SR can be used to work out how R's or F's point of view will work.

Here instead I tend to rather agree with PeterDonis. If I have understood PeterDonis well, with the concept of MCIF you can always convert R and F frames into an inertial frame at a specific event. Once this is done, you find that they disagree with the station frame and between themselves in terms of simultaneity and hence in terms of distance traversed by R and F. However, they do not disagree in terms of who won the race. They understand the spirit of the problem and so do not trust their simultaneity judgments on face value and work a little further. For example, what R should do is finding F’s proper time = dt^2 – dx^2. If the result is the same as the reading of his own clock, he agrees that the race was a draw.

Or did I catch it wrong? Would R and F, in spite of being non-inertial, make those calculations? Well, that was your discussion, which I don’t want to interrupt. Just thought that having this practical background might be helpful.

 

[Austin0]

Austin0:

Yes, it is. An MCIF *is* an inertial frame, by definition; it's just that the accerating object is only at rest in that frame momentarily, because it's accelerating. But considered just as an inertial frame, an MCIF is no different from any other.

OK it was simply to this, not inconsequential difference ,that I was refering. That regarding the frame it was only transitory and was in effect a series of MCIF's with differing metrics

#1* Events Rn and Fn+1 are simultaneous in the station frame for all n > 1.

#2* Events Rn and Fn+1 are *not* simultaneous in R's MCIF at Rn, or in F's MCIF at Fn+1, for all n > 1.

reduced to this,disregarding all other considerations, there has never been any disagreement. I am sure I have explicitly recognized this more than once.Two frames cannot agree on simultaneity regarding any two events
On #1 alone,, #2 is an automatic conclusion.

I don't know that I'd say there's no "logic" involved, although maybe "mathematics" would be a better word--but mathematics is just logic using equations and symbols as tools.

Perhaps I phrased that badly. Obviously the math was derived with impeccable logic, both mathematical and mental. I meant that having reached the point of applying the math, you are already beyond the point of exercising your own logical mind toward the question.

And again, please note that those two propositions don't talk about the "accelerating frame" itself--just the station frame and the two MCIFs, R's at Rn and F's at Fn+1--all inertial frames. I'm not asking here if you agree that the "accelerating frame" itself *has* to see things the same way at each event as the MCIF; I want to make sure first that you agree on how things will look in the MCIFs themselves. Since those are just inertial frames, there shouldn't be an issue, since you've already said you agree that relativistic kinematics gives correct predictions for how things look in inertial frames in relative motion. But I want to be sure before going on

Approached from the point of #'s 1 and 2 above, it appears to me to be straight forward application. Just looking at it mentally, it seems to me , we have R now behind ,so its clock is running behind ,exactly opposite the normal situation. F is ahead, with clocks running ahead.
They turn off the engines and resynch their clocks so that R is running ahead and F running behind and we have an inertial frame like any other ,except that it is exactly the same size as it was at rest in its original frame as observed in that original frame. And will continue to be so through any period of extended acceleration. As this is assumed to apply to a single ship as well , this presents an extraordinary picture. A frame that continues to attain greater and greater instantaneous velocities but remains the same length as observed from an inertial frame.
If you have done the math I am sure it all works out fine and I would like to hear any points I am missing.
So what is your interpretation of this picture?? Consider a single long ship.
What immediately occurs to me : The ship is extended beyond its rest length and Lorentz contracted back to exactly the same length?
or
The ship is not extended because Lorentz contraction is a result of specific application of force? But simultaneity, in effect , extends it so the stations are relatively contracted?
Or there is a flaw in the original logic [my own] and this does not describe an actual possible reality
All of these seem to have real problems so what is a better one??
And the real questions behind the query: Does the Lorentz contraction apply simply to material bodies or does it apply to extended frames and the space between bodies?
Is the phenomenon a purely physical one of forces, with implied stresses etc. or is it simply a stressfree result of the geometry of spacetime?
Is it fundamentally a temporal effect that is perceived as a spatial effect??
This one I have spent a lot of time playing with.
These questions do not seem to be clarified by this exercise. Maybe I should have left out the stations and simply gone directly to the physics??
thanks

 

[PeterDonis]

With your statement, you seem to be implying that SR means that the “specification” of the race (the rule for determining who wins) can be arbitrary: if you choose a coordinate-independent quantity, you’re doomed to disagreement; if instead you manage to specify an invariant, then you’ll get agreement; but the choice is whimsical, there is no physical reason to prefer one over the other.

I didn't mean to imply that there was no reason to prefer the invariant way of specifying who wins the race (i.e., preferring "lower elapsed proper time" as the criterion). Of course there is: invariants are the things everybody can agree on, so you avoid the ambiguity of specifying things in frame-dependent ways. I was just pointing out that you do, in fact, have to specify the invariant; you can't just say "whoever reaches the finish line first wins", without qualification, and expect everybody to agree, because those words are ambiguous. Saying "whoever has the least elapsed proper time when they cross their designated finish line, wins" is unambiguous.

Or did I catch it wrong? Would R and F, in spite of being non-inertial, make those calculations? Well, that was your discussion, which I don’t want to interrupt. Just thought that having this practical background might be helpful.

Of course R and F can each calculate what the other's proper time elapsed would be, given the appropriate data. Anyone can. That's not in question.

 

[atyy]

Here instead I tend to rather agree with PeterDonis. If I have understood PeterDonis well, with the concept of MCIF you can always convert R and F frames into an inertial frame at a specific event. Once this is done, you find that they disagree with the station frame and between themselves in terms of simultaneity and hence in terms of distance traversed by R and F. However, they do not disagree in terms of who won the race. They understand the spirit of the problem and so do not trust their simultaneity judgments on face value and work a little further. For example, what R should do is finding F’s proper time = dt^2 – dx^2. If the result is the same as the reading of his own clock, he agrees that the race was a draw.

Or did I catch it wrong? Would R and F, in spite of being non-inertial, make those calculations? Well, that was your discussion, which I don’t want to interrupt. Just thought that having this practical background might be helpful.

I think what I said agrees with all of your points and PeterDonis's too.

 

[Saw]

I didn't mean to imply that there was no reason to prefer the invariant way of specifying who wins the race (i.e., preferring "lower elapsed proper time" as the criterion). Of course there is: invariants are the things everybody can agree on, so you avoid the ambiguity of specifying things in frame-dependent ways. I was just pointing out that you do, in fact, have to specify the invariant; you can't just say "whoever reaches the finish line first wins", without qualification, and expect everybody to agree, because those words are ambiguous. Saying "whoever has the least elapsed proper time when they cross their designated finish line, wins" is unambiguous.

No, that's not the point. You're saying that the reason to prefer one specification over the other is just unambiguity. Unambiguity is not sufficient. With this rationale we could also say that the race is won by the ship whose driver is a lady. That's also unambiguous. (Well, not always..., but you catch the point). And unambiguity is necessary at the level of the ultimate meaning of the chosen concept, but only at that level. In fact, the specification might perfectly be: "whoever arrives earlier in the station frame". A different thing is that the ship frames may reach the same conclusion (not as to simultaneity, but as to the solution of the problem) through their own measurements of proper time. But as I said I don´t want to interrupt the discussion. If it's to going to be controversial, let's leave it aside. If you want to contest this, you can do it in the thread Elegant Universe Example, where this subject is already advanced.

Of course R and F can each calculate what the other's proper time elapsed would be, given the appropriate data. Anyone can. That's not in question.

ok, please answer Austin0.

 

[Saw]

I think what I said agrees with all of your points and PeterDonis's too.

Ah, then I misunderstood. Thanks for the comment.

 

[PeterDonis]

Two frames cannot agree on simultaneity regarding any two events
On #1 alone,, #2 is an automatic conclusion.

Ok, good.

Just looking at it mentally, it seems to me , we have R now behind ,so its clock is running behind ,exactly opposite the normal situation. F is ahead, with clocks running ahead.

I'm not sure what you mean by "opposite the normal situation". What is the "normal" situation I'm supposed to compare this one to, that comes out the opposite way?

But in any case, we haven't gotten to whose clock is running "behind" or "ahead" yet; all we've established so far is that events Rn and Fn+1 aren't simultaneous to either R or F, for n > 1. Now we take the next step: call the proper time elapsed on R's clock at event Rn, T-Rn. Call the proper time elapsed on F's clock at event Fn+1, T-Fn+1. These two proper times are equal, but let's keep separate symbols for them. Then we have the following:

* In R's MCIF at event Rn, the time coordinate of event Fn+1 is *less* than T-Rn.

* In F's MCIF at event Fn+1, the time coordinate of event Rn is *greater* than T-Fn+1.

This is a precise statement of what I think you were trying to get at by saying that R's clock is running behind and F's clock is running ahead. However, a I think a more natural interpretation for R and F to put on these observations would be:

* R, at event Rn, sees F pulling ahead of him, because in R's MCIF at event Rn, F has already passed event Fn+1--i.e., F has already passed his corresponding station.

* F, at event Fn+1, sees R falling behind him, because in F's MCIF at event Fn+1, R has not yet reached event Rn--i.e., R has not yet reached his corresponding station.

These interpretations are simply manifestations of the fact that the spatial separation of R and F, in the frame in which they start out--the station frame--*must* inevitably turn partially into time separation in any other frame in relative motion to the station frame. And the time separation must inevitably lead to "speed separation" in each observer's MCIF at any point along their worldlines.

Furthermore, all of the above will lead to a change in the distance between R and F, as measured in either of their MCIFs. For example, consider R's MCIF at event Rn (n > 1). What is the distance between R and F in that frame? Well, the "distance" between any two worldlines in a given frame is the spacelike interval between points on each of the worldlines that have the same time coordinate. But we've already seen that, whatever event on F's worldline has the same time coordinate as Rn in R's MCIF at Rn, it is *not* Fn+1--it is some event further along F's worldline than Fn+1. So the interval between events Rn and Fn+1 is irrelevant to answering the question, "how far away will F be from R when R passes station #n?"

And similarly, to answer the question, how far away R will be from F when F passes station n+1, we will have to find the event on R's worldline with the same time coordinate as Fn+1, in F's MCIF at Fn+1, and that event will not be as far along R's worldline as Rn. I won't do the math explicitly here (I can in a follow-up post if desired), but when you do all the calculations, it turns out that both distances, that between R and F at Rn and between F and R at Fn+1, are *larger* than the spatial separation between R1 and F2 in the station frame (which is the original separation between R and F, since those are the events where they start out). And of course this is consistent with R's observation that F is pulling away and F's observation that R is falling behind.

All this is also consistent with what is observed in the station frame. In that frame, the spatial separation between the worldlines remains constant. But in that frame, the ships are moving (for any time t > 0), so the separation between them in the station frame is Lorentz contracted, compared to the separation between them in either ship's MCIF (in which one ship is momentarily at rest).

Now, let's consider your suggestion:

They turn off the engines and resynch their clocks so that R is running ahead and F running behind and we have an inertial frame like any other ,except that it is exactly the same size as it was at rest in its original frame as observed in that original frame.

You failed to specify the most important thing: at precisely which events on their worldlines do R and F turn off their engines? It makes a difference.

Let's suppose that R turns off his engine at event Rn and F turns off his engine at event Fn+1 (i.e., they turn off their engines when they each pass their respective stations n and n+1, with the exact value of n being prearranged). This is what I think you meant to specify. Then, they resynchronize their clocks; R emits a light pulse towards F from event Rn, and F emits a light pulse towards R at event Fn+1.

What will happen? Of course their clocks will be out of sync. But to re-sync them, they have to know the distance between the ships, in what will now become their common (inertial) rest frame. What is that distance? They will find (and they can check, of course, by timing round-trip light pulses) that the distance is *not* the same as the original separation between them in the station frame; it is *larger*--in fact, just enough larger so that its Lorentz-contracted value, as it appears in the station frame, is the same as the original separation in the station frame. So it is *not* true that the separation between R and F, in this new inertial frame, will be "exactly the same size as it was at rest in its original frame". (It *is* true that the separation between R and F, in the *station frame*, will be the same as it always was--and this may have been what you meant by your statement of "exactly the same size" above, I'm not sure--as long as R and F each stop their engines at events which are simultaneous in the station frame, as I specified they did, above).

Armed with this (larger) distance value, R and F can each, by observing the "time stamps" on light pulses giving the time of emission, correcting those for light-travel time, and comparing those with the time of reception, correct their clock readings so that they are in sync. How they correct depends on what they want the new "origin" of time in their new common rest frame to be, but in general, R will have to advance his clock reading, or F will have to move back his clock reading, or some combination of the two, in order to get them in sync. (Btw, at the end of the synchronization, both clocks will have a common "origin of time"--R's clock will not be running ahead or F's running behind.)

As this is assumed to apply to a single ship as well , this presents an extraordinary picture.

No, it is *not* assumed to apply to a single ship as well--at least not by me. Everything I've said above applies only to the case of separate ships, with no physical connection between them, who each experience equal, constant proper acceleration, and therefore travel along the worldlines I've labeled R and F.

To apply any of what I've said to a single ship, you would have to set up a scenario such that the front end of the ship traveled along the same worldline as the "front" ship F in the scenario I've been discussing, and the rear end of the ship traveled along the same worldline as the "rear" ship R in that scenario. That can be done, of course, but it's going to look like a very unusual scenario. For example, here's one way of doing it:

Let there be a line of stations, numbered #1, #2, #3, etc., all moving inertially, and all at rest relative to one another. Let there be a long ship whose rear end, R, is located next to station #1, and whose front end, F, is located at station #2, at time t = 0 in the station frame. Let there be rocket engines at each end of the ship (R *and* F), and let them be identical in construction and performance, so that each imparts an equal, constant, proper acceleration to its end of the ship. At time t = 0 in the station frame, both rocket engines turn on, and remain on continuously thereafter.

In the scenario I've just stated, the worldlines of the rear and front ends of the ship *would* be the same as those, R and F, that I've been discussing. And in this scenario, the ship would gradually be stretched, and eventually, when the tension caused by the two rocket engines, each pushing its end of the ship, exceeded the tensile strength of the ship's hull, the ship would break apart.

But notice that to achieve that result, I had to put rocket engines at each end of the ship--not a typical design. :-) Suppose instead that I adopted a more normal rocket design principle:

Let there be a line of stations, numbered #1, #2, #3, etc., all moving inertially, and all at rest relative to one another. Let there be a long ship whose rear end, R, is located next to station #1, and whose front end, F, is located at station #2, at time t = 0 in the station frame. Let there be a rocket engine at the rear of the ship that, when turned on, imparts a constant proper acceleration to its end of the ship. At time t = 0 in the station frame, the rocket engine turns on, and remains on continuously thereafter.

In *this* scenario, the worldline of the *rear* end of the ship will be the same as the one, R, that I've been discussing. But as it stands, the scenario tells us *nothing* about how the *front* end of the ship will move! To draw any conclusions about that, we need to bring in some sort of physical assumption about how the force of the rocket engine at the rear of the ship gets transmitted through the ship's structure to the front of the ship. Different physical assumptions will result in different worldlines for the front of the ship, and that, in turn, will result in different predictions about what the ship will "look like" in its MCIF, and in the station frame--whether it will appear "Lorentz contracted" and by how much, and so on. All of that gets into what I called question #2 in my last post, and I haven't been discussing that up to now.

(Btw, one thing I haven't mentioned in any of the above: I've been quoting above how things would be observed in R's and F's MCIFs, without addressing the point of how the "accelerated frame" relates to the MCIFs--in fact I've simply been assuming that the view from the "accelerated frame" at a given event is the same as the view from the MCIF at that event. Do I need to go into how that works? If so, I will.)

And the real questions behind the query: Does the Lorentz contraction apply simply to material bodies or does it apply to extended frames and the space between bodies?
Is the phenomenon a purely physical one of forces, with implied stresses etc. or is it simply a stressfree result of the geometry of spacetime?
Is it fundamentally a temporal effect that is perceived as a spatial effect??

The question you're asking here is certainly a valid question, and hopefully will be somewhat clarified by what I'm saying. But the way you ask it may not be the best way to ask it. That's why I went to the trouble of separating the issue of "kinematics" from the issue of physical assumptions. Here's what I think is a better way to get at the points you're interested in.

(1) I would use the term "Lorentz contraction" *only* to refer to how a spacelike interval between a specific, defined pair of events appears in different frames which are in relative motion. In other words, the term should be used only for the "kinematic" aspects--the changes in "how things look" that are due *solely* to relative motion, *not* to any other physical assumptions.

(2) This means that the question of Lorentz contraction doesn't even come into play until you've already determined all the worldlines and events of interest. Lorentz contraction, defined as I recommend, is a "result of the geometry of spacetime", and that is all it is. It applies to any spacelike interval, once that interval is specified, regardless of whether the interval is occupied by empty space or by a material body (in the latter case, of course, the worldlines of all the points in the material body must already have been specified).

(3) All the "physical" questions about forces, stresses, etc. should be construed as questions about *how to specify the worldlines and events of interest*. In other words, they are questions about how to figure out which specific points and curves in the geometry of spacetime are the ones we are dealing with. This is where all the stuff about Born rigidity and so forth, and how reasonable various physical assumptions are about how bodies respond to stress, comes into play.

 

[PeterDonis]

If you want to contest this, you can do it in the thread Elegant Universe Example, where this subject is already advanced.

Ok, I'll look in that thread.

 

[PeterDonis]

Saw:

Ok, I'll look in that thread.

I looked and it looks like the thread has died. But I don't disagree with the position you were taking there: you were basically saying that (1) the "solution" (i.e., the "right" specification of who wins the race, for example) has to involve quantities that everyone can agree on, so there's no ambiguity, and (2) which quantities you choose has to be guided by the "spirit" of the problem. I agree with both of those statements. I was focusing on (1) in my comments, but I didn't mean to imply that (2) isn't important too.

 

[Austin0]

=PeterDonis;2358975] Originally Posted by Austin0
Two frames cannot agree on simultaneity regarding any two events
On #1 alone,, #2 is an automatic conclusion.

Ok, good.


Originally Posted by Austin0
Just looking at it mentally, it seems to me , we have R now behind ,so its clock is running behind ,exactly opposite the normal situation. F is ahead, with clocks running ahead.

I'm not sure what you mean by "opposite the normal situation". What is the "normal" situation I'm supposed to compare this one to, that comes out the opposite way?

I am well aware that I have to improve my grasp of accepted terminology. I am working on this. In this case the normal situation would be that relative to a another inertial frame the clocks in the rear would be running ahead of the clocks positioned forward in the direction of motion. As you point out here, that is exactly what we assume would transpire when they synch their clocks by the convention. The rear ship would set its clock ahead or the front ship would set its clock behind.

But in any case, we haven't gotten to whose clock is running "behind" or "ahead" yet; all we've established so far is that events Rn and Fn+1 aren't simultaneous to either R or F, for n > 1. Now we take the next step: call the proper time elapsed on R's clock at event Rn, T-Rn. Call the proper time elapsed on F's clock at event Fn+1, T-Fn+1. These two proper times are equal, but let's keep separate symbols for them. Then we have the following:

* In R's MCIF at event Rn, the time coordinate of event Fn+1 is *less* than T-Rn.

Fine or alternately; at event Rn the simultaneous location of F is Fn+1+? with T-F > T-Rn

* In F's MCIF at event Fn+1, the time coordinate of event Rn is *greater* than T-Fn+1.

OK or at Fn+1 the position of R is Rn minus (?) and the T-Rn-? is less than T-Fn+1

This is a precise statement of what I think you were trying to get at by saying that R's clock is running behind and F's clock is running ahead. However, a I think a more natural interpretation for R and F to put on these observations would be:

I assumed that a precise statement was not necessary and in fact you do seem to have figured out what I was saying in my casual way.

* R, at event Rn, sees F pulling ahead of him, because in R's MCIF at event Rn, F has already passed event Fn+1--i.e., F has already passed his corresponding station.

* F, at event Fn+1, sees R falling behind him, because in F's MCIF at event Fn+1, R has not yet reached event Rn--i.e., R has not yet reached his corresponding station.

Agreed. I was not assuming anything from the perspective of the ships, but only from our perspective of looking at the diagram and applying the assumption of simultaneity from that perspective, to spacetime positions of the ships.

These interpretations are simply manifestations of the fact that the spatial separation of R and F, in the frame in which they start out--the station frame--*must* inevitably turn partially into time separation in any other frame in relative motion to the station frame. And the time separation must inevitably lead to "speed separation" in each observer's MCIF at any point along their worldlines.

Agreed of course. From the point where it became a matter of simply applying the transformation and its implications to the ships, there is no longer any question. You have already answered any question of physics because the transformation itself predicts the complete spatial-temporal coordinate outcome. When I said I was sure your mathematical analysis was correct I did not mean that personally. As I see it, from this point it is purely the logic of mathematics. That any theorem or results validly derived from allowable arguments and axioms is bound to be true. I don't need to see or do the math to be "convinced" that it is going to be totally consistent . I already assume so.
All the rest; the spatial separation etc. is all immediately obvious as soon as you adopt the assumption of that simultaneity. If R is behind and F is ahead of course the distance in that perspective is going to be greater. Etc,etc

You failed to specify the most important thing: at precisely which events on their worldlines do R and F turn off their engines? It makes a difference.

Of course you are correct , it makes a difference. But I didnt see how the difference is relevant to the actual question here so I didnt specify. So if F cuts its engine at Fn+1 it will be before Rn and therefore R will continue acceleration for a longer period thus equalizing the velocities, I would assume. There are interesting detaila here I am sure , figuring out the relative time dilation between the ships etc. But I assume they also would work out consistently and don't add anything to these questions.

How they correct depends on what they want the new "origin" of time in their new common rest frame to be, but in general, R will have to advance his clock reading, or F will have to move back his clock reading, or some combination of the two, in order to get them in sync. (Btw, at the end of the synchronization, both clocks will have a common "origin of time"--R's clock will not be running ahead or F's running behind.)

Of course. That would only apply as observed from an inertial frame moving in the direction from F to R

No, it is *not* assumed to apply to a single ship as well--at least not by me. Everything I've said above applies only to the case of separate ships, with no physical connection between them, who each experience equal, constant proper acceleration, and therefore travel along the worldlines I've labeled R and F.

So I may have been taking JesseM's statement too literally.

In the scenario I've just stated, the worldlines of the rear and front ends of the ship *would* be the same as those, R and F, that I've been discussing. And in this scenario, the ship would gradually be stretched, and eventually, when the tension caused by the two rocket engines, each pushing its end of the ship, exceeded the tensile strength of the ship's hull, the ship would break apart.

AH HA! Finally ;-) From this it seems you are saying that the assumption of stretching does not come from the physics of acceleration re: propagation of momentum etc. but is derived from a conception of simultaneity. That this is a temporal translation but in this context is given a very physical interpretation.
I am not questioning this. My own "intuition" is that simultaneity is at the core of a lot of questions and could very well have physical implications , or at least as "physical" as any other phenomena.
I would guess then that you expect a steel rod , uniformly electromagnetically accelerated to stretch apart?

But notice that to achieve that result, I had to put rocket engines at each end of the ship--not a typical design. :-) Suppose instead that I adopted a more normal rocket design principle:

It would appear that Born rigid acceleration is an ideal abstraction impossible to even approximate in the real world. SO it would seem that any propulsion from the rear , a more normal rocket design as you put it , would automatically be as close to Born acceleration as we could get ?

Let there be a line of stations, numbered #1, #2, #3, etc., all moving inertially, and all at rest relative to one another. Let there be a long ship whose rear end, R, is located next to station #1, and whose front end, F, is located at station #2, at time t = 0 in the station frame. Let there be a rocket engine at the rear of the ship that, when turned on, imparts a constant proper acceleration to its end of the ship. At time t = 0 in the station frame, the rocket engine turns on, and remains on continuously thereafter.

In *this* scenario, the worldline of the *rear* end of the ship will be the same as the one, R, that I've been discussing. But as it stands, the scenario tells us *nothing* about how the *front* end of the ship will move! To draw any conclusions about that, we need to bring in some sort of physical assumption about how the force of the rocket engine at the rear of the ship gets transmitted through the ship's structure to the front of the ship. Different physical assumptions will result in different worldlines for the front of the ship, and that, in turn, will result in different predictions about what the ship will "look like" in its MCIF, and in the station frame--whether it will appear "Lorentz contracted" and by how much, and so on. All of that gets into what I called question #2 in my last post, and I haven't been discussing that up to now.


The question you're asking here is certainly a valid question, and hopefully will be somewhat clarified by what I'm saying. But the way you ask it may not be the best way to ask it. That's why I went to the trouble of separating the issue of "kinematics" from the issue of physical assumptions. Here's what I think is a better way to get at the points you're interested in.

The course of this thread is ample demonstration of the truth of that

(1) I would use the term "Lorentz contraction" *only* to refer to how a spacelike interval between a specific, defined pair of events appears in different frames which are in relative motion. In other words, the term should be used only for the "kinematic" aspects--the changes in "how things look" that are due *solely* to relative motion, *not* to any other physical assumptions.

(2) This means that the question of Lorentz contraction doesn't even come into play until you've already determined all the worldlines and events of interest. Lorentz contraction, defined as I recommend, is a "result of the geometry of spacetime", and that is all it is. It applies to any spacelike interval, once that interval is specified, regardless of whether the interval is occupied by empty space or by a material body (in the latter case, of course, the worldlines of all the points in the material body must already have been specified).

(3) All the "physical" questions about forces, stresses, etc. should be construed as questions about *how to specify the worldlines and events of interest*. In other words, they are questions about how to figure out which specific points and curves in the geometry of spacetime are the ones we are dealing with. This is where all the stuff about Born rigidity and so forth, and how reasonable various physical assumptions are about how bodies respond to stress, comes into play.

I understand what you are saying and certainly agree on the effects being the "result of the geometry of spacetime" but don't see the distinction between kinematics and physics you are implying.
On one hand you seem to be saying that contraction has no physical implications but then adopt assumptions of simultaneity from kinematics that then seem to dictate both contraction and the physical forces involved in acceleration.
Perhaps you could elaborate on this??
Thanks

 

[PeterDonis]

AH HA! Finally ;-) From this it seems you are saying that the assumption of stretching does not come from the physics of acceleration re: propagation of momentum etc. but is derived from a conception of simultaneity. That this is a temporal translation but in this context is given a very physical interpretation.

The stretching isn't an assumption; it's a logical consequence of specifying the worldlines R and F the way we've specified them (that both ends of the ship, R and F, experience equal constant proper acceleration). Once you've specified that, the stretching has to happen, by the laws of SR; you don't have to assume it.

I would guess then that you expect a steel rod , uniformly electromagnetically accelerated to stretch apart?

If the rod has a rocket engine at each end, with both engines imparting equal, constant proper accelerations to their respective ends, as with the ship I specified in my last post, then yes, it will stretch apart. I'm not sure what "electromagnetically accelerated" means here, but if it means that some electromagnetic force is applied all along the rod, as with the scenario I describe below for Born rigid acceleration, then no, the rod won't stretch apart.

It would appear that Born rigid acceleration is an ideal abstraction impossible to even approximate in the real world. SO it would seem that any propulsion from the rear , a more normal rocket design as you put it , would automatically be as close to Born acceleration as we could get ?

Neither of the scenarios I specified corresponds to Born rigid acceleration. You're right that Born rigid acceleration is an ideal abstraction, but it's certainly possible to construct a scenario that would realize it, which is no more physically impossible than the others we've been considering. :-) Here it is:

Let there be a line of stations, numbered #1, #2, #3, etc., all moving inertially, and all at rest relative to one another. Let there be a long ship whose rear end, R, is located next to station #1, and whose front end, F, is located at station #2, at time t = 0 in the station frame. Let the ship be divided into very short segments, running from R to F, and let each segment have its own rocket engine; each rocket engine is programmed to impart to its segment an acceleration equal to $\frac{c^2}{x}$, where x is the distance of the segment from the origin in the station frame at time t = 0. This means that, as we move from the rear end R to the front end F of the ship, the acceleration imparted to each segment gradually gets smaller, in inverse proportion to distance. At time t = 0 in the station frame, all of the rocket engines turn on, and they all remain on continuously thereafter.

In the above scenario, the ship as a whole will undergo Born rigid acceleration. Born rigid acceleration is a "nice" way to accelerate the ship because (1) in the MCIF of any given segment of the ship, at any point on that segment's worldline, the proper length of the ship appears unchanged from its initial value in the station frame at time t = 0; (2) because of this, the internal stresses within the ship stay the same as it moves. But as you can see, you need to arrange things very precisely in order to achieve this--and of course no real rocket ship would be designed to do this. (If you see a reference to Born rigid acceleration requiring a "conspiracy of forces", that's what it refers to.)

On one hand you seem to be saying that contraction has no physical implications but then adopt assumptions of simultaneity from kinematics that then seem to dictate both contraction and the physical forces involved in acceleration.
Perhaps you could elaborate on this??

I'm not saying that Lorentz contraction has no physical implications. For one thing, kinematics is a part of physics. I'm just trying to separate two stages of "solving" a problem that's posed. First, you analyze the specification of the problem to determine all the worldlines and events of interest. Then, you use relativistic kinematics to determine how those worldlines and events will appear in whatever frames you like. You could call both stages "physical" if you want to.

Also, I'm not "assuming" the kinematics, or the simultaneity relations, or the physical forces involved in acceleration, or anything of that sort, except insofar as I'm assuming that SR is correct. The laws of SR *require* all these things to be related in certain specific ways, once you've determined the worldlines and events involved. That's all I'm trying to emphasize when I talk about kinematics.

 

[Austin0]

Re: Is Distance/space Lorentz contracted?

--------------------------------------------------------------------------------

Originally Posted by Austin0
From this it seems you are saying that the assumption of stretching does not come from the physics of acceleration re: propagation of momentum etc. but is derived from a conception of simultaneity.

The stretching isn't an assumption; it's a logical consequence of specifying the worldlines R and F the way we've specified them #1(that both ends of the ship, R and F, experience equal constant proper acceleration). Once you've specified that, the stretching has to happen, by the laws of SR; you don't have to assume it.

Hi PeterDonis In this case [i.e. a single ship] we have not yet specified the world lines have we?

#1 This appears to make assumptions:
A) That simple application of force from one end would not result in equal acceleration at both ends.
B) That length contraction would not intrinsically occur as the result of instananeous velocities.[clock hypothesis]
C) It seems to be taking kinetic conclusions arrived at through consideration of two ships and then applying them to a different situation. Simply assigning world lines on that basis.
But I may be misinterpreting your statements so further clarification might help.

Originally Posted by Austin0
I would guess then that you expect a steel rod , uniformly electromagnetically accelerated to stretch apart?

If the rod has a rocket engine at each end, with both engines imparting equal, constant proper accelerations to their respective ends, as with the ship I specified in my last post, then yes, it will stretch apart. I'm not sure what "electromagnetically accelerated" means here, but if it means that some electromagnetic force is applied all along the rod, as with the scenario I describe below for Born rigid acceleration, then no, the rod won't stretch apart.

My knowledge of electromagnetic acceleration is minimal derived mostly from reading about railgun technology, current and hypothetical. From what I understand there are various methods of application depending on the timing and directions of fields. It seems that it is possible to, in effect, pull a rod , or push-pull or apply uniform fields , so I imagine it is also possible to control the force in the specific format required by Born acceleration.
So in this context do you still think a uniformly accelerated rod would deform?

Originally Posted by Austin0
It would appear that Born rigid acceleration is an ideal abstraction impossible to even approximate in the real world. SO it would seem that any propulsion from the rear , a more normal rocket design as you put it , would automatically be as close to Born acceleration as we could get ?

Let there be a long ship whose rear end, R, is located next to station #1, and whose front end, F, is located at station #2, at time t = 0 in the station frame. Let the ship be divided into very short segments, running from R to F, and let each segment have its own rocket engine; each rocket engine is programmed to impart to its segment an acceleration equal to , where x is the distance of the segment from the origin in the station frame at time t = 0. This means that, as we move from the rear end R to the front end F of the ship, the acceleration imparted to each segment gradually gets smaller, in inverse proportion to distance. At time t = 0 in the station frame, all of the rocket engines turn on, and they all remain on continuously thereafter.

In the above scenario, the ship as a whole will undergo Born rigid acceleration. Born rigid acceleration is a "nice" way to accelerate the ship because (1) in the MCIF of any given segment of the ship, at any point on that segment's worldline, the proper length of the ship appears unchanged from its initial value in the station frame at time t = 0; (2) because of this, the internal stresses within the ship stay the same as it moves. But as you can see, you need to arrange things very precisely in order to achieve this--and of course no real rocket ship would be designed to do this.

This raises a lot of interesting questions.
It would seem that application of force to the rear would represent maximal resistence to expansion. Now if compression was the issue, then it would make sense to apply additional forward force throughout the system. But that is not the object in this case, so I don't see how this calibrated application has any effect regarding the reduction of expansion. ?

If this does apply then what is the expected result for the contents of this vehicle??
Does this mean that a steel ruler hanging from the wall of Einsteins elevator would stretch like silly putty??
What of a ruler that has been accelerated while positioned transverse to the motion and is then rotated longitudinally?

Suddenly subjected to the cumulative acceleration of the total course?

I'm not saying that Lorentz contraction has no physical implications. For one thing, kinematics is a part of physics. I'm just trying to separate two stages of "solving" a problem that's posed. First, you analyze the specification of the problem to determine all the worldlines and events of interest. Then, you use relativistic kinematics to determine how those worldlines and events will appear in whatever frames you like. You could call both stages "physical" if you want to.

Also, I'm not "assuming" the kinematics, or the simultaneity relations, or the physical forces involved in acceleration, or anything of that sort, except insofar as I'm assuming that SR is correct. The laws of SR *require* all these things to be related in certain specific ways, once you've determined the worldlines and events involved. That's all I'm trying to emphasize when I talk about kinematics.

Agreed kinematics is a part of physics of course. It seems to me this question enters an area of a certain ambiguity.
For a second consider Newtonian kinematics. I am sure you would agree that the Galillean transforms had no physical implications or assumptions beyond the invariance of Newtonian kinematics. Given a set of coordinate events in one frame, there were no additional considerations of physics, per se, required to derive accurate events in another frame.
I think we agree that this is equally applicable to the Lorentzian transforms as applied to events between inertial frames.
But regarding the transitional state between inertial frames is a different situation.
To get coordinate events in any frame then requires consideration of physics, assumptions about, not only the physics of acceleration, but also about the meaning and interpretation of the Lorentzian effects, which did not need to be considered at all when dealing with inertial frames.
So IMHO it is not just a question of whether a particular set of assumptions is consistent with the fundamental constraints of SR, but whether or not it is the only set of assumptions that is consistent or the set that is the most consistent with the physics of acceleration as we know it from empirical testing to this point. I do not presume to have an answer to this, but it does appear to me that there are other possibilites.
Thanks. BTW this is starting to get more and more interesting

 

[PeterDonis]

In this case [i.e. a single ship] we have not yet specified the world lines have we?

In each of the single ship cases I specified, the worldlines of any point on the ship where I put a rocket engine were specified (because the accelerations that the rocket engines imparted to their locations on the ship were specified). But those were the *only* points whose worldlines were specified.

So, in the first case, with a rocket engine at the rear and front end of the ship, the worldlines of the rear and front end were specified, but the worldlines of intermediate parts of the ship were not. The front and rear end worldlines being specified were sufficient to conclude that the ship would stretch, but the exact worldlines of intermediate points would depend on the exact details of the stretching--i.e., how each segment responded to the tensile stress caused by the rear and front ends getting further apart.

In the second case, with a rocket engine only at the rear, as I said then, only the rear end's worldline was specified; the worldlines of all the other parts of the ship would depend on the details of how the force imparted at the rear end was transmitted through the ship. More on that below.

In the third case, with rocket engines all along the ship (Born rigid acceleration), the worldlines of all parts of the ship *were* specified (at least to the accuracy of the size of each "segment" of the ship that was assigned an engine). I had to do that because it's the only way to realize Born rigid acceleration--in effect, specifying that an object undergoes Born rigid acceleration *does* specify the worldlines of every part of the object.

A) That simple application of force from one end would not result in equal acceleration at both ends.

That's correct. It can't possibly, because it takes time for the information that the force has been applied to travel from one end of the ship to the other.

That information can't travel faster than the speed of light, so, for example, if the ship is one light-second long, then the front end can't possibly begin accelerating sooner than 1 second after the rear end does (as seen in the "station frame", the frame in which the ship is initially at rest). For any actual known material, it will take a lot longer than that for the applied force to propagate through the ship (because the speed of sound in all known materials, which is the speed at which applied forces propagate through the material, is many orders of magnitude less than the speed of light).

B) That length contraction would not intrinsically occur as the result of instananeous velocities.(clock hypothesis)

Once again, don't confuse the kinematic effect of Lorentz contraction, which requires that you have already specified the worldlines and events you're dealing with, with the dynamic effects like how objects respond to applied forces, which are what you use to *figure out* which worldlines and events you're dealing with.

The kinematic effect is simply this: if I know the length of a given object in a given frame, I also know its length in any other frame, just by applying the Lorentz transformations. But in order to know its length in any frame, I have to know the worldlines of both its ends. If I don't know those worldlines yet, I don't know its length. If I know only one particular pair of events on the worldlines of the object's ends--for example, I know the spatial position of its rear and front ends at time t = 0 in the station frame--then I only know its length in the frame in which that pair of events is simultaneous, at the particular time of those events--in this case, the station frame at time t = 0. I don't have enough information in that case to even apply the Lorentz transformation, because those two events won't be simultaneous in any frame that's moving relative to the station frame, so I can't use them to determine the object's length in any other frame. I have to find a pair of events on the worldlines of the object's ends that are simultaneous in the *moving* frame to determine its length in that frame, and if I don't know the worldlines of the ends, I can't do that.

Discussions of the Lorentz transformations in textbooks usually gloss over all this, by just specifying (often implicitly) all the worldlines that will be needed to solve whatever problems are posed, including performing whatever Lorentz transformations are required. But if you look "under the hood", you'll see that all the above stuff still applies; it's just hidden by the way the problem is stated. In other words: the kinematic effects, as I've defined them above, *do* "intrinsically occur"--but you have to know a lot to calculate what they are. Gaining that knowledge requires figuring out the dynamic effects first, so you can determine the worldlines and events you're dealing with, and the dynamic effects are *not* "intrinsic"--they depend on the physical assumptions you make.

C) It seems to be taking kinetic conclusions arrived at through consideration of two ships and then applying them to a different situation.

Nope--the only conclusions I'm drawing for each case are based on the worldlines that I know are specified in each case (I discussed which ones those were for each case above).

My knowledge of electromagnetic acceleration is minimal derived mostly from reading about railgun technology, current and hypothetical. From what I understand there are various methods of application depending on the timing and directions of fields. It seems that it is possible to, in effect, pull a rod , or push-pull or apply uniform fields , so I imagine it is also possible to control the force in the specific format required by Born acceleration. So in this context do you still think a uniformly accelerated rod would deform?

Once again, if the force is controlled the way I specified for Born rigid acceleration, then no, the rod would not deform. (I would think, as you do, that this is possible, but I don't know if any actual electromagnetic acceleration technologies do it in practice.) If the force is applied some other way, then the deformation of the rod would depend on how the force was applied.

It would seem that application of force to the rear would represent maximal resistence to expansion. Now if compression was the issue, then it would make sense to apply additional forward force throughout the system. But that is not the object in this case, so I don't see how this calibrated application has any effect regarding the reduction of expansion. ?

In the case of Born rigid acceleration, we *are* applying additional forward force throughout the system. At each point along the ship, we're applying just enough force to keep it "ahead" of the rear end of the ship by just the right amount, so that the total length of the ship remains constant (and internal stresses likewise remain constant). If we applied more force--for example, if we applied just as much force at the front end of the ship as at the rear--the ship would stretch, as in the first "single ship" case I specified. If we applied less force, the ship would compress, which is what would happen, at least initially, in the second "single ship" case I specified, where there was only a rocket engine at the rear end of the ship.

(How do I know the ship would initially compress in this case? Because, as I said above, the information that the force has been applied at the rear can't travel through the ship faster than the speed of light, so there will be an unavoidable minimum time delay before each part of the ship forward of the rear end starts accelerating. That means the ship will initially compress, as each segment, when it first starts accelerating, moves closer to the segment in front of it, that hasn't received the information to start accelerating yet.)

Does this mean that a steel ruler hanging from the wall of Einsteins elevator would stretch like silly putty??

If the elevator were accelerated hard enough, yes. Just as a steel ruler hanging from the ceiling in a room that was stationary in a strong enough gravitational field would stretch. Remember that in many of these cases, the implied accelerations we've been talking about are in the billions or trillions of g's.

What of a ruler that has been accelerated while positioned transverse to the motion and is then rotated longitudinally?

Accelerated how? Does the acceleration stop before the ruler is rotated?

Suddenly subjected to the cumulative acceleration of the total course?

I'm not sure I understand what you're asking here.

To get coordinate events in any frame then requires consideration of physics, assumptions about, not only the physics of acceleration, but also about the meaning and interpretation of the Lorentzian effects, which did not need to be considered at all when dealing with inertial frames.

The last statement here is incorrect; the meaning and interpretation of Lorentz contraction and time dilation is an integral part of figuring out how SR works for inertial frames. If Einstein had not been able to give each of these effects a direct physical interpretation in inertial frames in relative motion, I doubt he would have made much headway in getting people to consider SR. There is no fundamental difference here between inertial and accelerated motion, except that an object on an accelerated worldline "changes its inertial frame" as it moves--you have to use its MCIF at any given event to determine how things will look to it at that event. But as long as you know the object's worldline, its MCIF is fully determined at each event.

So IMHO it is not just a question of whether a particular set of assumptions is consistent with the fundamental constraints of SR, but whether or not it is the only set of assumptions that is consistent or the set that is the most consistent with the physics of acceleration as we know it from empirical testing to this point. I do not presume to have an answer to this, but it does appear to me that there are other possibilites.

Once more, I think it's important to keep emphasizing the distinction between kinematic and dynamic effects. As far as the kinematic effects are concerned (i.e., how things look in different frames, once you've already determined the specific worldlines and events you're dealing with), there is only *one* way to deal with them that is consistent with the fundamental constraints of SR (basically, the principle of relativity and the fact that the speed of light is the same in all inertial frames, plus the clock postulate). Once you have a specific set of worldlines and events, the way things look in any frame, inertial or accelerated, is fully determined by the postulates of SR (the three I just listed above) and the specifications of the worldlines and events. The only restriction is that spacetime has to be flat, at least to a good enough approximation for the problem you're dealing with.

However, that still leaves quite a bit of freedom in dealing with accelerated objects, because you still have to somehow determine the specific worldlines of the various parts of those objects, and that will depend on what assumptions you make about how forces applied to one part of the object are transmitted to other parts--the dynamic effects. There are *lots* of different assumptions you could make here that are consistent with the laws of SR, and each set of assumptions will result in a different set of worldlines for the various parts of the object. So in that sense, yes, there are certainly lots of different ways that an accelerated object could "look" that are all consistent with SR.

BTW this is starting to get more and more interesting

Good. :-)

 

[Austin0]

Originally Posted by Austin0
In this case [i.e. a single ship] we have not yet specified the world lines have we?

In each of the single ship cases I specified, the worldlines of any point on the ship where I put a rocket engine were specified (because the accelerations that the rocket engines imparted to their locations on the ship were specified). But those were the *only* points whose worldlines were specified.

So, in the first case, with a rocket engine at the rear and front end of the ship, the worldlines of the rear and front end were specified, but the worldlines of intermediate parts of the ship were not. The front and rear end worldlines being specified were sufficient to conclude that the ship would stretch, but the exact worldlines of intermediate points would depend on the exact details of the stretching--i.e., how each segment responded to the tensile stress caused by the rear and front ends getting further apart.

It seems to me that the above conclusions are justified only if:

A) It is justifiable to completely disregard the physics and existence of the intermediate body. I.e. That the motion and acceleration of the actual points of propulsion are completely independant of the connection with the total system.

B) You can apply an interpretation of acceleration as applied force and disregard the interpretation of acceleration as a change in the motion of a system.

C) That the Lorentz contraction can be completely disregarded as a factor in the physical situation regarding the relevant worldlines ,but can then, simply be applied as a kinematic result, ex post facto


In the second case, with a rocket engine only at the rear, as I said then, only the rear end's worldline was specified; the worldlines of all the other parts of the ship would depend on the details of how the force imparted at the rear end was transmitted through the ship. More on that below.

In the third case, with rocket engines all along the ship (Born rigid acceleration), the worldlines of all parts of the ship *were* specified (at least to the accuracy of the size of each "segment" of the ship that was assigned an engine). I had to do that because it's the only way to realize Born rigid acceleration--in effect, specifying that an object undergoes Born rigid acceleration *does* specify the worldlines of every part of the object.

Originally Posted by Austin0
A) That simple application of force from one end would not result in equal acceleration at both ends.

That's correct. It can't possibly, because it takes time for the information that the force has been applied to travel from one end of the ship to the other.

That information can't travel faster than the speed of light, so, for example, if the ship is one light-second long, then the front end can't possibly begin accelerating sooner than 1 second after the rear end does (as seen in the "station frame", the frame in which the ship is initially at rest). For any actual known material, it will take a lot longer than that for the applied force to propagate through the ship (because the speed of sound in all known materials, which is the speed at which applied forces propagate through the material, is many orders of magnitude less than the speed of light).

We are in complete agreement here regarding the propagation time. I would propose for the purposes of this discussion that Mach 1 be considered the practical limit. If the force applied is sufficient to make the speed of light relevant, I would say it is beyond the bounds of realism and billions of g's would imply inertial forces far beyond the ability of an addition of a few extra engines to counter.
Would you agree that as far as the principles involved are concerned ,the magnitude of the acceleration rate is not important?
WHile we agree on the importance of propagation time, we seem to have different ideas concerning the implications of this fact.
For discussion I will give a quick outline of my understanding and you or someone else with greater knowledge can point out any areas where I diverge from known physics.
Given that the magnitude of force is within the materials ability to transmit it fast enough, applied energy, momentum, propagates through the system, not as motion, but as a reciprocal occilation. At initial application ,there is no actual motion at the point of contact, the energy is simply passed along , with no net motion of either molecules or pressure.
It is not until the entire system has received enough energy that there is actual coordinate movement of the system as a whole.
As the magnitude is increased and exceeds the ability to transmit it fast enough, there is a build up , a delay at the point of introduction which could result in actual compression until the whole system reaches the point of motion. The initial compression you mentioned.
But is there any reason to expect this to remain once the system is in motion and the momentum is uniform throughout the system and additional applied energy is constant?
Or to assume that the intermolecular tensile forces would not reach some kind of equilibrium with the applied force??
Even possibly a sustained occilation?
Even more important, is there any reason to expect ,not only a sustained compression but an incrementally increasing compression or expansion over time?
My own thought is that this would only be reasonable if it was a dynamic rate of acceleration.
Obviously the situation becomes more complex with more elastic subjects.
Say a spring. Where the vector of applied force would not be aligned with the path of propagation and the path length would be longer relative to the mass and system length.
Torsion effects etc etc. would seem to lead to more intitial compression or expansion but not neccessarily any more of a sustained or cumulative effect.


Originally Posted by Austin0
B) That length contraction would not intrinsically occur as the result of instananeous velocities.(clock hypothesis)

Once again, don't confuse the kinematic effect of Lorentz contraction, which requires that you have already specified the worldlines and events you're dealing with, with the dynamic effects like how objects respond to applied forces, which are what you use to *figure out* which worldlines and events you're dealing with.

The kinematic effect is simply this: if I know the length of a given object in a given frame, I also know its length in any other frame, just by applying the Lorentz transformations.

QUOTE] I don't have enough information in that case to even apply the Lorentz transformation, because those two events won't be simultaneous in any frame that's moving relative to the station frame, so I can't use them to determine the object's length in any other frame.[/QUOTE]

In other words: the kinematic effects, as I've defined them above, *do* "intrinsically occur"--but you have to know a lot to calculate what they are.

Gaining that knowledge requires figuring out the dynamic effects first, so you can determine the worldlines and events you're dealing with, and the dynamic effects are *not* "intrinsic"--they depend on the physical assumptions you make.

I am not completely sure what the implications are of all the above quotes as some of them seem to be in agreement with what I have been saying while others seem to disagree.
But one thing seems sure, taken all together they appear ro agree with me when I said that the situation is not as simple as it might appear.


Originally Posted by Austin0
It would seem that application of force to the rear would represent maximal resistence to expansion. Now if compression was the issue, then it would make sense to apply additional forward force throughout the system. But that is not the object in this case, so I don't see how this calibrated application has any effect regarding the reduction of expansion. ?

In the case of Born rigid acceleration, we *are* applying additional forward force throughout the system. At each point along the ship, we're applying just enough force to keep it "ahead" of the rear end of the ship by just the right amount, so that the total length of the ship remains constant (and internal stresses likewise remain constant).

You seem to have missed the point of what I was saying.
In this case the proposition is: Single point thrust from the rear of the system would result in system expansion. Expansion is the "problem"
Logically it would seem to follow that a physical solution to this problem would be a counter thrust applied to the front. A compressive thrust against the direction of motion.
Do you disagree with this logic?
It would also seem to follow that the addition of forward propulsion would not only not reduce expansion it would reduce the compressive effect of a single rear drive.
It would make sense if you wanted to reduce compression.
DO you disagree with this logic??

If

we applied more force--for example, if we applied just as much force at the front end of the ship as at the rear--the ship would stretch, as in the first "single ship" case I specified. If we applied less force, the ship would compress, which is what would happen, at least initially, in the second "single ship" case I specified, where there was only a rocket engine at the rear end of the ship.

(How do I know the ship would initially compress in this case? Because, as I said above, the information that the force has been applied at the rear can't travel through the ship faster than the speed of light, so there will be an unavoidable minimum time delay before each part of the ship forward of the rear end starts accelerating. That means the ship will initially compress, as each segment, when it first starts accelerating, moves closer to the segment in front of it, that hasn't received the information to start accelerating yet.)

Originally Posted by Austin0
What of a ruler that has been accelerated while positioned transverse to the motion and is then rotated longitudinally?

Accelerated how? Does the acceleration stop before the ruler is rotated?

Consider either case.
A ruler that has not been Born accelerated longitudinally during the course of a long acceleration but is then reoriented during acceleration.
And then the same ruler a second after the engines are turned off.
What do expect the result to be in either case?

Originally Posted by Austin0
Suddenly subjected to the cumulative acceleration of the total course?

I'm not sure I understand what you're asking here.

In the course of a prolonged acceleration according to the hypothesis , the system has undergone coordinate contraction as well as resisted assumed proper expansion due to the action of a specific physical acceleration , directional force. A ruler that has been translated while oriented transversely has not experienced this directed force along its length.
If rotated so that that length is aligned with the acquired motion [cumulative acceleration]
it should not be contracted according to the hypothesis unless you assume that it instantly
undergoes an equivalent acceleration through rotation, in which case I guess you could expect it to stretch and break unless it had Born rockets attached throughout to control the rotation?
Or other?

Originally Posted by Austin0
To get coordinate events in any frame then requires consideration of physics, assumptions about, not only the physics of acceleration, but also about the meaning and interpretation of the Lorentzian effects, which did not need to be considered at all when dealing with inertial frames.

The last statement here is incorrect; the meaning and interpretation of Lorentz contraction and time dilation is an integral part of figuring out how SR works for inertial frames. If Einstein had not been able to give each of these effects a direct physical interpretation in inertial frames in relative motion, I doubt he would have made much headway in getting people to consider SR. There is no fundamental difference here between inertial and accelerated motion, except that an object on an accelerated worldline "changes its inertial frame" as it moves--you have to use its MCIF at any given event to determine how things will look to it at that event.

Out of context again.
So here is a more complete context.
PeterDonis

That's why I went to the trouble of separating the issue of "kinematics" from the issue of physical assumptions.

The kinematic effect is simply this: if I know the length of a given object in a given frame, I also know its length in any other frame, just by applying the Lorentz transformations
Original Austin0
For a second consider Newtonian kinematics. I am sure you would agree that the Galillean transforms had no physical implications or assumptions beyond the invariance of Newtonian kinematics. Given a set of coordinate events in one frame, there were no additional considerations of physics, per se, required to derive accurate events in another frame.
I think we agree that this is equally applicable to the Lorentzian transforms as applied to events between inertial frames.
_______________________________________________________________________

SO I was actually agreeing with you and somehow this was reinterpreted into implying that I thought there was no physical interpretation or meaning to Lorentz effects in SR.
In fact you seem to be contradicting yourself, since you have said more than once that physical assumptions were irrelevant to the application of the coordinate transforms.

However, that still leaves quite a bit of freedom in dealing with accelerated objects, because you still have to somehow determine the specific worldlines of the various parts of those objects, and that will depend on what assumptions you make about how forces applied to one part of the object are transmitted to other parts--the dynamic effects. There are *lots* of different assumptions you could make here that are consistent with the laws of SR, and each set of assumptions will result in a different set of worldlines for the various parts of the object. So in that sense, yes, there are certainly lots of different ways that an accelerated object could "look" that are all consistent with SR.

I am happy to see an area of at least partial agreement ;-)
We haven't yet touched on the basis for the Born hypothesis. I hope you don;t get tired of this topic before we do as I am sure you are a lot more knowledgeable and I am very curious.
Thanks

 

[PeterDonis]

A) It is justifiable to completely disregard the physics and existence of the intermediate body. I.e. That the motion and acceleration of the actual points of propulsion are completely independant of the connection with the total system.

You're correct that, in specifying that the rocket engines at the front and rear end of the ship each imparted the same constant proper acceleration to their ends of the ship, I was assuming that the engines could do that independently of the forces exerted on the front and rear ends by other parts of the ship. In any real case, that would mean the engines would have to be controlled very precisely, to maintain the same constant proper acceleration; for example, as the ship started to stretch, the segment of the ship just to the rear of the front end would start pulling back on the front end, so the rocket engine at the front end would have to increase its rate of fuel burn *just* enough to compensate for this extra pull, in order to maintain the same constant proper acceleration. Since that's not physically impossible (however unlikely it might be in practice), it doesn't invalidate my specification of the scenario, since it's just a thought experiment.

B) You can apply an interpretation of acceleration as applied force and disregard the interpretation of acceleration as a change in the motion of a system.

This isn't a matter of interpretation; the two have to be connected, because of conservation of momentum. The applied force is

$$F = \frac{d P}{d \tau}$$

where P is the energy-momentum 4-vector of the small segment of the ship that's being accelerated, and $\tau$ is the proper time of that same small segment. But we have

$$P = m U$$

where m is the rest mass of the segment and U is its 4-velocity; so we have

$$F = m \frac{d U}{d \tau} = m A$$

where A is the proper acceleration (i.e., the rate of change of 4-velocity with respect to proper time). This is the relativistic version of Newton's second law, and if momentum is conserved, it *has* to hold.

C) That the Lorentz contraction can be completely disregarded as a factor in the physical situation regarding the relevant worldlines ,but can then, simply be applied as a kinematic result, ex post facto

Again, this isn't an assumption, it's a *definition* I've made in order to make it easier to avoid confusion between what I've been calling kinematic and dynamic effects. You can define "Lorentz contraction" to include those dynamic effects, if you want to, but then you'll have to come up with a different name for the kinematic ones, because they'll still be separate, conceptually, from the dynamic ones.

I would propose for the purposes of this discussion that Mach 1 be considered the practical limit.

Mach 1 *is* the limit--the speed of sound in a given substance is the limit of how fast applied forces can be propagated in that substance. But the speed of sound varies widely according to the substance. The speed of sound in air, which is what people usually refer to as "Mach 1", is pretty slow as sound speeds go, about 340 m/s, or 10^-6 x the speed of light. In water the sound speed is about 5 times that, about 1500 m/s, and in steel it's about 20 times that in air, about 6,000 m/s.

Sound speeds in plasmas (such as, for example, the Sun) can actually get up to a few percent of the speed of light. For example, using the "ion sound velocity" formula from http://en.wikipedia.org/wiki/Plasma_parameters#Velocities", with parameters typical of the Sun's core ($\gamma$ = 5/3 for a non-relativistic plasma, $T_e$ = 1,500 eV, Z = 1 and $\mu$ = 1 because the Sun is mostly hydrogen), yields a sound speed of about 5 x 10^7 cm/s, or about 2/10 of a percent of the speed of light; and the Sun is actually a fairly cool plasma--larger stars have significantly hotter cores, and therefore higher sound speeds.

If the force applied is sufficient to make the speed of light relevant, I would say it is beyond the bounds of realism and billions of g's would imply inertial forces far beyond the ability of an addition of a few extra engines to counter.

The muons that were trapped and their lifetimes measured to check the clock postulate experienced accelerations up to 10^18 g. There's nothing physically impossible about such accelerations, and someday we may learn how to make macroscopic objects that are able to withstand them; the laws of physics don't prohibit that. Anyway:

Would you agree that as far as the principles involved are concerned ,the magnitude of the acceleration rate is not important?

Yes.

Given that the magnitude of force is within the materials ability to transmit it fast enough,

The speed of transmission of forces through an object (i.e., the sound speed) doesn't depend on the magnitude of the force. Small forces don't get transmitted any faster than large ones.

*How* the force gets transmitted does depend greatly on the physical constitution of the object--see next item.

At initial application ,there is no actual motion at the point of contact, the energy is simply passed along , with no net motion of either molecules or pressure.

Unless the point of application of the force is physically restrained in some way so that it can't move (i.e., if there's no counterbalancing force to work against the applied force), it *will* move upon application of the force. That is, the individual atoms or molecules of the object will move; at the atomic/molecular level, of course, the applied force is transmitted by interactions between the atoms/molecules--more precisely, by the electrical repulsion between the electrons in the atoms/molecules.

As the first "layer" of atoms/molecules in the object move in response to the applied force, they will, of course, move closer to the next "layer" in the object, and will thus exert a force on that next layer. In other words, the applied force starts a "wave" of force moving through the object--more precisely, a longitudinal wave (i.e., a sound wave) of alternating compression and expansion, as each layer pushes against the next, which then responds by moving away and pushing against the next in turn.

If we assume that the applied force at one end of the object continues to be applied, then once the initial wave has traveled the length of the object, it will start to be damped out. This is because as each layer pushes against the next, not all of the energy in the push gets converted into motion of the next layer; some of the energy goes into the internal degrees of freedom in the atoms/molecules--i.e., it gets converted into heat. If the material is very stiff (e.g., very high tensile steel--or even better, carbon nanotubes), the wave will be damped out quickly; if the material is soft, it will take longer for the wave to be damped out. While the wave is damping, the object will be oscillating, expanding and compressing with gradually decreasing amplitude. Once the wave is damped out, the object will be in a state (assuming the force continues to be applied at one end) in which there is a compressive stress all along its length, and in which its material is somewhat hotter than it was before the force was applied (because some of the applied energy got converted into heat during the damping).

In this case the proposition is: Single point thrust from the rear of the system would result in system expansion.

As the above analysis shows, it wouldn't; it would result in compression.

To get coordinate events in any frame then requires consideration of physics, assumptions about, not only the physics of acceleration, but also about the meaning and interpretation of the Lorentzian effects, which did not need to be considered at all when dealing with inertial frames.

...

For a second consider Newtonian kinematics. I am sure you would agree that the Galillean transforms had no physical implications or assumptions beyond the invariance of Newtonian kinematics. Given a set of coordinate events in one frame, there were no additional considerations of physics, per se, required to derive accurate events in another frame.
I think we agree that this is equally applicable to the Lorentzian transforms as applied to events between inertial frames.
__________________________________________________ _____________________

SO I was actually agreeing with you and somehow this was reinterpreted into implying that I thought there was no physical interpretation or meaning to Lorentz effects in SR.
In fact you seem to be contradicting yourself, since you have said more than once that physical assumptions were irrelevant to the application of the coordinate transforms.

I know that we agree that converting coordinates from one *inertial* frame to another is just "kinematics"; but you appeared to be saying that that was *not* true of accelerating frames. I'm saying that, since an "accelerating frame" at any given event on an accelerated worldline is just the MCIF at that event, what applies to inertial frames applies equally to "accelerating frames". Converting coordinates from an inertial frame to an "accelerated frame" is just converting from the inertial frame to the MCIF, which is another inertial frame. So it's just "kinematics" in both cases. All the physics is in how you determine *which* accelerated worldlines you're dealing with. As long as we agree on that, I'm good.

We haven't yet touched on the basis for the Born hypothesis. I hope you don;t get tired of this topic before we do as I am sure you are a lot more knowledgeable and I am very curious.

I'm certainly not tired of the topic. If you have a question about the Born hypothesis, please ask.

 

[Austin0]

So, in the first case, with a rocket engine at the rear and front end of the ship, the worldlines of the rear and front end were specified, but the worldlines of intermediate parts of the ship were not. The front and rear end worldlines being specified were sufficient to conclude that the ship would stretch, but the exact worldlines of intermediate points would depend on the exact details of the stretching--i.e., how each segment responded to the tensile stress caused by the rear and front ends getting further apart.

=PeterDonis;2367574]You're correct that, in specifying that the rocket engines at the front and rear end of the ship each imparted the same constant proper acceleration to their ends of the ship, I was assuming that the engines could do that independently of the forces exerted on the front and rear ends by other parts of the ship. In any real case, that would mean the engines would have to be controlled very precisely, to maintain the same constant proper acceleration; for example, as the ship started to stretch, the segment of the ship just to the rear of the front end would start pulling back on the front end, so the rocket engine at the front end would have to increase its rate of fuel burn *just* enough to compensate for this extra pull, in order to maintain the same constant proper acceleration. Since that's not physically impossible (however unlikely it might be in practice), it doesn't invalidate my specification of the scenario, since it's just a thought experiment.

Here you are basing conclusions of both physics and resultant worldlines on an already arrived at assumption of stretching. Well, we know the stretching occurs, just look at the worldlines. Well we know these would be the worldlines because we know that stretching must occur.


Originally Posted by Austin0
B) You can apply an interpretation of acceleration as applied force and disregard the interpretation of acceleration as a change in the motion of a system.

This isn't a matter of interpretation; the two have to be connected, because of conservation of momentum. The applied force is

$$F = \frac{d P}{d \tau}$$

where P is the energy-momentum 4-vector of the small segment of the ship that's being accelerated, and $\tau$ is the proper time of that same small segment. But we have

$$P = m U$$

where m is the rest mass of the segment and U is its 4-velocity; so we have

$$F = m \frac{d U}{d \tau} = m A$$

where A is the proper acceleration (i.e., the rate of change of 4-velocity with respect to proper time). This is the relativistic version of Newton's second law, and if momentum is conserved, it *has* to hold.

I was not suggesting that conservation of momentum wouldn't hold. I would assume that the overall acceleration of the system would be the sum of force applied at both ends.
I was talking about the apparent implication of what you were saying : that the only points of equal proper acceleration were the points of actual application of the force. Disregarding the possibility that the overall system change in velocity could be equivalent throughout , which would mean equal proper acceleration if that were the case . Would you disagree? That if accelerometer readings were equivalent at all parts of the ship, this would mean equal proper acceleration.

That information can't travel faster than the speed of light, so, for example, if the ship is one light-second long, then the front end can't possibly begin accelerating sooner than 1 second after the rear end does

Mach 1 *is* the limit--the speed of sound in a given substance is the limit of how fast applied forces can be propagated in that substance. But the speed of sound varies widely according to the substance. The speed of sound in air, which is what people usually refer to as "Mach 1", is pretty slow as sound speeds go, about 340 m/s, or 10^-6 x the speed of light. In water the sound speed is about 5 times that, about 1500 m/s, and in steel it's about 20 times that in air, about 6,000 m/s.

Well I completely disagree with your statement that the speed of sound is the limit of propagation of momentum. Apparently you do too as you were the one who brought up the actual upper limit of c.
BTW Do you seriously think I might be unaware that the speed of sound is dependant on the properties of the medium? :-(

Original Austin0
I would say it is beyond the bounds of realism and billions of g's would imply inertial forces far beyond the ability of an addition of a few extra engines to counter.

The muons that were trapped and their lifetimes measured to check the clock postulate experienced accelerations up to 10^18 g. There's nothing physically impossible about such accelerations, and someday we may learn how to make macroscopic objects that are able to withstand them; the laws of physics don't prohibit that. Anyway:

I made no statement regarding the possibility of greater accelerations but simply commented that at those accelerations a Born rigid ship wouldn't be either rigid or survive. That it was unrealistic to think so.

Original AUstin0
Given that the magnitude of force is within the materials ability to transmit it fast enough, applied energy, momentum, propagates through the system, not as motion, but as a reciprocal oscillation.

The speed of transmission of forces through an object (i.e., the sound speed) doesn't depend on the magnitude of the force. Small forces don't get transmitted any faster than large ones.

Do you think I was suggesting that loud noises move faster than soft ones??
As I understand it, the magnitude of force or momentum is a function of both acceleration and mass in one case, and velocity and mass in the other. So propagation is not just dependant on the speed of transmission which is of course constant, but also volume so to speak..
As an extreme case : a planetoid rear ends the ship at .3c. In this case wouldn't you agree that the applied momentum would be beyond the materials ability to transmit it fast enough and the resultant acceleration would reach the front of the ship as motion much faster than the speed of sound?
Or at the other end. If the momentum is being imparted by a number of very small particles
we would assume that with few particles, the momentum would simply be dissapated as incoherent heat. NO motion. Increasing the number would reach a point where the energy would reach the entire system as oscillation which would build up into infintesimal actual system motion. Like ideally slow adiabatic expansion. At some magnitude the energy input [mass*v] is greater than can be simply transmitted and actual motion starts at the point of introduction, resulting in stress and compression. Further increases go up the scale to torquing of the system or actual disruption

Unless the point of application of the force is physically restrained in some way so that it can't move (i.e., if there's no counterbalancing force to work against the applied force), it *will* move upon application of the force. That is, the individual atoms or molecules of the object will move; at the atomic/molecular level, of course, the applied force is transmitted by interactions between the atoms/molecules--more precisely, by the electrical repulsion between the electrons in the atoms/molecules.

Of course the atoms will move but not neccessarily net motion.
You may be right about the repulsion between electrons but my assumption would be that the internal electrostatic and nuclear forces between the nucleus and the electron shells, ionization due to nucleus displacement, repulsion between the nucleus and electrons, etc
would be equally relevant if not actually predominant, given that the majority of inertial masss resides in the nucleus

As the first "layer" of atoms/molecules in the object move in response to the applied force, they will, of course, move closer to the next "layer" in the object, and will thus exert a force on that next layer. In other words, the applied force starts a "wave" of force moving through the object--more precisely, a longitudinal wave (i.e., a sound wave) of alternating compression and expansion, as each layer pushes against the next, which then responds by moving away and pushing against the next in turn.

This is of course what I was referring to by reciprocal oscillation. But two points.
In the transmission of sound there is no net motion of the molecules . Is there real motion of the pressure wave? IMHO, No. It is a back and forth motion.
The internal atomic forces do work at the speed of light, as far as I know, so the reciprocal reactions of these forces would be happening almost instantly compared to the slow propagation rate.So right from the beginning there would be oscillation inherent in the propagation.
The only thing actually moving is a net translation of energy , momentum, until it is sufficient to actually move the system.

If we assume that the applied force at one end of the object continues to be applied, then once the initial wave has traveled the length of the object, it will start to be damped out. This is because as each layer pushes against the next, not all of the energy in the push gets converted into motion of the next layer; some of the energy goes into the internal degrees of freedom in the atoms/molecules--i.e., it gets converted into heat. If the material is very stiff (e.g., very high tensile steel--or even better, carbon nanotubes), the wave will be damped out quickly; if the material is soft, it will take longer for the wave to be damped out.

You may be right here also but my assumption would be that the softer, the more degrees of freedom that a material has, the faster the coherent vibrations would be damped.
I could easily be wrong :-)
BTW after I sent that post I realized i hadn't mentioned heat dissapation and almost went back to edit as I knew you were going to jump on it, but naaH

While the wave is damping, the object will be oscillating, expanding and compressing with gradually decreasing amplitude. Once the wave is damped out, the object will be in a state (assuming the force continues to be applied at one end) in which there is a compressive stress all along its length, and in which its material is somewhat hotter than it was before the force was applied (because some of the applied energy got converted into heat during the damping).

Of course the dissapation of energy would be taking place from the beginning throughout the system .
There is no disagreement that there would be a certain energy loss but would it be significant??
There is also no disagreement that there would be a certain compressive stress,
after all what is an accelerometer? In its most primitive form it is simply a spring scale
and as such is just measuring inertial force through compression. But this hypothesis says that over time, the compression would increase. That the stretching or compression would continue to incrementally increase and that this would be sustained after acceleration was terminated.That is what I am questioning.
If we use Einsteins elevator as an example how significant is the compression factor??
How significant is it here, when raisng a rigid rod from horizontal to vertical??
You never did answer my question about what you thought would happen on a Born ship raising a rod in this manner.

My understanding of the basis of Born acceleration is that it is founded on hyperbolic geometry and an interpretation of accelerated lines of simultaneity. Are there other important principles involved?
I don't remember any analysis along the lines of this thread , did I just miss that part??
Thanks

 

[PeterDonis]

Austin0:

Here you are basing conclusions of both physics and resultant worldlines on an already arrived at assumption of stretching. Well, we know the stretching occurs, just look at the worldlines. Well we know these would be the worldlines because we know that stretching must occur.

No--I'm basing the *conclusion* that there is stretching on the *stipulation* that both the front and the rear ends of the ship experience the same constant proper acceleration. That stipulation is physically possible (though not very practical, as I said), so I'm allowed to make it in a thought experiment.

I was not suggesting that conservation of momentum wouldn't hold. I would assume that the overall acceleration of the system would be the sum of force applied at both ends.

Remember how I specified the scenario: I specified that each of the two rocket engines, the rear one and the front one, imparted *equal, constant proper acceleration* to its segment of the ship. As I said in my last post, to do that in practice would require fantastically precise control of the rocket engines, in order to adjust their actual thrust to compensate for the forces exerted by other parts of the rocket. I only gave an example for the front end of the rocket last time, but the same would apply to the rear as well.

In other words, I was specifying the acceleration of each end of the ship, which in turn specifies the *net* force on each end of the ship--but I did *not* go into any details about how that net force results from the combination of the force exerted by the rocket engines and the internal forces exerted by parts of the ship on each other. I agree this is not a very practical scenario, but as I said, it's not physically impossible.

That if accelerometer readings were equivalent at all parts of the ship, this would mean equal proper acceleration.

Yes, that's true. In the scenario I specified, with rocket engines at the front and rear ends of the ship, accelerometers at the front and rear ends would each give the same readings. I didn't discuss what accelerometers at intermediate points on the ship would read, but if they all also gave the same readings, then proper accelerations would be equal at all those intermediate points as well as at the front and rear ends. And if that were the case, the ship would gradually stretch until it broke apart when the stretching exceeded the tensile strength of its hull.

Well I completely disagree with your statement that the speed of sound is the limit of propagation of momentum. Apparently you do too as you were the one who brought up the actual upper limit of c.

I brought up the upper limit of c as an upper limit to the *possible sound speed* in a material; in other words, it's not possible for the sound speed in a material to be faster than c. I never said that propagation of force in a particular material could be faster than the sound speed in that material. It can't.

As an extreme case : a planetoid rear ends the ship at .3c. In this case wouldn't you agree that the applied momentum would be beyond the materials ability to transmit it fast enough and the resultant acceleration would reach the front of the ship as motion much faster than the speed of sound?

If the sound speed in the material of the ship were slower than .3c (which of course it would be), then the ship would be crushed against the planetoid. If that counts as the acceleration reaching the front of the ship faster than the sound speed in the ship, then yes, it would. Most physicists would probably say that the concept of "speed of sound" wouldn't apply in a case like this, where the structure of the material was destroyed. (I'm assuming here that the planetoid is much larger than the ship.)

Or at the other end. If the momentum is being imparted by a number of very small particles we would assume that with few particles, the momentum would simply be dissapated as incoherent heat. NO motion.

This would violate conservation of momentum. The particles impacting the rear of the ship have *some* forward momentum, and that has to get translated into forward momentum of the ship itself, however small it is. I'm assuming that the ship is out in empty space, with no other forces acting.

You may be right about the repulsion between electrons but my assumption would be that the internal electrostatic and nuclear forces between the nucleus and the electron shells, ionization due to nucleus displacement, repulsion between the nucleus and electrons, etc would be equally relevant if not actually predominant, given that the majority of inertial mass resides in the nucleus

You're correct that I was ignoring the internal details of forces within each atom, and just treating the atoms as single objects. That's a pretty good approximation for the case we were considering, but of course you're right that the forces have to be transmitted internally between the electron shells and the nucleus or the atom as a whole won't move.

In the transmission of sound there is no net motion of the molecules . Is there real motion of the pressure wave? IMHO, No. It is a back and forth motion.

That depends on the driving force. For ordinary sound in, for example, air, the driving force is oscillatory--something, like a speaker membrane or a person's vocal cords, is vibrating back and forth, with no net motion, so there's no net motion in the sound wave either. In the case of the ship, the driving force is not oscillating; it's constant in a particular direction. In that case, the driving force does result in net motion immediately at the end of the ship where the force is applied, and the transmission of the sound wave through the ship transmits the net motion.

You may be right here also but my assumption would be that the softer, the more degrees of freedom that a material has, the faster the coherent vibrations would be damped.

I would agree that more degrees of freedom in the material ought to mean more places for the energy of vibration to go, hence faster damping, but I'm not sure softer materials have more degrees of freedom. I'd have to refresh my memory about the details of how bulk properties like the stiffness of the material are derived from microscopic properties of the atoms or molecules.

But this hypothesis says that over time, the compression would increase. That the stretching or compression would continue to incrementally increase and that this would be sustained after acceleration was terminated. That is what I am questioning.

In the case I was discussing here--a rocket engine only at the rear end of the ship--I didn't mean to imply that the compression would continue to increase. It wouldn't; it would reach a steady state and then not increase any further. In that steady state, there would be a compressive *stress* along the ship's length, but it would be opposed by the inter-atomic or inter-molecular forces, which would have increased because the atoms or molecules were now closer together. So there would be a new equilibrium and no further compression.

Also, I didn't mean to imply that the compression would remain if the acceleration was terminated; I didn't discuss that case (I was only discussing what would happen while the force continued to be applied). If the acceleration was terminated, of course the material would expand again, by the reverse process to that which compressed it: the information that the acceleration had ceased would take time to propagate from the rear to the front of the ship, so segments of the ship more towards the front would continue to accelerate longer, so a wave of initial expansion would travel throughout the ship. It would oscillate between expansion and compression again, but with no force being applied at the rear, the oscillations would eventually damp out and the ship would be at the same length (in its rest frame) as it was originally.

(By the way, all of the above assumes that the stresses induced in the ship don't exceed the elastic limit of its material. If they do, the ship's structure might be permanently deformed so it wouldn't return to its original length when the applied force was removed.)

In the case where there was a rocket engine at the front *and* rear of the ship, the stretching would continue to increase for as long as the rocket engines remained on--or until the ship broke apart. If the engines were turned off while the ship was still intact, the ship would still be stretched, but if the elastic limit of the material was not exceeded, it would shrink back to its original length (in its rest frame).

If we use Einsteins elevator as an example how significant is the compression factor??

I haven't calculated it specifically; in general it would be proportional to the acceleration and inversely proportional to the strength of the material. If you want a more specific formula I can try to dig one up.

How significant is it here, when raisng a rigid rod from horizontal to vertical??
You never did answer my question about what you thought would happen on a Born ship raising a rod in this manner.

I haven't considered this situation yet because adding the transverse dimension adds complications that make things more difficult to analyze, and I'm not sure it would add anything to the basic picture we already have. In general, I don't think the compression *ratio* (the ratio of original length to compressed length) would change, but of course once the rod is raised from horizontal to vertical the compression will be along the longer dimension of the rod. So in general, when the rod is horizontal it will be compressed along its shorter dimension and the longer dimension will be unchanged; once it is raised to vertical, and everything has time to equilibrate again, it will be compressed along its longer dimension and the shorter dimension will be what it originally was (i.e., before any force was applied at all).

Also, all of the above applies to the case where the force is applied at the *rear* only. This case is *not* the same as Born rigid acceleration. Born rigid acceleration would make the case of the rod rotating from horizontal to vertical *much* more complex to analyze exactly, because every atom in the rod would need to travel along a very precise path. I *think* the end result would be that, both horizontally and vertically, the rod's dimensions would remain the same as its unaccelerated dimensions; but that's only based on the general property of Born rigid acceleration that it's supposed to do that. I haven't actually calculated the precise accelerations that would have to be imparted to each little piece of the rod to make that come about.

My understanding of the basis of Born acceleration is that it is founded on hyperbolic geometry and an interpretation of accelerated lines of simultaneity. Are there other important principles involved?
I don't remember any analysis along the lines of this thread , did I just miss that part??

I don't think there's been a detailed discussion of Born acceleration or Born rigidity yet in this thread. There's a discussion of it http://www.mathpages.com/home/kmath422/kmath422.htm" that covers the main points. The original reason why Born proposed the idea was to find some kind of relativistic version of the concept of a "rigid body". The property he wanted to try and preserve was that each atom in the body would maintain the same spatial distance from its neighboring atoms (so that the acceleration would be "stress-free"); given that requirement, relativistic kinematics was enough to determine the worldlines of each part of the body, as shown in the page I linked to above. As I've said, it would be practically impossible to realize Born rigid acceleration for a real object, since it would require very precise application of force to each little piece of the object.

 

[Austin0]

Austin0:

No--I'm basing the *conclusion* that there is stretching on the *stipulation* that both the front and the rear ends of the ship experience the same constant proper acceleration. That stipulation is physically possible (though not very practical, as I said), so I'm allowed to make it in a thought experiment.

I have no quibbles about how you want set up your parameters or small details of practicallity. But given your condition of same constant proper acceleration at each end, I missed how this leads to inevitably increasing stretching. :-)




Originally Posted by Austin0
Or at the other end. If the momentum is being imparted by a number of very small particles we would assume that with few particles, the momentum would simply be dissapated as incoherent heat. NO motion.

This would violate conservation of momentum. The particles impacting the rear of the ship have *some* forward momentum, and that has to get translated into forward momentum of the ship itself, however small it is. I'm assuming that the ship is out in empty space, with no other forces acting.

I also am assuming in empty space. But I do not see how there would be any violation of conservation of momentum.
My assumption is that the momentum from a few particles would be converted from linear motion into molecular vibration [heat] and ultimately radiated into space as infrared photons. That in losing coherence it would also lose any unified momentum vector and would be nullified in that regard. Individual motions pointing in all directions.
ANything wrong with this picture?
____________________________________________________________________________-
Originally Posted by Austin0
In the transmission of sound there is no net motion of the molecules . Is there real motion of the pressure wave? IMHO, No. It is a back and forth motion.

That depends on the driving force. For ordinary sound in, for example, air, the driving force is oscillatory--something, like a speaker membrane or a person's vocal cords, is vibrating back and forth, with no net motion, so there's no net motion in the sound wave either. In the case of the ship, the driving force is not oscillating; it's constant in a particular direction. In that case, the driving force does result in net motion immediately at the end of the ship where the force is applied, and the transmission of the sound wave through the ship transmits the net motion.

What you are saying about coherent sound production is obviously relevant to the pattern of transmitted sound [momentum] in air.
But it seems to me that the reciprocal oscillatory nature of momentum propogation is independant of the driving force and stems directly from the 3rd law of motion.
That this does not just apply to the propulsion itself, but at every intermediate transference of momentum from particle to partcle. Obviously in a molecular lattice structure the structure itself will impose certain periodicities and larger coordinations ,harmonics etc.,
but will not detract from this fundamental action-reaction inherent reciprocity.
A thought picture:
Take the classic series of suspended identical metal balls. One end ball is elevated and released gaining momentum until impact with the first ball in the line. The momentum passes from the moving ball into the next ball in line and then through the series until encountering the last ball. It enters that ball and the momentum is then translated into motion.
It passed through the intermediate balls without resultant motion. It propagated throughout the last ball without coordinate motion until the whole amount of energy was absorbed and then the whole ball moved as a whole system. It seems like we can logically infer that there was no motion at the point of introduction because as soon as that point of contact actually moved there could be no further transfer of momentum. Yet we know that the total amount of momentum in the last ball is equal to the amount in the first ball minus the small amount of loss through dissapation.
SO it would seem there could be no net motion until the transfer was complete.
Is there something here I am not seeing?

A space launch: the engines introduce thrust for a considerable period before actual motion occurs. My assumption is that a lot of momentum has to propagate through the system to overcome the opposite acceleration of gravity and then additional momentum has to go into and throughout the system to actually overcome inertia and result in system motion.
Am I seeing this wrong??.

I would agree that more degrees of freedom in the material ought to mean more places for the energy of vibration to go, hence faster damping, but I'm not sure softer materials have more degrees of freedom. I'd have to refresh my memory about the details of how bulk properties like the stiffness of the material are derived from microscopic properties of the atoms or molecules.

You're probably right about there being some weird polymers that are both soft and resonant but i think in general bronze bells ring longer than paper ones ?

In the case I was discussing here--a rocket engine only at the rear end of the ship--I didn't mean to imply that the compression would continue to increase It wouldn't; it would reach a steady state and then not increase any further..

Can we consider this initial compression/contraction and consider it is not really what we are talking about with this question and so disregard it from this point?

Also, I didn't mean to imply that the compression would remain if the acceleration was terminated; I didn't discuss that case (I was only discussing what would happen while the force continued to be applied). If the acceleration was terminated, of course the material would expand again, by the reverse process to that which compressed it:

It seems that this question involves the concept of permanent change.
Fundamental to SR is the concept that contraction does not occur within a system in relative motion but is a condition observed from other frames.
Practically speaking that means no perceptible change in internal geometry or metric.
It seems to me that the Born hypothesis is based on the idea that to preserve this condition through the transition between relative velocities [ acceleration], that specific structured force must be applied.
From what I gotten from you, it seems like the difference between Born acceleration and simple acceleration from the rear, is that the rear propulsion would produce some slight initial
compression that would disappear after reaching a new velocity. Wouldn't this also conform to the expectations of SR , with the small difference that, during the transition there would be a small temporary difference in internal geometry. SR already considers accelerated systems as being radically different from inertial ones and Born acceleration in any case is not realistically achievable , so the question would seem to be what is the internal basis of the hypothesis that would make it inevitable as a reality?

I don't think there's been a detailed discussion of Born acceleration or Born rigidity yet in this thread. There's a discussion of it http://www.mathpages.com/home/kmath422/kmath422.htm" that covers the main points. The original reason why Born proposed the idea was to find some kind of relativistic version of the concept of a "rigid body". The property he wanted to try and preserve was that each atom in the body would maintain the same spatial distance from its neighboring atoms (so that the acceleration would be "stress-free"); given that requirement, relativistic kinematics was enough to determine the worldlines of each part of the body, as shown in the page I linked to above. As I've said, it would be practically impossible to realize Born rigid acceleration for a real object, since it would require very precise application of force to each little piece of the object

No there has not been a discussion of the hypothesis. I meant did any of the reasoning regarding worldlines etc that have come up in this thread , enter into the original formulation of the hypothesis. I have seen the link before and have questions regarding the interpretation of accelerated lines of simultaneity that is portrayed there.
More later Thanks

 

[PeterDonis]

Austin0:

But given your condition of same constant proper acceleration at each end, I missed how this leads to inevitably increasing stretching.

Because that condition means that the front and rear ends of the ship, R and F, follow exactly the same worldlines as the separate ships, R and F, in our previous scenario, where we agreed that the proper distance between the ships, in either ship's MCIF, would increase--ship F would "pull away" from ship R, and ship R would "fall behind" ship F (these are just two different ways of saying the same thing). If R and F are the front and rear ends of a single ship, but follow the same worldlines as before, then the proper distance between them will still increase--which means that the ship as a whole must stretch.

The only difference when R and F are two ends of a single ship is that the intermediate parts of the ship exert a force on the front and rear ends, so the condition of "same constant proper acceleration" does *not* imply "same constant rocket thrust", as it did in the case of the separate ships. With a single ship, as I've said before, the rocket thrusts at the ends will need to vary to compensate for the internal forces exerted by the parts of the ship on each other, so that the *net* force at the front and rear ends remains equal and constant, since it's the *net* force that determines the proper acceleration--i.e., what would be measured by an accelerometer.

My assumption is that the momentum from a few particles would be converted from linear motion into molecular vibration [heat] and ultimately radiated into space as infrared photons. That in losing coherence it would also lose any unified momentum vector and would be nullified in that regard. Individual motions pointing in all directions.
ANything wrong with this picture?

Yes. A "unified momentum vector", by which I assume you mean a net momentum vector pointing in a particular direction, *cannot* be "nullified"--it *cannot* be converted into individual motions pointing in all directions, with no net bias in any particular direction. That violates conservation of momentum.

Take the classic series of suspended identical metal balls. One end ball is elevated and released gaining momentum until impact with the first ball in the line. The momentum passes from the moving ball into the next ball in line and then through the series until encountering the last ball. It enters that ball and the momentum is then translated into motion.

Here the driving force is a single initial impulse, not a continuous applied force. That's why the behavior is different. For your thought picture to match the situation of the continuously accelerating rocket, you would have to have a continuous push on the ball at one end of the series. That would impart a net motion to the whole set of balls--at least until you've pushed them so far that they break away from their suspension points.

A space launch: the engines introduce thrust for a considerable period before actual motion occurs.

That's because the rocket is held down by clamps to the launch pad. As soon as the clamps are released, the rocket starts moving. The clamps are there because it's not safe to let the rocket move when the thrust to weight ratio is too low; there's too much chance that it would topple. So the rocket is held down until the main engines are all at full thrust. (On the space shuttle, the SRBs ramp up to full thrust fast enough that this delay isn't necessary.)

Check out http://science.ksc.nasa.gov/shuttle/countdown/count.html" . Note the "SRB HOLDDOWN RELEASE COMMAND" at exactly T minus 00 minutes 00 seconds, and liftoff at that same moment.

Can we consider this initial compression/contraction and consider it is not really what we are talking about with this question and so disregard it from this point?

Yes, that's fine; the steady-state condition is really what we're interested in.

Fundamental to SR is the concept that contraction does not occur within a system in relative motion but is a condition observed from other frames.

Only if you're considering "contraction" to be the *kinematic* effect--i.e., you already know all the worldlines and events, and you're just figuring out how they would look from different frames. But if you start mixing in the dynamic effects--the internal forces that parts of an object exert on other parts, which can *change* the worldlines that the various parts travel on--then you're mixing together two different effects, and you have to be very precise in distinguishing which observations you're making are due to the fact that the parts of an object are traveling along different worldlines, and which are due to the way those worldlines look in different frames.

From what I gotten from you, it seems like the difference between Born acceleration and simple acceleration from the rear, is that the rear propulsion would produce some slight initial
compression that would disappear after reaching a new velocity. Wouldn't this also conform to the expectations of SR , with the small difference that, during the transition there would be a small temporary difference in internal geometry.

Everything I've said conforms to the expectations of SR; indeed, SR is the theory I'm using to determine all the effects I'm talking about.

With regard to the quote above, are you now talking about scenarios where objects are accelerated only for a limited period, and the acceleration is then removed? In any case like that, generally speaking, assuming that the elastic limit of the material is not exceeded (which can be a big "if"--see below), an object that is accelerated to a given velocity (as seen from its original rest frame) and then left to move inertially thereafter, will have the same dimensions in its new inertial rest frame as it had in its original rest frame, regardless of how it was accelerated--provided only that we allow enough time for dynamic effects such as waves of compression and expansion in the material to propagate and damp themselves out. The details of the acceleration will be important only if you want to know the *precise* worldlines that each part of the object will follow in getting from its original state of inertial motion to its final one.

Note that qualification I gave, that the elastic limit of the material is not exceeded. That means that the general prediction I just made--that objects which are accelerated will return to their original dimensions (in their new rest frames) when the acceleration is removed--is *not* just a prediction of SR alone. It also requires knowledge of the physics of elastic materials. Non-elastic materials will have different behavior, even though the same laws of SR can be used to help predict the behavior in both cases. That's why I've been so insistent on maintaining the distinction between "kinematic" effects (which are *purely* due to the laws of SR) and "dynamic effects" (which depend on other physical theories in addition to SR).

 

[Austin0]

Originally Posted by Austin0
. If the momentum is being imparted by a number of very small particles we would assume that with few particles, the momentum would simply be dissapated as incoherent heat. NO motion.

This would violate conservation of momentum. The particles impacting the rear of the ship have *some* forward momentum, and that has to get translated into forward momentum of the ship itself, however small it is. I'm assuming that the ship is out in empty space, with no other forces acting.

I also am assuming in empty space. But I do not see how there would be any violation of conservation of momentum.
My assumption is that the momentum from a few particles would be converted from linear motion into molecular vibration [heat] and ultimately radiated into space as infrared photons. That in losing coherence it would also lose any unified momentum vector and would be nullified in that regard. Individual motions pointing in all directions.
ANything wrong with this picture?


Y

es. A "unified momentum vector", by which I assume you mean a net momentum vector pointing in a particular direction, *cannot* be "nullified"--it *cannot* be converted into individual motions pointing in all directions, with no net bias in any particular direction. That violates conservation of momentum.

Hi PeterDonis It would appear that either my understanding of conservation of momentum is incomplete or there is a miscommunication here and we are talking about different things.
Would you agree that the statements I made above are a description of certain aspects of both entropy and damping , which in itself is a specialized case of entropy??

That when we were talking about degrees of freedom we were talking about entropic dispersal possibilities, microstates?

Considering a single particle:
It transfer its momentum to the system as a discrete definable vector. But inside the lattice, that definite locality and direction is immediately distributed. Initially having a generalized forward direction, as it propagates it would spread over a wider front and be redirected through degrees of freedon in other directions. Besides this, parts of the energy would be continually being localized as molecular vibration, heat. Translational motion halted.
Entropy.

Would you disagree?

a net momentum vector pointing in a particular direction, *cannot* be "nullified"--it *cannot* be converted into individual motions pointing in all directions, with no net bias in any particular direction.

Isn't this simply a question of boundary conditions?
Using your extreme example of a rod light seconds long. If one end is subjected to an impact [momentum] that is small relative to the overall mass, would you expect that momentum to ever reach the other end at all??
Or would it be expected that it would be dissipated as internal vibration and heat long before reaching the other end??
If this magnitude of momentum was sustained would it be expected to change this basic entropic resistance?
Would you disagree that it seems likely that there would be a threshold of minimum input of momentum /time required to overcome this inherent entropic resistance??
WOuldn't this apply to any system it just being a quantitative question??

____________________________________________________________________________



Take the classic series of suspended identical metal balls. One end ball is elevated and released gaining momentum until impact with the first ball in the line. The momentum passes from the moving ball into the next ball in line and then through the series until encountering the last ball. It enters that ball and the momentum is then translated into motion.
It passed through the intermediate balls without resultant motion. It propagated throughout the last ball without coordinate motion until the whole amount of energy was absorbed and then the whole ball moved as a whole system. It seems like we can logically infer that there was no motion at the point of introduction because as soon as that point of contact actually moved there could be no further transfer of momentum. Yet we know that the total amount of momentum in the last ball is equal to the amount in the first ball minus the small amount of loss through dissapation.

Here the driving force is a single initial impulse, not a continuous applied force. That's why the behavior is different. For your thought picture to match the situation of the continuously accelerating rocket, you would have to have a continuous push on the ball at one end of the series. That would impart a net motion to the whole set of balls--at least until you've pushed them so far that they break away from their suspension points.

Would you disagree with the following view?
The driving force in this case is a continuous applied force , albeit of very short duration. The momentum passes from the first ball to the second not as a single instantaneous pulse but as a sustained transference through the point of contact.

Couldn't we in principle extend this time interval greatly up to some limit, simply by increasing the size of the spheres??

AS far as the principles involved are concerned the duration is not really important is it?

Couldn't we, from a frame at rest wrt the intermediate spheres, plot a clear picture of the complete sequence of events??
Couldn't we also graph the momentum itself, independant from the masses involved?
Give it both a spatial location as well as a defined velocity for any given instant? [with a certain statistical uncertainty regarding location].
So at T= (diameter/speed of sound) the momentum would be located, essentially , within sphere #2 etc, etc.
Culminating in its location within the final sphere before any coordinate motion of that sphere??

Now I may be seriously mistaken but I see several apparently fundamental priciples of momentum and inertia demonstrated here.
That, disregarding entropy, momentum simply passes through mass as long as there is an open pathway.
That inertia resists actual system motion until the momentum has propagated completely through that system.
That if the momentum of the final sphere is equal to the momentum of the initial sphere then the conservation of momentum would seem to dictate that none of it remains in the intermediate spheres as motion other than residual vibration.

It is understood that these are special conditions : identical mass and composition, perfectly symmetric transfer paths etc. That changing these parameters , for instance increasing the initial momentum significantly etc. would result in interactions that would not be so neat , but do you see any reason why these basic priciples wouldn't fundamentally apply as generalizations?

Or some basic understanding that I am missing?

If I have errors of understanding here I definitely want to get them right.

That's because the rocket is held down by clamps to the launch pad. As soon as the clamps are released, the rocket starts moving. The clamps are there because it's not safe to let the rocket move when the thrust to weight ratio is too low; there's too much chance that it would topple. So the rocket is held down until the main engines are all at full thrust. (On the space shuttle, the SRBs ramp up to full thrust fast enough that this delay isn't necessary.)

Check out http://science.ksc.nasa.gov/shuttle/countdown/count.html" . Note the "SRB HOLDDOWN RELEASE COMMAND" at exactly T minus 00 minutes 00 seconds, and liftoff at that same moment.

Yes I was lacking in knowledge of the mechanics and thought that the gantry supports were basically passive restraints against torquing SO BAD analogy.

Thanks for your patience

 

[PeterDonis]

Considering a single particle:
It transfer its momentum to the system as a discrete definable vector. But inside the lattice, that definite locality and direction is immediately distributed. Initially having a generalized forward direction, as it propagates it would spread over a wider front and be redirected through degrees of freedon in other directions. Besides this, parts of the energy would be continually being localized as molecular vibration, heat. Translational motion halted.
Entropy.

Would you disagree?

Yes. *That would violate conservation of momentum.* You *cannot* take a "discrete definable vector" of momentum and make it disappear. You can't dissipate it as heat. You can't redistribute it into other internal degrees of freedom. The total momentum of the system--"system" meaning the single particle and the larger object that it hits--*must* be the same after the collision as before. That's what conservation of momentum means. If the total momentum was a "discrete definable vector" before the collision, it must still be a "discrete definable vector" after the collision.

This is such a basic point that I want to make sure it's clear in this simple example before discussing any of the other examples in your post. Let me write down explicitly what the momentum looks like before and after the collision, in the frame in which the large object is initially at rest:

Before collision: Single particle with mass m and moving to the right (positive x-direction) with velocity v. Large object with mass M at rest.

Total momentum before collision: $m v$. Note that this is a vector in the positive x-direction (the same direction as v). Also note that this is a non-relativistic expression; if you want to look at the relativistic case, we can, but it doesn't change anything essential to what we're discussing right now.

After collision: Object with mass m + M moving to the right (positive x-direction) with velocity w.

Total momentum after collision: $\left( m + M \right) w$. Again, this is a vector in the positive x-direction (the same direction as w, which is the same direction as v).

Equating the momentum before and after the collision, as required by conservation of momentum, we have:

$$m v = \left( m + M \right) w$$

or, rearranging terms,

$$w = \frac{m}{m + M} v$$

So w will be much smaller than v, but it will *not* be zero, and since the object is now alone in empty space with no other objects to collide with, its velocity will remain w; there's nowhere for it to "dissipate" to, because interactions internal to an object can't change its total momentum.

Before going any further, please let me know your thoughts on the above.

 

[Austin0]

Yes. *That would violate conservation of momentum.* You *cannot* take a "discrete definable vector" of momentum and make it disappear. You can't dissipate it as heat. You can't redistribute it into other internal degrees of freedom. The total momentum of the system--"system" meaning the single particle and the larger object that it hits--*must* be the same after the collision as before. That's what conservation of momentum means. If the total momentum was a "discrete definable vector" before the collision, it must still be a "discrete definable vector" after the collision.

This is such a basic point that I want to make sure it's clear in this simple example before discussing any of the other examples in your post. Let me write down explicitly what the momentum looks like before and after the collision, in the frame in which the large object is initially at rest:

Before collision: Single particle with mass m and moving to the right (positive x-direction) with velocity v. Large object with mass M at rest.

Total momentum before collision: $m v$. Note that this is a vector in the positive x-direction (the same direction as v). Also note that this is a non-relativistic expression; if you want to look at the relativistic case, we can, but it doesn't change anything essential to what we're discussing right now.

After collision: Object with mass m + M moving to the right (positive x-direction) with velocity w.

Total momentum after collision: $\left( m + M \right) w$. Again, this is a vector in the positive x-direction (the same direction as w, which is the same direction as v).

Equating the momentum before and after the collision, as required by conservation of momentum, we have:

$$m v = \left( m + M \right) w$$

or, rearranging terms,

$$w = \frac{m}{m + M} v$$

So w will be much smaller than v, but it will *not* be zero, and since the object is now alone in empty space with no other objects to collide with, its velocity will remain w; there's nowhere for it to "dissipate" to, because interactions internal to an object can't change its total momentum.

Before going any further, please let me know your thoughts on the above.

There is no disagreement on the abslolute application of the fundamental conservation of mass and energy.
There is no disagreement on the truth of inertial motion or the conservation of momentum as applied to inertial motion.
there is no disagreement on the validity of the formula for force and momentum as applied to macrosystems in general.
There is no disagreement that principles and mathematical descriptions of physics do not exist if they are not validated to a high dgree of accuracy.

But you have presented this question in the context of absolutes.
Of abstract principles divorced from the real world.
So given a complex system regarded as an abstract point particle. GIven a system ideally isolated from the world with no photon transmission possible.
Then you are abslutely right.
Whether any principle, no matter how valid, can be quantitatively applied to real world systems in this absolute way is another question.

But this is not the real point of disagreement .
Viewed in this absolute way , the mathematical expression, having been derived from consideration of interactions between point particles or ideal elastic bodies defines an instantaneous change in momentum in reaction to force.
You seem to apply this to a complex system as a local reality. I.e. a ship with multiple points of applied momentum can be viewed as a set of separate systems. The momentum applied locally results in different velocities throughout the overall system because motion starts immediately at the source . Or at least this is what it seems to mean to me you have been saying .
My view is that the global condition is predominant and energy, momentum, enters the system and is propagated throughout the system which accelerates as a whole.
Do you have any other information that would help clarify this question?

 

[PeterDonis]

You seem to apply this to a complex system as a local reality. I.e. a ship with multiple points of applied momentum can be viewed as a set of separate systems. The momentum applied locally results in different velocities throughout the overall system because motion starts immediately at the source . Or at least this is what it seems to mean to me you have been saying .
My view is that the global condition is predominant and energy, momentum, enters the system and is propagated throughout the system which accelerates as a whole.
Do you have any other information that would help clarify this question?

Both of the views you have stated above are true. If you view the system as a single system, sitting in empty space, with no interactions with other systems other than the specific ones you've defined in the statement of the problem (for example, 1 rocket engine pushing on the rear of a spaceship, or a small particle hitting a larger object), then as a single system (e.g., the spaceship or the larger object), conservation of momentum applies just as I have stated it.

If, on the other hand, you view the system as a composite of subsystems (e.g., the spaceship as a composite of little segments, each with its own motion), then conservation of momentum applies to each individual subsystem, *and* to the system that's the composite of all the subsystems. The law of conservation of momentum for each individual system applies as I have stated it, but you have to obtain the *net* force on each individual system (e.g., each individual piece of the spaceship) to apply the law of conservation of momentum to it.

The law of conservation of momentum for the composite system (e.g., the whole spaceship) is then the sum of all the laws of conservation of momentum for the individual systems. What you will find when you do the sum is that all of the internal forces between parts of the composite system cancel out, and you're left with just the net external force on the whole system (e.g., the rocket engine pushing on the spaceship, or the small particle hitting the large object), which must balance with the net change in its total momentum, according to the law as I have stated it.

So *both* versions of conservation of momentum--the version that applies to the whole system, and the version that applies to each individual subsystem--are valid, and consistent with each other.

 

[Austin0]

=PeterDonis;2380008] Both of the views you have stated above are true.
So *both* versions of conservation of momentum--the version that applies to the whole system, and the version that applies to each individual subsystem--are valid, and consistent with each other

As far as I can see they are somewhat mutually exclusive. Certainly the expected end results appear to be.

Given: A ship with thrust applied to the front and back, with greater thrust at one end.

#1 The momentum from the front propagates through the ship to the back, while the momentum from the back propagates through to the front and then results in system motion . So the differential is equalized and the net acceleration is also equal at the front and the back.

#2 The momentum propagates locally from the source as motion and results in greater velocity at the end with greater thrust.

__________________________________________________________________________

If we assume that the applied force at one end of the object continues to be applied, then once the initial wave has traveled the length of the object, it will start to be damped out. This is because as each layer pushes against the next, not all of the energy in the push gets converted into motion of the next layer; some of the energy goes into the internal degrees of freedom in the atoms/molecules--i.e., it gets converted into heat. . While the wave is damping, the object will be oscillating, expanding and compressing with gradually decreasing amplitude. Once the wave is damped out, the object will be in a state (assuming the force continues to be applied at one end) in which there is a compressive stress all along its length, and in which its material is somewhat hotter than it was before the force was applied (because some of the applied energy got converted into heat during the damping).

Where did this energy come from?

Would it completely stop after initial adjustment to acceleration??

Would the system be getting continually hotter??

__________________________________________________________________________--
THE SPHERES

Here the driving force is a single initial impulse, not a continuous applied force. That's why the behavior is different. For your thought picture to match the situation of the continuously accelerating rocket, you would have to have a continuous push on the ball at one end of the series. That would impart a net motion to the whole set of balls--at least until you've pushed them so far that they break away from their suspension points..

Would you disagree with the following view?
The driving force in this case is a continuous applied force , albeit of very short duration. The momentum passes from the first ball to the second not as a single instantaneous pulse but as a sustained transference through the point of contactCouldn't we in principle extend this time interval greatly up to some limit, simply by increasing the size of the spheres??

AS far as the principles involved are concerned the duration is not really important, is it?

Couldn't we, from a frame at rest wrt the intermediate spheres, plot a clear picture of the complete sequence of events??
Couldn't we also graph the momentum itself, independant from the masses involved?
Give it both a spatial location as well as a defined velocity for any given instant? [with a certain statistical uncertainty regarding location].

So at T= (diameter/speed of sound) the momentum would be located, essentially , within sphere #2 etc, etc.
Culminating in its location within the final sphere before any coordinate motion of that sphere??

Now I may be seriously mistaken but I see several apparently fundamental priciples of momentum and inertia demonstrated here:

That, disregarding entropy, momentum simply passes through mass as long as there is an open pathway.

That inertia resists actual system motion until the momentum has propagated completely through that system.

That if the momentum of the final sphere is equal to the momentum of the initial sphere then the conservation of momentum would seem to dictate that none of it remains in the intermediate spheres as motion other than residual vibration.

It is understood that these are special conditions : identical mass and composition, perfectly symmetric transfer paths etc. That changing these parameters , for instance increasing the initial momentum significantly etc. would result in interactions that would not be so neat , but do you see any reason why these basic priciples wouldn't fundamentally apply as generalizations?

Or some basic understanding that I am missing?

 

[Austin0]

Yes. *That would violate conservation of momentum.* You *cannot* take a "discrete definable vector" of momentum and make it disappear. You can't dissipate it as heat. You can't redistribute it into other internal degrees of freedom. The total momentum of the system--"system" meaning the single particle and the larger object that it hits--*must* be the same after the collision as before. That's what conservation of momentum means. If the total momentum was a "discrete definable vector" before the collision, it must still be a "discrete definable vector" after the collision.

This is such a basic point that I want to make sure it's clear in this simple example before discussing any of the other examples in your post.

So w will be much smaller than v, but it will *not* be zero, and since the object is now alone in empty space with no other objects to collide with, its velocity will remain w; there's nowhere for it to "dissipate" to, because interactions internal to an object can't change its total momentum.

Before going any further, please let me know your thoughts on the above.

AM I incorrect in my understanding, that kinetic enrgy in its many forms including heat,
results in an increase in inertial mass??
If this is the case, how does this fit into the momentum conservation equation?
It would appear that any part of the directed applied momentum that was internally transformed into kinetic energy, would then contribute to the overall system conservation of momentum as an increase in inertial mass rather than, neccessarily, an increase in system velocity.
Is there some principle I am unaware of that would negate this concept??
Are we going to discuss any of the other examples in my post??