Can an Infinitely Stiff Rod Defy the Speed of Light in Relativity?

[PeterDonis]

I don't agree with your statement. Assuming a massless, infinitely stiff reference framework has no effect on the calculation of the experiment. See attachment.

I guess you didn't read carefully enough about the Bell Spaceship Paradox. What you have tried to set up appears to be the same situation as that paradox, with your "accelerating observer" and "observer xo" as the two spaceships. However, you have included a number of things in your formulation of the problem which are either not consistent or not stated properly, so it's hard to be sure what you intended.

First, you state:

The acceleration of observer xo will be such that he appears stationary at the same instant that the original accelerating observer appears stationary in any inertial reference frame, and the coordinate of observer xo will be exactly xo when compared to the original observer.

As this is stated, it is logically impossible, because of the phrase "in any inertial reference frame". I *think* that what you meant to say is that the two accelerating observers remain a distance xo apart as seen in *the original inertial frame*, i.e., the one in which they are both at rest before they start accelerating. In other words, xo's worldline is identical to the original accelerating observer's worldline, except that it's displaced in space by a distance xo, as seen from the original inertial frame. That much is fine: but if that is the case as seen in the original inertial frame, it will *not* be the case as seen in *any* other inertial frame, *or* as seen by the two accelerated observers themselves. In particular, if the two observers remain a distance xo apart as seen in the original inertial frame, they will *separate* as seen by each other; meaning that if a rope were connecting the two observers, such that the rope had a normal unstretched length of xo, the rope would stretch as they accelerated. That's the whole point of the Bell Spaceship Paradox, and it invalidates any attempt to set up a "rigid accelerating reference frame" this way.

Next, you calculate the "value of acceleration for observer xo" to be *different* than for your accelerating observer. But if both observers follow identical worldlines, just displaced in space, as seen in the original inertial frame, then the acceleration they feel must be equal. If the accelerations they feel are different, then they can't stay the same distance apart as seen in the original inertial frame.

So as it stands, what you posted, if I'm understanding your intention correctly, doesn't invalidate anything I said; it just illustrates some pitfalls in trying to analyze accelerating frames. If you meant to set up the scenario differently than how I interpreted it above, then you may want to clarify what you intended and revise your formulation accordingly.

 

[PeterDonis]

If you meant to set up the scenario differently than how I interpreted it above, then you may want to clarify what you intended and revise your formulation accordingly.

To expand on this, I'll go ahead and comment on the other obvious possible interpretation of what you posted. This is that you intended to have "observer xo" accelerate such that he stops accelerating at the instant he reaches object A, and at that instant, the following are true:

(1) Observer xo has the same velocity beta as the original accelerated observer relative to the original inertial frame (the one in which both were at rest before they started accelerating); this means that observer xo, at that instant, is at rest (once again) relative to the original accelerated observer.

(2) Observer xo, in the new inertial frame in which he and the original accelerated observer are now both at rest (once he reaches object A--this frame is moving at beta relative to the original inertial frame), is still at distance xo from the original accelerated observer.

The above is certainly physically possible, but let me point out some things about this setup. For brevity, I'll refer to the original accelerated observer as O, the original inertial frame (in which O and xo are at rest before they start accelerating) as frame A (because object A is at rest in it), and the new inertial frame, in which both O and xo are at rest after xo stops accelerating, as frame B.

(a) There is no way for O and xo to communicate with each other to coordinate starting and stopping their accelerations as specified above; they both have to prearrange the accelerations and independently start and stop them at the right events. This is because communication signals are limited to the speed of light, and the event pairs of O and xo starting their accelerations, and O and xo stopping their accelerations, are each spacelike separated (they have to be, because each pair of events is simultaneous in some inertial frame--A in the first case, B in the second), so O and xo cannot communicate to coordinate them. So physically, though you can specify O and xo's worldlines as you have, you *cannot* attribute their following these worldlines to the presence of an "infinitely rigid structure" between them that keeps them moving in concert. Such a structure is physically impossible. You have to explicitly specify that O and xo have agreed in advance to carry out a pre-planned acceleration profile that happens to work out as you describe.

This may seem like semantics, but it's important because it makes clear that, once you are dealing with accelerated frames, you can't use the same "rigid structure of rods and clocks" method to visualize what's going on as is often taught for inertial frames in SR. You have to adopt a more abstract viewpoint.

(b) O and xo do *not* remain the same distance apart while they are accelerating, in *any* inertial frame. In frame A, they start out a distance xo apart, and end up a distance xo \ gamma apart, where gamma is 1 \ sqrt(1 - beta^2), the relativistic length contraction/time dilation factor associated with velocity beta. In frame B, they start out a distance gamma xo apart, and end up a distance xo apart. (Note that in defining these distances, one also has to be careful about specifying exactly which events on O's and xo's worldlines are being used to measure the distances. Drawing a spacetime diagram would make all this a lot clearer.) Similar remarks would apply to any other inertial frame. Only when both O and xo are moving inertially will their separation, as seen in any inertial frame, remain constant.

(c) O and xo *do* remain a distance xo apart in an "accelerated frame" defined such that O's worldline is at the origin, for the period during which O is accelerating. In this "accelerated frame", O and xo both start accelerating and stop accelerating at the same time, and O and xo remain a distance xo apart (if "distance" is defined appropriately--there are actually caveats to this too, because there isn't a unique notion of "distance" in an accelerated frame; I won't go into that now). However, in between those two events, the time elapsed on xo's clock is *greater* than the time elapsed on O's clock! So this "accelerated frame" has a weird kind of time associated with it, whose rate of flow varies with distance from the origin. That also means that, once you pick a time coordinate for this "accelerated frame", only one observer's actual proper time will match that time coordinate. For example, if we pick O's proper time to define the time coordinate of the accelerated frame, then xo's proper time will flow faster than coordinate time does.

(d) The "accelerated frame" I just spoke of can be extended indefinitely to the right (i.e., in the direction from O to xo and beyond), by adding more observers at distances "in between" O and xo, and then beyond xo to the right, but it *cannot* be extended indefinitely to the left (i.e., in the direction from xo back to O and beyond). At a distance 1 / a_0 to the left of O, the "accelerated frame" breaks down; it can no longer assign unique time and space coordinates to events. One way of seeing this is to note that, for an observer at distance 1 / a_0 to the left of observer O to keep up with O and xo, that observer would have to have infinite acceleration (equivalently, he would have to move at the speed of light). So this "accelerated frame" cannot cover the entire spacetime the way an inertial frame can.

For more info, Google on "Rindler coordinates", or check out the Wiki page on them:

http://en.wikipedia.org/wiki/Rindler_coordinates

Edit: I should note that I was using units in which c = 1 above, i.e., distance in light-seconds (because you've been using seconds for time). To convert distances to meters, just read c / a_0 instead of 1 / a_0 above (since you've defined a_0 as the acceleration felt by observer O, divided by c).

 

[?????]

I guess you didn't read carefully enough about the Bell Spaceship Paradox. What you have tried to set up appears to be the same situation as that paradox, with your "accelerating observer" and "observer xo" as the two spaceships. However, you have included a number of things in your formulation of the problem which are either not consistent or not stated properly, so it's hard to be sure what you intended.

First, you state:



As this is stated, it is logically impossible, because of the phrase "in any inertial reference frame". I *think* that what you meant to say is that the two accelerating observers remain a distance xo apart as seen in *the original inertial frame*, i.e., the one in which they are both at rest before they start accelerating. In other words, xo's worldline is identical to the original accelerating observer's worldline, except that it's displaced in space by a distance xo, as seen from the original inertial frame. That much is fine: but if that is the case as seen in the original inertial frame, it will *not* be the case as seen in *any* other inertial frame, *or* as seen by the two accelerated observers themselves. In particular, if the two observers remain a distance xo apart as seen in the original inertial frame, they will *separate* as seen by each other; meaning that if a rope were connecting the two observers, such that the rope had a normal unstretched length of xo, the rope would stretch as they accelerated. That's the whole point of the Bell Spaceship Paradox, and it invalidates any attempt to set up a "rigid accelerating reference frame" this way.

Next, you calculate the "value of acceleration for observer xo" to be *different* than for your accelerating observer. But if both observers follow identical worldlines, just displaced in spac.e, as seen in the original inertial frame, then the acceleration they feel must be equal. If the accelerations they feel are different, then they can't stay the same distance apart as seen in the original inertial frame.

So as it stands, what you posted, if I'm understanding your intention correctly, doesn't invalidate anything I said; it just illustrates some pitfalls in trying to analyze accelerating frames. If you meant to set up the scenario differently than how I interpreted it above, then you may want to clarify what you intended and revise your formulation accordingly.

What I stated is what I meant. Let me put it another way. You state that it is logically impossible for the ends of an infinitely stiff object to have different accelerations, but relativity is often not logical. It is mathematically imperative that the two ends of the object have different accelerations if they are to have the same velocity during the acceleration. That is how my attachment is written. What I have given you in the attachment is a modified Bell Paradox. If you calculate the original Bell Paradox, obviously the two ships will keep getting farther apart as each ship views the other. If they keep the same distance apart in the stationary reference frame, then they must get farther apart as each views each other. Length contraction alone guarantees this.

There are probably thousands of books written on relativity. Some I have read multiple times in detail and some I have skimmed through. The idea I have just related about acceleration is in all of them, though rarely is it stated explicitly. I am not the only person to make this statement. I have seen publications where other people state the same thing, including on this web site. It is a fact in relativity, although maybe not a well known fact. But don't take my word for it. Do the calculation yourself using whatever means you like.

I do not necessarily like the idea of infinitely stiff objects. You have to keep remembering the "infinitely stiff" assumption as you go through your calculation and keep trying to assess how this assumption is affecting the outcome of your analysis. But it is useful in some cases; I would even offer that it is incredibly useful as the above example illustrates. The infinitely stiff object will appear stationary simultaneously in any inertial reference frame along its entire length at its original undistorted dimensions, although clock readings down its length will vary as I have indicated.

 

[?????]

I have searched for a particular reference I remember from this website about different accelerations at the ends of an accelerating rod, but have not found it.

Of some support to what I stated is your reference Bell's Paradox link. Here are some passages from that article.

[How did the ships get farther apart, if they maintained the same constant acceleration at all times?] In the (t',x') coordinate system, dx'/dt' = 0 at t'=0 for the left-hand curve, but dx'/dt' > 0 at t'=0 for the right-hand curve. [The co-moving observers say the pursuing ship is momentarily at rest, but the pursued ship is moving, thanks to that old relativistic standby, failure of simultaneity. So the pursued ship is "pulling away".]

This first picture interprets "two ships with the equal constant accelerations" to mean "constant for the co-moving observers, and equal in the lab frame". Note that the lab frame says that the accelerations are not constant, and the co-moving observers say the accelerations are not equal! (More precisely, any particular co-moving observer says this. The phrase "the co-moving observers" does not refer to a single frame of reference, unlike the phrase "the lab frame".) The lab frame says the ships maintain a constant distance from each other; the co-moving observers don't agree.

 

[?????]

I think it would be useful if I presented an argument against the statements that I made in previous posts. Let's say you assume you have an infinitely stiff rod of length L. Now you do an experiment where you push on one end of the rod. The rod is infinitely stiff. Does that mean that the other end of the rod instantly accelerates when you start pushing on one end? Or does the other end of the rod have to wait a period of time L/c so that the signal of acceleration gets to it? The answer: I don't know. So, when you caution people about the perils of using infinitely stiff anything, I support you fully on this. I hope I didn't give the impression of anything else.

Now, let me argue in favor of my previous statements. The original observer starts accelerating. An observer can be located at position xo and accelerate as shown in my post with the results given in my post. So we can do this for other observers. In fact, we can have an infinite number of observers lined up between positions zero and L that accelerate in the prescribed manner. These observers all simultaneously become stationary in any inertial reference frame as described. To me, this is functionally equivalent to an accelerating reference frame. And one of the descriptions that you might use for this frame is that it is infinitely stiff. But now the question of when the end of the frame at position L accelerates has been answered. For me, this is much better than using the inertial reference frame the observer is instantaneously in. Because the inertial frame doesn't contain the time information that the accelerating frame does. If you could successfully define an accelerating frame as I have done, you have some interesting time differences between the clocks of this accelerating frame and the clocks of any inertial reference frame the accelerating frame is instantly at rest in. The results of calculations of the speed of Object A are interesting (to me) when you do the analysis as I have presented. Object A never goes faster than c, even though it appears at first that Object A should easily exceed c.

About 50% of this website posts consist of people trying to clarify what they said in previous posts. I too am guilty of being a bit too casual in the statements that I have made. The position I have offered is not usual in the publications most people read.

 

[PeterDonis]

I have searched for a particular reference I remember from this website about different accelerations at the ends of an accelerating rod, but have not found it.

I think you are talking about this page from the Usenet Physics FAQ:

http://math.ucr.edu/home/baez/physics/Relativity/SR/spaceship_puzzle.html

I'm not disputing anything about how the Bell Spaceship Paradox works, nor am I saying that either version (the one described in the web page, or the "modified" one that you describe) is physically impossible. They're both physically possible; they're just different, and since the wording of your formulation of the scenario was ambiguous, I couldn't be sure which "version" of the scenario you intended. Your follow-up remarks make it clear that you intended the "modified" version, where the two ships remain at the same distance apart as seen by each other (the Usenet Physics FAQ page calls this the "second picture"). That's fine, as long as you bear in mind the observations I raised in post #32 about this version of the scenario.

You state that it is logically impossible for the ends of an infinitely stiff object to have different accelerations, but relativity is often not logical.

That's not what I said. I said that it is logically impossible for the two ends to have different accelerations *and* remain at the same distance from each other *as seen in the original inertial frame*. That's true. And relativity *is* always logical; it's a consistent mathematical system. It's just not always based on intuitively obvious premises.

It is mathematically imperative that the two ends of the object have different accelerations if they are to have the same velocity during the acceleration.

Now you're getting ambiguous again. Velocity relative to what? I *think* you mean "velocity relative to the MCIF at any given event along the accelerating observer's worldline", but if so, you should state that explicitly instead of assuming that "velocity" unqualified has a well-defined meaning.

If you calculate the original Bell Paradox, obviously the two ships will keep getting farther apart as each ship views the other. If they keep the same distance apart in the stationary reference frame, then they must get farther apart as each views each other. Length contraction alone guarantees this.

Agreed. This was the point Bell made in his original paper about the paradox. (It's in his book, "Speakable and Unspeakable in Quantum Mechanics", by the way, which I highly recommend.) Again, I wasn't disputing any of this; I was just trying to figure out which version of the scenario you intended.

I do not necessarily like the idea of infinitely stiff objects. You have to keep remembering the "infinitely stiff" assumption as you go through your calculation and keep trying to assess how this assumption is affecting the outcome of your analysis.

Wouldn't it be better to recognize that, since "infinitely stiff" is physically impossible (since it would imply a sound speed greater than the speed of light), you should leave it out of your analysis altogether, and recognize explicitly that to realize the scenario you describe, you need a family of observers executing pre-planned acceleration profiles that are related in a particular way? You can express everything you need to express about this scenario without having to postulate anything at all about hypothetical massless objects linking the various observers. You just need to describe each observer's worldline, and that's easy to do; the page I linked to above writes down the appropriate equations.

But it is useful in some cases; I would even offer that it is incredibly useful as the above example illustrates. The infinitely stiff object will appear stationary simultaneously in any inertial reference frame along its entire length at its original undistorted dimensions, although clock readings down its length will vary as I have indicated.

As I just noted, you can derive all of these consequences without ever postulating the infinitely rigid "framework" at all. A fleet of rocket ships each executing the appropriate pre-planned acceleration profile will work just as well, plus it won't violate any physical laws as the infinitely stiff object does. The scheme of varying the acceleration profile with distance goes by the name of "Born rigid acceleration", and there's a good page about it here:

http://www.mathpages.com/home/kmath422/kmath422.htm

 

[harrylin]

What I stated is what I meant. Let me put it another way. You state that it is logically impossible for the ends of an infinitely stiff object to have different accelerations, but relativity is often not logical. [..]

I'm not sure about GR, but SR is always logical as far as I can tell - just as logical as classical physics. If you think otherwise, it's probably due to some kind of misunderstanding. In particular Bell's spaceship thought experiment is very logical.

 

[ghwellsjr]

Object A never goes faster than c, even though it appears at first that Object A should easily exceed c.

Object A never accelerates so how could it go faster than its initial speed, let alone c?

I already told you:

The details of how the acceleration takes place have nothing to do with this problem. It won't matter whether the acceleration is constant over the 0.1 second interval or all occurring at any instant of time during that interval. All that matters is the difference in speed between the beginning of the interval and the end of the interval.

You need to pay attention to posts #2 and #6.

I also provided you a detailed explanation of how you can view your scenario in two different inertial reference frames, one in which the observer is stationary at that start of your scenario and one in which the observer is stationary after he accelerates. In both of these reference frames, Object A never accelerates.

Please go back and read post #29.

 

[PeterDonis]

Let's say you assume you have an infinitely stiff rod of length L. Now you do an experiment where you push on one end of the rod. The rod is infinitely stiff. Does that mean that the other end of the rod instantly accelerates when you start pushing on one end? Or does the other end of the rod have to wait a period of time L/c so that the signal of acceleration gets to it? The answer: I don't know.

No, the answer is: "Your assumption of an infinitely stiff rod is physically impossible, so the scenario as you state it is impossible and doesn't require an answer. If the scenario is reformulated properly, to give the rod a speed of sound as high as possible, equal to the speed of light, the other end of the rod does not start accelerating instantly; it has to wait a period of time L/c."

I guess you could imagine an alternate universe where the laws of physics were different and infinitely stiff objects were possible; but you would then have to construct *some* logically consistent system of laws in order to predict what would happen there. Posting it here would probably violate forum rules, though.

These observers all simultaneously become stationary in any inertial reference frame as described.

This is the sloppy language that I was objecting to at the start. What you mean to say, I believe, is that the observers simultaneously become stationary, for an instant, in the MCIFs at each event along any single observer's worldline.

To me, this is functionally equivalent to an accelerating reference frame.

Yes, with the limitations described in my post #32 and in the pages I linked to in my last post.

And one of the descriptions that you might use for this frame is that it is infinitely stiff.

Not if you have any respect for physical laws, IMO. But even if you mean it just as a "technical term" for the acceleration profile, there's a perfectly good way of describing what's going on without ever having to postulate the infinitely stiff frame, so why do it? It's just going to obfuscate the issue.

For me, this is much better than using the inertial reference frame the observer is instantaneously in. Because the inertial frame doesn't contain the time information that the accelerating frame does.

A single MCIF doesn't, no. The set of all of them taken together does, for a given worldline--but *only* for that worldline (because the rate of time flow varies from worldline to worldline). The full description of the scenario has to include *all* the MCIFs along *all* the worldlines, with their associated proper times. There is no way to describe this using a single "accelerating frame"; any set of "accelerating coordinates" for this family of observers has to pick one observer's worldline as its "standard" for time. So no matter how you describe this scenario, you're going to have to accept that no single "frame" can capture everything about it.

Another good web page that discusses all this is Greg Egan's, here:

http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

 

[Dale]

relativity is often not logical

This is false. Relativity is perfectly and completely logical. It is often not intuitive, and it often violates common-sense, but it is always completely logical.

 

[Dale]

Let's say you assume you have an infinitely stiff rod of length L. Now you do an experiment where you push on one end of the rod. The rod is infinitely stiff. Does that mean that the other end of the rod instantly accelerates when you start pushing on one end? Or does the other end of the rod have to wait a period of time L/c so that the signal of acceleration gets to it? The answer:

See the new FAQ on the topic:
https://www.physicsforums.com/showthread.php?p=3537287#post3537287