Unraveling the Speed of Light Puzzle: A Thought Experiment Exploring Relativity

[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