Bell's spaceship paradox unknown? Interpretation?

[rupcha]

TL;DR Summary

Why is "Bell's spaceship paradox" not more widely known? What are accepted interpretations?

Recently, I spent some time trying to get an intuitive understanding of special relativity.
(I am not a physicist, only took a few physics lectures in the mid-90s)

It all went well until I tried to imagine accelerating objects with non-zero length.
Specifically, I tried to imagine what a spaceship would look like if it was (brutally) decelerating from relativistic speeds to a standstill.

I expected it to “grow” out of its back towards the front (like an accordion) and obviously having its proper length after coming to a stop. But at the same time, it seemed that it should have grown beyond its proper length by factor of gamma.
After some frustration and thinking that I must have misunderstood some core aspect of SR, I finally found out about "Bell’s spaceship paradox".

While that cleared up the dilemma at hand, some questions keep bugging me:

1. Why is Bell’s spaceship paradox / the described effect not more widely known or communicated? Why did it take until the 60s/70s to be mentioned at all?
Does the effect even have a name? “Relativistic stress” seems to have been used but not widely accepted?

To me, it it was one of the most interesting and surprising effects of SR I have come upon so far. So I am just very surprised that almost all explanations of SR (including e.g. Gerthsen) do not mention it at all.

2. Interpretation?
Solving the “paradox” was one thing. You can either have identical proper acceleration and drift apart or have higher proper acceleration in the back than in the front in order to stay at the same distance/length.

But is there any accepted interpretation of the (seemingly?) emerging forces / accelerations?
Are they considered tidal forces? Is space expanding (for the accelerating body) in the direction of acceleration?

Thankful for everyone who can shine some light on any of those questions.

I am only just trying to wrap my head around all this. And then I thought about circular motion, which all but made my head explode. So I will leave that for a (much) later post.

Rupert

 

[PeterDonis]

1. Why is Bell’s spaceship paradox / the described effect not more widely known or communicated? Why did it take until the 60s/70s to be mentioned at all?

This is not a physics question, it's a historical question, so it is really off topic here. As a general observation, the fact that a particular theory becomes established does not mean all of its logical consequences are immediately known. It can take a long time to work those out.

I am just very surprised that almost all explanations of SR (including e.g. Gerthsen) do not mention it at all.

You would have to ask the authors about that. Again, that's not really a physics question. (Note that to the extent this is a question about physics pedagogy, it would be on topic in one of the education forums here at PF. There might be frequent users there who could give more insight.)

But is there any accepted interpretation of the (seemingly?) emerging forces / accelerations?

Um, spacetime geometry and worldlines? I'm not sure what kind of "interpretation" you are looking for.

Are they considered tidal forces?

No. Spacetime in the Bell spaceship scenario is flat.

Is space expanding (for the accelerating body) in the direction of acceleration?

No.

 

[PeroK]

But is there any accepted interpretation of the (seemingly?) emerging forces / accelerations?
Are they considered tidal forces? Is space expanding (for the accelerating body) in the direction of acceleration?

It all depends on how you propose to accelerate the spaceship. Ultimately, as with many so-called paradoxes in SR, it's just the relativity of simultaneity in a new scenario.

 

[Ibix]

Why is Bell’s spaceship paradox / the described effect not more widely known or communicated?

The maths of non-inertial frames is quite a lot more complex than inertial frames, in both pre-relativistic and relativistic physics. I suspect it's simply a case of avoiding the scarier maths in introductory courses. But that's just a guess.

But is there any accepted interpretation of the (seemingly?) emerging forces / accelerations?

An object undergoing constant proper acceleration traces out a hyperbola in spacetime. Hyperbolae and the Lorentz transforms in Minkowski spacetime are analogous to circles and rotations in Euclidean spacetime: if the origin of the transforms is the center of the curve, the curve does not change under the transform. So a set of nested hyperbolae with the same center don't change under a Lorentz boost - that's what you called the "higher proper acceleration in the back" case. So that case is the definition of "nothing changes about your fleet of ships as you accelerate" as you see it. Any other case (e.g. Bell's scenario) involves the ships drifting apart and the tension in the string is an obvious consequence.

I think that's the most fundamental explanation available without invoking expansion tensors (which is the same thing in more technical language).

 

[A.T.]

But is there any accepted interpretation of the (seemingly?) emerging forces / accelerations?

The forces are not seemingly emerging. They are frame invariant physical facts. But the "cause" of them can be frame dependent. In the intial restframe the fields between the atoms contract, so they cannot span the same total length anymore, which is held constant. In other frames the distance between the rockets increases, because the rockets don't accelerate synchronously.

Is space expanding (for the accelerating body) in the direction of acceleration?

It depends what that means exactly. When you extend the Bell spaceship paradox to circular motion, you get the Ehrenfest paradox. There, one explanation is that in a rotating rest frame the spatial geometry is hyperbolic, so space is effectively expanded along the tangential direction.

 

[PeterDonis]

When you extend the Bell spaceship paradox to circular motion, you get the Ehrenfest paradox.

Kinda sorta. There are many differences between the two scenarios. To name just the most important, in the Ehrenfest paradox the rotation is stationary--everything stays the same distance apart. There is nothing analogous to the stretching and breaking of the string in the Bell spaceship paradox.

There, one explanation is that in a rotating rest frame the spatial geometry is hyperbolic, so space is effectively expanded along the tangential direction.

This is a very misleading way of putting it. The "spatial geometry" you refer to is, in technical language, a quotient space; there is no spacelike slice in the spacetime that corresponds to it. Also, this quotient space geometry does not change with time--again, in technical language, the expansion of the rotating congruence of worldlines is zero, whereas the expansion of the Bell spaceship congruence of worldlines is positive. What you are calling "space is effectively expanded" in the rotating case, in other words, is (a) a different kind of "space", and (b) not changing with time; both of those are significant disanalogies with the Bell spaceship paradox.

 

[Sagittarius A-Star]

To name just the most important, in the Ehrenfest paradox the rotation is stationary--everything stays the same distance apart.

According to the headline of Ehrenfest's article yes, but within the text he described an increase of the angular velocity:

In fact: let a relative-rigid cylinder of radius $R$ and height $H$ be given. A rotation about its axis which is finally constant, will gradually be given to it.

Source of the English translation:
https://en.wikisource.org/wiki/Tran..._of_Rigid_Bodies_and_the_Theory_of_Relativity

Original in German:
https://de.wikisource.org/wiki/Gleichförmige_Rotation_starrer_Körper_und_Relativitätstheorie

Einstein added later additional insight related to non-Euclidean geometry:

It therefore follows that $U/D > \pi$.

https://en.wikipedia.org/wiki/Ehrenfest_paradox#Einstein_and_general_relativity

 

[PeterDonis]

within the text he described an increase of the angular velocity

Yes, you're right, he did talk about the impossibility of rigidly spinning up a cylinder from zero to nonzero angular velocity. This is true, but the details of how the spin-up process occurs and what particular departures from rigidity occur cannot be derived solely from kinematic considerations, as the expansion of the Bell spaceship congruence can.

Also, Ehrenfest's discussion of spinning up a cylinder was part of his attempt to deal with the issue of the non-Euclidean spatial geometry of the rotating (at a constant angular velocity) cylinder--what you refer to as "a wrong conclusion related to the radius". (Note that he says the spin up only occurs until a final constant angular velocity is reached--in the Bell spaceship paradox there is no final constant state that is reached.) He was trying to figure out solely from kinematic considerations what would happen to the geometry of the cylinder--and, as above, we now know that's impossible. (Note that Einstein's discussion also does not fully recognize this impossibility.)

 

[FactChecker]

It all depends on how you propose to accelerate the spaceship. Ultimately, as with many so-called paradoxes in SR, it's just the relativity of simultaneity in a new scenario.

Good point. Suppose the front and rear of the spaceship experiences exactly the same timing and amount of acceleration in the spaceship time synchronization. Then those two ends do not accelerate at the same time in the Earth (or "stationary") time synchronization.

expected it to “grow” out of its back towards the front (like an accordion) and obviously having its proper length after coming to a stop. But at the same time, it seemed that it should have grown beyond its proper length by factor of gamma.

If the OP calculations are off by a factor of $\gamma$, it seems very likely that something like that has been overlooked. The calculations are beyond my (casual amateur) ability, especially since the accelerating clocks would have to be continuously synchronized some how. But the overlooked instantaneous factor of $\gamma$ for synchronized clocks might be easier to spot. The amateur's rule of thumb is "Ahead is behind and behind is ahead." That would mean that a "stationary" observer would say that the rear accelerated earlier than the front, which would shorten the length.

 

[PeterDonis]

Suppose the front and rear of the spaceship experiences exactly the same timing and amount of acceleration in the spaceship time synchronization.

You are assuming that there is such a thing as "the spaceship time synchronization". But for the Bell spaceship case, there is no such thing.

 

[PeterDonis]

If the OP calculations are off

The OP has not shown any calculations. The OP remark about gamma is just hand-waving, and does not appear to be relevant to the actual questions they are asking anyway.

 

[rupcha]

Thanks a lot for all the answers - and sorry for the parts which were off-topic here.

While all kinds of new questions are popping into my head about variations of the topic (like rockets going back and forth and circular paths), I need to do more mulling over before I'll be able to even ask those.

But I think I'm all set for the "basic case".

Thanks again,

Rupert

If the OP calculations are off by a factor of γ, ...

I don't believe my calculations were off, I was merely describing how I stumbled upon the paradox.

 

[Meir Achuz]

Start with Wikipedia, and read the references therein.

 

[PeroK]

But I think I'm all set for the "basic case".

The basic is case is two particles initially at rest a set distance apart, which accelerate in the direction of their initial separation. After a period of acceleration they attain the same velocity in the original rest frame. They cannot be the same distance apart in the original rest frame as they are in the inertial frame in which they are finally at rest. This can be seen from the concept of length contraction.

A synchronized acceleration in the original rest frame is different from a synchronized acceleration in any inertial frame moving relative to this frame. This includes frames instantaneously comoving with one of the particles, say. This is the relativity of simultaneity.

Two ships joined by a rope are essentially just a line of a large number of particles, in this context. And all the questions of internal forces etc. arise as a consequence of the basic case of two particles above.

There is no paradox.

 

[pervect]

TL;DR Summary: Why is "Bell's spaceship paradox" not more widely known? What are accepted interpretations?

Recently, I spent some time trying to get an intuitive understanding of special relativity.
(I am not a physicist, only took a few physics lectures in the mid-90s)

It all went well until I tried to imagine accelerating objects with non-zero length.
Specifically, I tried to imagine what a spaceship would look like if it was (brutally) decelerating from relativistic speeds to a standstill.

I expected it to “grow” out of its back towards the front (like an accordion) and obviously having its proper length after coming to a stop. But at the same time, it seemed that it should have grown beyond its proper length by factor of gamma.
After some frustration and thinking that I must have misunderstood some core aspect of SR, I finally found out about "Bell’s spaceship paradox".

While that cleared up the dilemma at hand, some questions keep bugging me:

1. Why is Bell’s spaceship paradox / the described effect not more widely known or communicated? Why did it take until the 60s/70s to be mentioned at all?
Does the effect even have a name? “Relativistic stress” seems to have been used but not widely accepted?

To me, it it was one of the most interesting and surprising effects of SR I have come upon so far. So I am just very surprised that almost all explanations of SR (including e.g. Gerthsen) do not mention it at all.

2. Interpretation?
Solving the “paradox” was one thing. You can either have identical proper acceleration and drift apart or have higher proper acceleration in the back than in the front in order to stay at the same distance/length.

But is there any accepted interpretation of the (seemingly?) emerging forces / accelerations?
Are they considered tidal forces? Is space expanding (for the accelerating body) in the direction of acceleration?

Thankful for everyone who can shine some light on any of those questions.

I am only just trying to wrap my head around all this. And then I thought about circular motion, which all but made my head explode. So I will leave that for a (much) later post.

Rupert

I think what you are trying to imagine is the acceleration and deacceleration of a rigid object.

The author who explored this was Max Born, and the associated concept is known as "Born Rigidity". The wiki article on this topic is https://en.wikipedia.org/wiki/Born_rigidity

It is more clearly described as "rigid motion" rather than a "rigid object", because the notion of a force acting instantaneously across a space-like interval is not compatible with special relativity. However, the notion of an object "holding it's shape" does exist. A force at a point may not allow this motion, but a properly distributed array of forces may allow an object to "hold it's shape".

What Born provides is a mathematical precise description of what the words "hold it's shape" actually means.

Born actually had a couple of notions of rigid motion. The second one is not really well-described by my description of "holding shape", but it solves some cases where the required motion does not exist with the first definition, for instance the case of rotating objects undergoing acceleration.

Ignoring the rotational issues, which relate to another well known paradox, the Ehrenfest paradox, Born rigidity has some other interesting consequences, such as a maximum length of a rigid object. If the front of an object is accelerating or deaccelerating at 1g, where g is the acceleration of an Earth gravity, at a distance of c^2 / g, which is approximately 1 light year, the acceleration required to maintain rigidity would be infinite, which is impossible.

The Bell spaceship paradox is just one interesting example of the notions and limitations of notions of the concept of rigidity in special realtivity, from my point of view.

My own point of view is also that the notion of a rigid object is just not a good approximation to any physical problem in special relativity, so that for any realistic case, matter is just not strong enough to hold together when deaccelerated rapidly to or from relativistic speeds - one is better off, in the long run, learning the techniques of non-rigid motion. Fluid mechanics, where one totally ignores the rigidity of a substance, is a much better approximation for any sort of physical high speed impacts, the forces that hold an object together are pretty much negligible for such high velocity impacts.

The philsophical problem I see is that many people base their notion of physics on the physics of rigid objects. I agree this is simpler, but at some point, if one is serious about physics, one needs to consider learning methods that allow one to do physics for non-rigid objects. This does leave people who have based their entire knowledge of physics on the assumption that rigid objects are fundamental in a bit of a quandry, I'm not sure I have a way out of this quandry that does not involve quite a large amount of work.

 

[PeterDonis]

Born actually had a couple of notions of rigid motion.

The only one I'm aware of is Born rigidity. What other one are you referring to?

 

[pervect]

The only one I'm aware of is Born rigidity. What other one are you referring to?

From the wiki: https://en.wikipedia.org/wiki/Born_rigidity

The concept was introduced by Max Born (1909),[1][2] who gave a detailed description of the case of constant proper acceleration which he called hyperbolic motion. When subsequent authors such as Paul Ehrenfest (1909)[3] tried to incorporate rotational motions as well, it became clear that Born rigidity is a very restrictive sense of rigidity, leading to the Herglotz–Noether theorem, according to which there are severe restrictions on rotational Born rigid motions. It was formulated by Gustav Herglotz (1909, who classified all forms of rotational motions)[4] and in a less general way by Fritz Noether (1909).[5]

As a result, Born (1910)[6] and others gave alternative, less restrictive definitions of rigidity.

Emphasis mine. The 1910 paper appears to be

Born, Max (1910), "Zur Kinematik des starren Körpers im System des Relativitätsprinzips" [Wikisource translation: On the Kinematics of the Rigid Body in the System of the Principle of Relativity], Göttinger Nachrichten, 2: 161–179

which I have not read. My personal feeling is that if the condition in Born's first paper is not met, the object does not hold it shape. This doesn't necessarily make the other notions useless, though. Unfortunately, I haven't seen an English paper discussing the alternatives in detail, and I only have some speculations on how this is probably done, which it would be best not to post.

 

[PeterDonis]

My personal feeling is that if the condition in Born's first paper is not met, the object does not hold it shape.

That must be true because the Born rigidity condition (the "very restrictive" one) is the condition required for the object to hold its shape.

This doesn't necessarily make the other notions useless, though.

I have not seen anything that applies any notion of rigidity other than standard Born rigidity, which restricts rigid motions to those allowed by the Herglotz-Noether theorem. From a quick read of Born's 1910 paper, the one the Wikipedia article references for the "less restrictive" definitions, Born appears to be trying to develop a classical model of fundamental particles like the electron, so I suspect that once QM was developed such models were discarded, and the "less restrictive" definitions of rigidity were discarded along with them.

 

[PAllen]

That must be true because the Born rigidity condition (the "very restrictive" one) is the condition required for the object to hold its shape.


I have not seen anything that applies any notion of rigidity other than standard Born rigidity, which restricts rigid motions to those allowed by the Herglotz-Noether theorem. From a quick read of Born's 1910 paper, the one the Wikipedia article references for the "less restrictive" definitions, Born appears to be trying to develop a classical model of fundamental particles like the electron, so I suspect that once QM was developed such models were discarded, and the "less restrictive" definitions of rigidity were discarded along with them.

I’ve seen a few over the years. The end of the Wikipedia article mentions an interesting idea I hadn’t seen before:

https://arxiv.org/abs/1307.1914

 

[PeterDonis]

I’ve seen a few over the years. The end of the Wikipedia article mentions an interesting idea I hadn’t seen before:

https://arxiv.org/abs/1307.1914

I think I've seen this before, though it's been a while: basically, as I understand it, the idea is that if you only require the motion of the boundary of a "world tube" to be Born rigid, you can add additional degrees of freedom.

 

[Sagittarius A-Star]

Born, Max (1910), "Zur Kinematik des starren Körpers im System des Relativitätsprinzips" [Wikisource translation: On the Kinematics of the Rigid Body in the System of the Principle of Relativity], Göttinger Nachrichten, 2: 161–179

This article can be found here:
https://gdz.sub.uni-goettingen.de/i...":0.495,"y":0.499},"view":"toc","zoom":0.586}
via:
http://histmath-heidelberg.de/homo-heid/gdz/born.htm

Citation out of it by Ehrenfest:

M. Born[6], in a later paper, sticks to the idea (in opposition to Planck) that the extension of the rigidity concept to relativity theory is necessary, and he develops – by reseting his original rigidity definition as too narrow – a new rigidity definition. There he accepts, that any rigid body must be imagined as having – with respect to the rigidity definition – a preferred point once and for all.[7] Yet also in this way, by far not all difficulties of the rotation problem can be totally resolved: As demonstrated by Born, a stationary (!) rotation about a fixed axis takes place here, so that the resting observer sees the multi-axial layers rotating with larger angular velocity as the peripheral layers, thus the "rigid" body (starting from the axis) increasingly stirs itself.[8]

Source:
https://en.wikisource.org/wiki/Tran...'s_Treatment_of_Born's_Definition_of_Rigidity