# Franklin, Lorentz contraction, Bell's spaceships, and

• bcrowell
In summary: This suggests the need for a deﬁnition of ‘length’ that is the same for any state of uniform motion. This would correspond to the use in relativity of ‘proper time’ and ‘invariant mass’ for time and mass, but the terms “proper length’ and ‘invariant length’ have... not been explicitly introduced in relativity.This is a quote from the article, which is discussing why it is necessary to introduce a deﬁnition of "length" that is the same in any state of uniform motion. This quote further supports the idea that "length" is not a
bcrowell
Staff Emeritus
Gold Member
Franklin, "Lorentz contraction, Bell's spaceships, and ..."

In the discussion of a new FAQ entry on the Bell spaceship paradox, the following paper came up:

Jerrold Franklin, "Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity," Eur. J. Phys. 31 (2010) 291, http://arxiv.org/abs/0906.1919

Since the paper contains some serious mistakes, I thought it would be helpful to start a thread here about it in order to collect some discussion and keep it publicly available for people who might otherwise be in danger of being misled by some of Franklin's claims. I initially thought, "Oh, nice, a detailed discussion of the Bell spaceship paradox that isn't behind a paywall." Then PAllen pointed out serious problems with it. The following two quotes, with my added labels [A] and , are both on p. 2:

[A] The measured length of a moving object depends on the ‘particular way’ in which it is measured.

It should also be pointed out that another classically reasonable method of
measuring the length is to take a photograph of a moving object and compare
it with a photograph of the same object at rest. As Terrell[3] showed some time
ago, the photograph would show an object that is somewhat rotated, but of
the same shape and dimensions as it had at rest. Indeed, the photograph of a
moving sphere would show a sphere of the same size.

A is a general claim, and B appears to be intended as an example that demonstrates it to be true.

B is simply wrong. There is a group at ANU, for example, that has made a series of educational videos such as this one http://youtube.com/watch?v=JQnHTKZBTI4 , which I often show to my classes. In it, you can see very clearly that a shape such as a cube does not retain "the same shape and dimensions." I have a treatment of the cube as an example in section 6.5 of my SR book, http://www.lightandmatter.com/sr/ . The calculations are summarized and explained there, and anyone who's interested in the details of the calculation can examine my computer code here https://github.com/bcrowell/special_relativity/tree/master/ch06/figs/cube-sources . I'm pretty sure my calculations are right, since the resulting shape looks just like the one seen in the ANU video. Franklin has mischaracterized Terrell's work.

A could be argued to be true in the sense that optical measurements do not simply show the longitudinal length contraction by 1/γ. But the fact that Franklin supports it using a completely incorrect statement suggests that he thinks it is true in some broader sense, which is false. When we talk about the length of an extended body in relativity, we mean, simply as a matter of definition, the distance between two events on that body's world-sheet, with the two events being simultaneous. This definition gives a consistent result, longitudinal contraction by 1/γ, regardless of the details of the procedure used for carrying out the measurement. Optical measurements simply don't implement the definition, since the observer receives light rays simultaneously that were not emitted simultaneously in his frame.

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I have a question and a comment. On page 5 of the paper, Franklin says the following:

Bell’s paradox was that his intuition told him the cable would break, yet there was no change in the distance between the ships in system S. He suggested resolving the paradox by stating that a cable between the ships would shorten due to the contraction of a physical object proposed by Fitzgerald and Lorentz, while the distance between the ships would not change.

But this isn't really what the proposed solution entailed. Rather it was recognized that, in frame S, the "natural" or "equilibrium" length of the cable would undergo (continuous) length contraction whereas the actual length of the cable would be held constant by the initial conditions of the problem. In other words there are two different lengths at play in frame S itself: the equilibrium length and the actual length. Only the former undergoes length contraction, not the latter, which is why the cable eventually snaps. Therefore...

This resolution however contradicts special relativity which allows no such difference in any measurement of these two equal lengths.

Franklin claims this (on the same page) but I don't really understand what he means. What supposed contradiction between SR and the aforementioned resolution is he referring to when he mentions "measurements of these two equal lengths"?

WannabeNewton said:
What supposed contradiction between SR and the aforementioned resolution is he referring to when he mentions "measurements of these two equal lengths"?

I think he is basically trying to enforce a definition of the term "length" that he hasn't really thought through. Consider these quotes:

This suggests the need for a deﬁnition of ‘length’ that is the same for any state of uniform motion. This would correspond to the use in relativity of ‘proper time’ and ‘invariant mass’ for time and mass, but the terms “proper length’ and ‘invariant length’ have already been used in the literature with other meanings. The term we recommend for length is ‘rest frame length’, which we deﬁne as the length a moving object has after a Lorentz transformation to its rest system. If length is to be considered a physical attribute of an object, then this physical attribute should be the rest frame length. This length, of course, would not be changed by uniform motion.

(p. 3, 3rd paragraph)

We conclude that the length of an object can be measured only in its rest system. If the object is moving, making a Lorentz transformation to its rest system is the only way to get a reliable measurement of its length. This is generally true of intrinsic properties of physical objects that are not Lorentz invariants.

(p. 3, bottom, to p. 4, very top)

I say his attempted definition isn't really thought through because of the following:

(1) He is trying to draw an analogy between his concept of "length" and proper time and invariant mass. But proper time and invariant mass are different types of things to begin with. Invariant mass is a Lorentz scalar; proper time is the integrated path length along a timelike curve. Even if we restrict the curve to be a geodesic (which is what "uniform motion" implies), proper time is still a different kind of invariant from invariant mass.

(2) If we leave out invariant mass and consider the analogy between proper time (in "uniform motion") and his concept of "length", the obvious thing to do would be to define "length" as the integrated path length along a spacelike geodesic. But I'm pretty sure the terms "proper length" and "invariant length" have been used to refer to precisely that, so I'm not sure what he's talking about when he says those terms are already used with other meanings.

(3) If we define "length" as the integrated path length along a spacelike geodesic, then it's obvious that picking different geodesics results in different "lengths". That's what the standard definition of "length", the one that shows length contraction in relative motion, does: in frame S, you pick spacelike geodesics that are curves of constant ##t##, but in frame S', you pick spacelike geodesics that are curves of constant ##t'##. But for some reason I can't fathom, Franklin apparently can't or won't take this view, so he ties himself in knots trying to make his concept of "length" work while still making it an "invariant".

(4) But, he's apparently confused about what "invariant" means. The obvious way to define his "rest frame length" is to pick the spacelike geodesic that is a curve of constant time in the object's instantaneous rest frame, and use integrated path length along that geodesic. But this is an integral invariant in the same way proper time is; yet he talks (second quote above) about "intrinsic properties of physical objects that are not Lorentz invariants". Wait, what? I though the whole *point* of relativity was that anything that's an "intrinsic property" must be Lorentz invariant: it's either a straightforward scalar like invariant mass, or it's a contraction of vectors and tensors with no free indexes (i.e., a scalar formed by contraction), or it's an integral of one of the above along a well-defined curve.

(5) Furthermore, the way he actually tries to define his "length" does *not* use a single spacelike geodesic. Instead he tries to argue that we can still make position measurements of two ends of an object at different "times" (he's sloppy about which frames the times are taken in), and use them to compute his "length" because it doesn't make any difference. But of course that's only true if the motion of the object(s) in question describes a Killing congruence, and the motion of the Bell spaceships does not: yet he claims this method works for the Bell spaceship scenario! (See p. 5 and the top of p. 6.)

Bottom line, I think this paper causes far more confusion than it solves, if indeed it solves any.

PeterDonis said:
But of course that's only true if the motion of the object(s) in question describes a Killing congruence, and the motion of the Bell spaceships does not: yet he claims this method works for the Bell spaceship scenario!

A Killing congruence means a congruence whose tangent vectors are a Killing field? Is characterizing the motion of the object as a Killing congruence equivalent to saying that it's Born-rigid, which I guess is the same as saying that the congruence has a vanishing expansion scalar?

In Franklin's defense, the title of his paper does say "in special relativity." Are some of the distinctions in your #3 irrelevant in flat spacetime?

bcrowell said:
A Killing congruence means a congruence whose tangent vectors are a Killing field?

Yes.

bcrowell said:
Is characterizing the motion of the object as a Killing congruence equivalent to saying that it's Born-rigid

I believe so, yes.

bcrowell said:
which I guess is the same as saying that the congruence has a vanishing expansion scalar?

Vanishing expansion *and* shear; only the proper acceleration and vorticity can be nonzero.

bcrowell said:
In Franklin's defense, the title of his paper does say "in special relativity." Are some of the distinctions in your #3 irrelevant in flat spacetime?

No; the example I gave, of "length" changing when you change inertial frames, is one of the standard SR textbook examples in flat spacetime (though they don't always phrase it the way I did).

PeterDonis said:
I think he is basically trying to enforce a definition of the term "length" that he hasn't really thought through. Consider these quotes:

Thanks Peter!

Franklin seems to also find inadequate the usual definition of length in terms of distance between simultaneous events relative to a given frame because the Terrell rotation, according to him, serves as an example of length measurement in which an object is simply rotated but retains its dimensionality. He concludes from this that the usual definition of length is ambiguous. But the only reason the Terrel rotation in the context of relativity leads to an object that is simply rotated but unchanged in dimensionality is the length contraction that cancels out the optical elongation, which itself relies on the usual definition of length measurement e.g. in terms of radar signals and Einstein simultaneity. The same effect in a non-relativistic context but with finite signal propagation speed would lead to a rotation + an elongation so his Terrel rotation argument in view of the inadequacy of the usual definition of length measurement makes little to no sense to me, especially in light of what Ben said in post #1 regarding the optical effects of a passing cube in the relativistic context.

It seems to me the paper uses very specific situations to form general remarks but resorts to situations that are exceptions rather than the rule.

As regards the Hergolt-Noether theorem:

The definition I see from several papers is:

all rotational shear-free and expansion-free flows (rotational Born-rigid flows) in Minkowski spacetime are generated by Killingvector fields (isometric flows).

But I don't understand why the flows are specified as rotational flows. What about irrotational flows?

I did find the following (in http://arxiv.org/pdf/0802.4345v1.pdf)

A sharp and useful criterion for whether a rigid motion is Killing or not is given by the following

Lemma 20.
Let u be a normalised timelike vector field on a region ##\Omega \subseteq M##. The motion generated by u is Killing iff it is rigid and ##a^♭## is exact on ##\Omega##

I don't understand what they mean by a^b, or exact, and I find the notation hard to follow. So I feel like I might be missing something here.

pervect said:
But I don't understand why the flows are specified as rotational flows. What about irrotational flows?

Not all rigid irrotational flows in flat space-time need to be generated by Killing fields. This is discussed in the proof of the theorem. Only rigid rotational flows in flat space-time necessarily have to be generated by Killing fields.

pervect said:
I don't understand what they mean by a^b, or exact, and I find the notation hard to follow. So I feel like I might be missing something here.

##a^{\flat}## is the musical isomorphism of ##a##. In index notation it just means ##a_{\mu}##. Exact means ##a^{\flat} = df## or in index notation ##a_{\mu} = \nabla_{\mu}f## for some smooth scalar field ##f##.

WannabeNewton said:
Not all rigid irrotational flows in flat space-time need to be generated by Killing fields. This is discussed in the proof of the theorem. Only rigid rotational flows in flat space-time necessarily have to be generated by Killing fields.

I'm looking for a specific example of an irrotational flow that isn't generated by a Killing field. My intuition is telling me to consider a Born-rigid spaceship whose proper accleleration is a function of time, but I haven't worked out the details.

pervect said:
I'm looking for a specific example of an irrotational flow that isn't generated by a Killing field.

See theorem 22 (page 43) in the paper.

The FAQ entry referred to in the OP is now posted:

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PeterDonis said:
I think he is basically trying to enforce a definition of the term "length" that he hasn't really thought through. . . But, he's apparently confused about what "invariant" means. . .

It seems to me that he is just saying what the group said in an earlier thread on length about how proper length is invariant (which he considers to be a physical attribute), https://www.physicsforums.com/showthread.php?t=732122. DaleSpam wrote "it is correct that length is frame variant while proper length [is] frame invariant." And PAllen wrote "Proper length of an object is invariant, and is the same in any reference frame."

Franklin writes on page 3: "the length of an object can be measured only in its rest system." PAllen writes "Proper length is only directly measured in the object's rest frame. In other frames it is computed (from several measurements)."

I am not sure what he means by "proper length" having been used in other ways, and thus why he feels the need to use a different term "rest length." Perhaps it is to emphasize his view of the physical significance of the Lorentz equation, which he treats as an important part of the paper (particularly at the bottom of page 1 and the top of page 2) . . .

JVNY said:
Franklin writes on page 3: "the length of an object can be measured only in its rest system." PAllen writes "Proper length is only directly measured in the object's rest frame. In other frames it is computed (from several measurements)."

Yes, and proper length in this sense was what I was referring to in point (4) of my post as the integrated path length along a spacelike geodesic in the object's instantaneous rest frame. I certainly agree that proper length defined this way is an invariant; but as I pointed out in my post, I understand this to be the standard definition of the term, but Franklin apparently doesn't, plus he appears to be saying that his "rest length" is *not* an invariant when he talks about "intrinsic properties that are not invariants". So if he's really trying to say that proper length as defined above is an invariant, he's not doing a very good job of it.

When I started this thread, I emailed Franklin to point out the error, and pointed him to this thread. Today I received a long, angry email from him in reply, in which he was not willing to concede that he had made an error in his characterization of the Terrell rotation. I had suggested that he post a corrected version of the paper on arxiv, with this mistake fixed, but he seems to have reacted defensively and is clearly not going to do that.

bcrowell said:
A could be argued to be true in the sense that optical measurements do not simply show the longitudinal length contraction by 1/γ. ...
Terrell rotation is often quoted as demonstrating that length contraction cannot be photographed, but this is only true in the specialised case of a sphere. A photograph of a long narrow rod moving parallel to its long axis would clearly show a contracted length. The 'rotation' is also more of a shear than a rotation.

I spent some more time looking through the Franklin paper and found more material that seems to mischaracterize other people's papers. On p. 3:

One other point to be considered is whether strains and stresses can be induced by Lorentz contraction, as is contended in Refs. [1,2,4,5]. Our answer to this is clear from the previous discussion. Just as a 3D rotation of an object does not induce strain, a 4D rotation (Lorentz transformation) will not induce strain and consequent stress.

The references are:

[1] Lorentz H A 1892 Versl. Kon. Akad. Wetensch. 1 74
[2] FitzGerald G F 1889 Science 13 390
[4] Bell J S 1993 Speakable and Unspeakable in Quantum Mechanics (Cambridge: Cambridge University Press) , pp 67-68
[5] Dewan E and Beran M 1959 Am. J. Phys. 27 517

It's a little silly IMO to lump 19th century papers by Lorentz and FitzGerald together with 20th century papers.

Franklin appears to be mischaracterizing the contents of the Bell article. Franklin is basically arguing that if there is no stress in one frame there is no stress in any other frame frame. (A similar argument is made by Petkov, http://arxiv.org/abs/0903.5128 , on p. 4.) But I don't see anything in Bell that claims otherwise. The most relevant passage would seem to be the following:

Lorentz invariance alone shows that for any state of a system at rest there is a corresponding 'primed' state of that system in motion. But it does not tell us that if the system is set anyhow in motion, it will actually go into the 'prime' of the original state, rather than into the 'prime' of some *other* state of the system at rest. In fact, it will generally do the latter. A system set brutally in motion may be bruised, or broken, or heated, or burned. For the simple classical atom similar things could have happened if the nucleus, in stead of being moved smoothly, had been jerked. The electron could be left behind completely.

Bell is clearly not claiming that under a Lorentz transformation, an unstressed body becomes a body under mechanical stress. He is clearly just saying that if the body undergoes noninertial motion, due to some external force, there will be dynamical effects.

## What is Franklin?

Franklin refers to Benjamin Franklin, a famous American scientist, inventor, and statesman. He is best known for his contributions to electricity, including the invention of the lightning rod.

## What is Lorentz contraction?

Lorentz contraction, also known as length contraction, is a phenomenon described by Albert Einstein's theory of relativity. It states that objects in motion will appear shorter in the direction of their motion when observed by an outside observer.

## What are Bell's spaceships?

Bell's spaceships, also known as the Bell spaceship paradox, is a thought experiment proposed by physicist John Bell. It aims to demonstrate the effects of time dilation and length contraction in Einstein's theory of relativity.

## What is the relationship between Franklin and Lorentz contraction?

There is no direct relationship between Franklin and Lorentz contraction. However, both are important figures in the history of science and have made significant contributions to our understanding of the physical world.

## How do Bell's spaceships relate to Einstein's theory of relativity?

Bell's spaceships illustrate the concepts of time dilation and length contraction, which are fundamental principles in Einstein's theory of relativity. The thought experiment helps us understand the effects of these phenomena in a simplified way.

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