SRT is completely erroneous since it is based on the wrong kind of transformations: they have lost the scale factor characterizing the Doppler effect (which defines the asymmetry between approach and removal). First, Lorentz considered a more general form of transformations (with a scale factor), but then he, and also Poincare and Einstein equated it 1 without proper grounds. Their form was artificially narrowed, the formulas became incorrect. This led to a logical contradiction of the theory, to unsolvable paradoxes. Accordingly, GRT is also incorrect. For more details, see my brochure "Memoir on the Theory of Relativity and Unified Field Theory" (2000):

https://www.researchgate.net/publication/339090652_Memoir_on_the_Theory_of_Relativity_and_Unified_Field_Theory

The ``paradox", as always, is due to ambiguous choice of words. What matters are the transformations. In special relativity spacetime distances are invariant under global Lorentz transformations, spatial distances are not.

In general relativity global Lorentz transformations arr not symmetries.

It's wrong to state that under special relativity the string should break. The two spaceships are separated by a space-like distance. Any Lorentz transformation will preserve this property. The question is, whether a Lorentz transformation exists, that can relate the frame in which the two spaceships are separated by a distance that's less than the maximal length of the string, to one in which their separation is greater. Since the answer is affirmative, the string will break.

In general relativity the same result is obtained, in the absence of horizons, since general coordinate transformations can't change the property that the separation between two events is space-like, light-like or time-like.

Stam Nicolis

Not sure why you say "It's wrong to state that under special relativity the string should break", since you then go on to give an argument under SR for why the string breaks.

That the string length is space-like does not seem to me sufficient to argue that the string also breaks under GR. After all, every string length is space-like, but most strings don't break.

Have the field equations of GR ever been solved in the case of Bell's paradox? I haven't been able to find any exact analysis of the paradox under GR. Do you know of one? Whether Rindler coordinates can be used seems debatable - using them begs the question of whether the equivalence principle doesn't apply.

My point is that the reason the string breaks is an illustration of special relativity, but special relativity isn't the cause; the cause is the assumption about the existence of a maximal length for the string. That property, along with the property that the spacetime distance between the ends of the string is invariant under global Lorentz transformations, but the spatial part isn't, implies that there can exist a reference frame where the spatial distance is greater than the maximal length of the string, while the spacetime distance is the same. Since all frames, that can be related by a global Lorentz transformation, are equivalent, this means the string doesn't break-it can't exist as such at all.

In GR a similar argument holds, since a fixed distance isn't consistent with invariance under diffeomorphisms.

What this ``paradox" illustrates is that it’s not possible to impose a fixed spatial distance in any way that’s consistent with invariance under Lorentz transformations. So the correct statement is that imposing that the string can have a maximal length in one frame implies that it will be broken in all frames.

Which implies that it doesn't make sense asking about solutions of equations that assume the existence of a fixed length and impose invariance under diffeomorphisms.

Stam Nicolis

OK, you're quibbling over "cause", which is odd, since I didn't use that word.

Your argument seems to prove way too much: if "imposing that the string can have a maximal length in one frame implies that it will be broken in all frames", then mustn't everything break?

The point isn’t that the string will break; it’s that a string with a fixed maximal length can’t exist at all, if one imposes Lorentz invariance or invariance under diffeomorphisms. And the reason is that a fixed maximal length isn’t consistent with Lorentz invariance, since the spatial part of a spacetime distance isn’t invariant under global Lorentz transformations or diffeomorphisms. Constraints can be imposed on quantities that are invariant under the transformations, not on quantities that aren’t.

I think one should remember that no one part of the ships-plus-string complex is ever moving at anything approaching light speed relative to any other part. It is acceleration that breaks strings and I don't see that the special theory or Lorentz transformations should get involved in the argument. Except, of course, right at the end when the ships-plus-string complex has reached its steady relativistic speed relative to observers in the original inertial rest frame, who then measure its length as being contracted.

David Thomas Cornwell

So, e.g., you don't think Rindler space can be used to analyze this paradox?

Can you refer me to any analysis of this paradox using GR? As far as I can tell from web searching, this paradox still has not been put to rest definitively because of the difficulty of solving the GR field equations.

I don't know of any helpful analysis of this paradox using GR, but this was never really my field. However, if memory serves me correctly, Rindler coordinates are used to describe what I once knew as Born acceleration in which there is a steady acceleration of a non-inertial frame. My point would then be that there are two possibilities: either the string breaks when the thrusters are fired, most probably with the initial jerk up to the constant acceleration, or it doesn't. If it does not break than, when the acceleration stops and the complex is travelling at near light speed, in its then inertial frame it will have its original length and will be measured by observers in what was its initial inertial frame as contracted. If it does break, the accelerations of the two ships remaining the same they will end up the same distance apart in the final inertial frame and going at the same speed relative to the initial one; and both they and the gap between them will still be measured as contracted by initial inertial frame observers.

David Thomas Cornwell

Most accounts I've seen, including Bell's, have the string breaking quickly. The idea that string might not break till acceleration stops seems odd, especially since the acceleration need not be constant - it could gradually taper off to 0.

You say "both they and the gap between them will still be measured as contracted by initial inertial frame observers", but the gap is by design kept constant w.r.t. the initial frame. Of course the moving observers will measure the gap as larger, hence the string breaking.

It seems that Bell's paradox still awaits a definitive account. That seems very odd to me, as a non-physicist. Shouldn't this apparently simple problem be settled, and made a standard exercise in relativity textbooks? I guess it's not so simple, given that the definitive account would depend on solving the GR equations.

Andrew, I think the answer to your last might be that, as this is not a research topic, active researchers take no real interest in such paradoxes. It is the sort of thing that retired physicists like me return to after careers elsewhere and include in books for a non-specialist audience. Another, similar topic is Ehrenfest’s Paradox or, even, Schrödinger’s Cat, where there is also no consensus. I personally published something on Bell’s paradox in 2005 while I was still working as an administrator – incidentally, I reread that paper yesterday and was pleased that I concentrated on acceleration but not with the way I seemed to view contraction.

I agree with you wholeheartedly that one can switch the acceleration both on and off very gently; and the string could be given a high enough tensile strength and pre-stressed to ensure it would not break. I would repeat and would like your comment upon my claim that neither special nor general relativity theory is required in deciding whether the string breaks. To repeat, no part of the complex is ever travelling at relativistic speeds relative to any other part.

I do not agree with your “the gap is by design kept constant w.r.t. the initial frame”. Where do you get this from?

Suppose:

A the ships are joined by an iron girder and fire their engines identically, starting at the same time in the initial rest frame; according to identical on board clocks, identically situated on the ship that is fore and the one that is aft, they continue to fire their thrusters on identical patterns and switch them off at the same on board time. The length of the complex as measured in the final rest frame will be what it was before the period of acceleration; in the initial rest frame it will be measured as contracted.

B the ships are joined by a spider’s thread and fire their engines identically, starting at the same time in the initial rest frame; according to identical on board clocks, identically situated on the ship that is fore and the one that is aft, they continue to fire their thrusters on identical patterns and switch them off at the same on board time. PROVIDED THE SPIDER’ THREAD DOES NOT BREAK the length of the complex as measured in the final rest frame will be what it was before the period of acceleration; in the initial rest frame it will be measured as contracted.

C the ships are joined by a spider’s thread and fire their engines identically, starting at the same time in the initial rest frame; according to identical on board clocks, identically situated on the ship that is fore and the one that is aft, they continue to fire their thrusters on identical patterns and switch them off at the same on board time. IF THE SPIDER’ THREAD DOES BREAK AT SOME POINT then, since it is so light its absence can make no difference to the dynamics of the two ships, the length of the complex, INCLUDING THE GAP, as measured in the final rest frame will be what it was before the period of acceleration; in the initial rest frame the length of the complex, INCLUDING THE GAP, will be measured as contracted.

I think I ought to answer more directly the question posed concerning the exposure or non-exposure of differences between the special and general theories. I don’t think it does

David, I know this is not considered an important research topic. But the nature of gravity very much is, so I, perhaps naively, would think that easy questions about gravity should be gotten out of the way before tackling difficult ones (like quantum gravity). And I repeat that this seems like an excellent problem for testing one's understanding of relativity.

Where do you get "no part of the complex is ever travelling at relativistic speeds relative to any other part"? I'm not sure this is true. If the gap growing in the moving frames, then the speed is nonzero; what keeps it from becoming relativistic?

That “the gap is by design kept constant w.r.t. the initial frame” seemed to me to part of Bell's own formulation of the problem. Have you read "How to teach special relativity"? And isn't that implicit in your formulation, that the ships "fire their engines identically, starting at the same time in the initial rest frame; according to identical on board clocks, identically situated on the ship that is fore and the one that is aft, they continue to fire their thrusters on identical patterns and switch them off at the same on board times"?

Your 3 scenarios:

A: By an "iron girder", do you mean that this connection cannot break or change length? I don't think that's possible. You seem to be describing Rindler space, in which accelerations are not the same at different positions.

What do you mean by "final rest frame"? Do you mean the final inertial frame of the moving objects after acceleration has ceased?

B: If the thread does not break, that seems to me an argument for a difference between GR and SR here. If Rindler space, which is based on SR, is a valid model, then the thread breaks.

C: Once the thread breaks, then we back to SR, because only the thread could cause a distortion of space-time in the gap. In that case the gap grows in the moving frames.

C:

Thank you for reminding me of that chapter in that famous book by John Bell. I reread the opening few pages last night. I had forgotten that he held very unorthodox views on the special theory. I am doing a calculation and will send a fuller rey shortly.

Looking forward to your results. I don't know if Bell actually _held_ unorthodox views of SR, but he defended them as tenable.

After a number of SS posts on RG [and all Shevchenko-Tokarevsky’s papers], which are read by some “silent readers”, who, including, modify correspondingly Wiki articles

– as that, say, happened in https://en.wikipedia.org/wiki/Twin_paradoxarticle,

- where earlier full stop, but well popular in mainstream physics and the SR textbooks, “solution” of twin paradox, where the twin traveler really ages poor homebody by his frame’s “lines of simultaneity” [more about what this “solution” is see https://www.researchgate.net/publication/322798185_The_informational_model_twin_paradox], was presented in full form, now that is a short passage “Relativity of simultaneity”

- and the article about the Bell paradox https://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox#cite_note-mac-38

is modified also. First of all in the article really – since, as that is rigorously shown in the SS posts [and essentially, though not quite directly in the SS&VT link above] application of the GR aimed at solution of both paradoxes above is really quite senseless,

- the corresponding section in the Bell paradox wiki article disappeared – as also in the twin paradox article above disappeared the section “twin paradox GR solution”, where the also too strange Tolman solution [more see the SS&VT paper linked above] was described.

Now the Bell paradox article looks as correct, and the solution in the article

“…Consequently, in the case of Born rigidity, the constancy of length L' in the momentary frame implies that L in the external frame decreases constantly, the thread doesn't break. However, in the case of Bell's spaceship paradox the condition of Born rigidity is broken, because the constancy of length L in the external frame implies that L' in the momentary frame increases, the thread breaks (in addition, the expression for the distance increase between two observers having the same proper acceleration becomes also more complicated in the momentary frame). ….”

- is, in certain sense, correct; just since in it in the SR term “Born rigidity” is applied – “Born rigidity” relates to concrete bodies length, [in https://en.wikipedia.org/wiki/Born_rigidity] as that

“…length of the rigid body in momentary co-moving inertial frames measured by standard measuring rods (i.e. the proper length) is constant and is therefore subjected to Lorentz contraction in relatively moving frames …..”

- what is the principally questionable in the SR, since in the SR it is postulated that in moving in a stationary reference frame reference frames just space is contracted, and just this “contracted space” contracts every – “rigid” and “non-rigid” bodies.

And this term appeared in this Bell paradox wiki article since in the SS&VT Planck scale informational physical model, in this case it is enough to read

https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics

- yet in 2010 it was rigorously scientifically proven that the “fundamental relativistic properties and effects” , including any “space contraction”, really fundamentally cannot, and so don’t exist,

- while really only lengths of moving rigid enough bodies/systems of the bodies are contracted; and it is explained – why and how that happens. So, say, the arms of M&M interferometer – as all other rigid enough bodies on Earth – were/are well contracted in accordance with Lorentz transformations, the satellites on their orbits and, say, clocks on Earth surface, compose, because of Earth Gravity, quite rigid systems, etc.; what is the physical sense of Lorentz transformations see the last link.

Etc. more see the links, here only note again – application in this case of the GR is senseless.

Cheers

Sergey: Are you saying the whole geometric approach to gravity is misguided? That's how I interpret "...space contraction, really fundamentally cannot, and so don’t exist..."?

Do you share the Fitzgerald interpretation of SR according to which is just the material structures, rather than space itself, which are distorted?

Do you GR can be reduced to, derived from, or interpreted by, SR Lorentz tranforms?

That would be a very provocative take.

Andrew Dabrowski

“…Sergey: Are you saying the whole geometric approach to gravity is misguided? That's how I interpret "...space contraction, really fundamentally cannot, and so don’t exist..."?

Do you share the Fitzgerald interpretation of SR according to which is just the material structures, rather than space itself, which are distorted?...”

- again – see SS post above – as that rigorously provenin the SS&VT Planck scale informational physical model, Matter’s spacetime is fundamentally absolute, fundamentally flat, fundamentally continuous, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct), utmost universal “kinematical” metrics is [5]4D metrics (cτ,X,Y,Z,ct),

- while the used now in physics spacetime has metrics 4D Euclidian (ct,X,Y,Z) in “Newton mechanics” and in Lorentz 1904 theory, and 4D (ict,X,Y,Z) Minkowski and pseudo Riemannian spaces in the standard SR and GR, where really [mostly] time dimension ict is imaginary mathematically dimension.

Both, the SR/GR spacetimes are fundamental illusions of the authors, who had only some transcendent imaginations about what “space” and “time” are – what looks as rather evident, nobody and never observed imaginary either space or time; and the “geometric approach” to both – the real mechanics and gravity is also nothing else than some illusion.

Really fundamentally nothing in Matter fundamentally can impact on the spacetime, and the spacetime fundamentally cannot impact on anything in Matter. Spacetime fundamentally is nothing else than some infinite “empty container” where everything in Mater exists and happens.

Correspondingly that

“ Do you GR can be reduced to, derived from, or interpreted by, SR Lorentz tranforms?

That would be a very provocative take. ……”

- is rather vague passage. There is no any sense in some “reducing of GR to SR”, why? – see above, moreover, really Gravity is fundamentally nothing else than some fundamental Nature forces, which in a few traits is similar to Electric Force.

Though, since the SR also based on the really extremely mighty Galileo-Poincaré relativity principle – the Lorentz transformations follow from which, despite the fundamental flaws above, it is well applicable at description and analysis of EM interaction, and real Gravity theory will be consistent with the transformations as well.

In this above there is nothing provocative, that is only real science; and, since the SS&VT model becomes be more and more known in physics, in physics in some time really correct Gravity theory will be developed instead the GR now as standard physical theory.

Besides the above this [and Electric and at east Nuclear Forces theories] theory must be based on the SS&VT initial Planck scale models, more see

https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces , and

https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force, where, including, it is rigorously scientifically shown that at least these 3 Forces are similar in that act by the same scheme.

Cheers

I did a calculation from first principles of the trajectory of a particle moving under a constant acceleration (in a well-defined sense). My conclusion is that from the point of view of observers in the initial inertial frame from which two such particles are launched simultaneously, the particles will remain the same distance apart as they increase speed. Consequently, once the acceleration is turned off, in their final inertial frame they will be further apart than they were initially. I think this contradicts one of my previous statements in this discussion. Of course, this calculation concerns particles and not spaceships or strings. But I argue that it can be used to support my contention that the string will either break immediately or never, which was my point of view in my 2005 paper. A friend of mine in Canada has kindly agreed to check my calculation but is currently fighting some very low temperatures and cannot do so for some days. As I will be out of communication for the next few weeks I will now add my calculation and interpretation of what it means to my "preprints" list.

David:

Thanks, that sounds right to me, except I think the string would break by the time acceleration ceases, since the gap in the moving frame is longer than the string. Don't you agree?

It would be nice to be able to determine definitively if and when the string breaks. :)

I think if the string does not break, that would demonstrate the greater power of GR in modeling this scenario.

Sergey:

Your understanding of physics is muich deeper than my own, and extends to new cutting edge theories which I cannot evaluate.

May ask you, very simply: In the Bell Paradox scenario, do you believe that the string breaks or not? I believe what you said earlier implies that it does break, but I want to be clear.

Andrew,

“……Sergey:

Your understanding of physics is muich deeper than my own, and extends to new cutting edge theories which I cannot evaluate.

May ask you, very simply: In the Bell Paradox scenario, do you believe that the string breaks or not? I believe what you said earlier implies that it does break, but I want to be clear.….”

- sorry for a late answer, I was last days rather busy. Again, the answer in detail is pointed in the SS&VT Planck scale informational physical model, in this case it is enough to read

https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics - see the SS post on page 2, 4 days ago now, so more see the link – and SS posts in the thread; here a brief comment.

Again, in the model above it is rigorously proven that there is no “space contraction” and etc. in the SR’s “fundamental relativistic properties and effects”,

[besides – Matter’s spacetime is fundamentally absolute one with utmost universal [5]4D metrics (cτ,X,Y,Z,ct), where cτ,X,Y,Z compose 4D space, ct is just real time dimension. In mainstream physics the spacetime has metrics (ct,X,Y,Z), i.e. really space cτ-dimension is postulated as the time dimension; so, say, in the SR antiparticles don’t exist, etc. At that everything that exists and happens in Matter has real parameters’ values only in the absolute, i.e. that are at rest in the absolute 3DXYZ space [and which as real are measured in absolute frames, instruments of which are at rest in the space];

- while in any measured in a moving in the space reference frame the parameters’ values are correspondingly unreal, nonetheless, because of the really extremely mighty Galileo-Poincaré relativity principle, using of these values in everyday physical experiments and theories is well adequate to the reality; so the SR was/is “confirmed” in innumerous experiments.

However everything is utmost clear if is considered in an absolute frame, so further just this frame is used.].

Really only impacted, and so accelerated up to some 3D speed V; and only rigid, bodies are “contracted” in 3D space [and, since most of interactions happen only 3D space, that is experimentally observed], what happens because of that all particles are some “gyroscopes” [why? see the linked paper], which all constantly and always move in the 4D space only with 4D velocities that have identical absolute values be equal to the speed of light, c, [bold means 4D vector].

At that “gyroscopes’ rotation axes” are always directed along 4D articles’ motion direction, so if a particle is at rest in 3D space, it moves only along ct-axis with the speed of light, and in the particle [which is a system [of FLEs, again see the paper], intrinsic processes in which tick with maximal rate. If an particle after an impact moves also in 3D space, say, along X-axis, its speed in ct-axis decreases in accordance with Pythagoras theorem in Lorentz factor, intrinsic processes rate correspondingly decreases in this factor as well,

- and the “gyroscope” turns in (X, cτ) plane on corresponding angle. If particles constitute a rigid body, they turn whole body in the plane. At inertial motion the body’s 4D length L is the same as it was at body’s rest in 3D space, however the, again, just observed, its 3D projection is lesser than L in Lorentz factor – what is just “length contraction”;

Besides, say, back and front body’s length clocks showings are different – front clock is “younger” than the back one on the Voigt-Lorentz decrement –VL/c2[by this way the just “relativity of simultaneity” appears].

In the Bell paradox the system of two distant on, say, LB, rigid bodies “ships” is accelerated, however the ships don’t compose rigid system. So, while every ship turns in 3D space, and their observed lengths are contracted at motion, the whole system doesn’t turn, and so the distance between the ships remains be LB.

So since all particles in the string turn, so, if the string isn’t too strong to make the system be rigid, it breaks.

Besides at that, when in the ships clocks along L have different showings, and so if in the absolute frame some events in the ships are simultaneous, but according to the on board clocks are non-simultaneous,

- but if some events happen simultaneously in absolute frames in different ships, and in identical points in the shifts, say, on their back ends, these events will be simultaneous in the ships’ clocks showings as well. I.e. in this case “relativity of simultaneity” disappears, what again contradicts with the SR, if Bell would knew that, there would be two Bell paradoxes.

Cheers

Once more: The statement that, at some initial time, two objects are connected by a string, of length less than a fixed size and that this string will break, if the two objects are moving at the same velocity, ``due to Lorentz contraction", at some later time, is wrong.

What happens is that the constraint of a fixed maximal size for the string implies that there always exists a frame, where the string exceeds this size. Therefore such a string-of fixed maximal size-can’t exist in any frame.

if the object are moving at the same velocity nothing breaks because they belong to the same inertial frame.

The paradox is considered to raise when the objects have the same proper acceleration.

Once more, there isn't any paradox, just the confusion brought about by sloppy formulations.

If the objects are subject to acceleration, Lorentz contraction is even more irrelevant. What's relevant, in that case, are tidal forces. These, indeed, enter in competition with the electrostatic forces that ensure the existence of the string and it is they that can lead to the string breaking in time. That can be found straightforwardly by solving the geodesic equation(s).

In fact it is not Lorentz contraction to play any role

Andrew Dabrowski ,

the paradox which Bell raised is a peculiar effect due special relativity according to which if two objects have the same proper acceleration, they cannot maintain distance. IN the case they keep the same distance there will be a difference in the proper acceleration.

That is a consequence of an application of the Lorentz T. as coordinate transformations formulated by Einstein and Poincare''.

By writing the LT in this equivalent form

t'=gamma-1 t - vx'/c2

the term vx'/c2 which is a coordinate time is considered to be responsible of that weird effect.

If we consider the version of LT according to Lorentz where vx'/c2 is not a coordinate time, and since between the two systems there is no exchange of EM waves, then the transformation will be just

t'=gamma-1 t

in such case the effect would be absent.

Then is it appropriate to apply SR in this problem?

Stam Nicolis

You don't seem to have read Bell's paper. It's called "How to teach special relativity", and can be found in the collection "Speakable and Unspeakable in Quantum Mechanics". Bell may have had some unorthodox views, but he was not a sloppy thinker.

By the "geodesic equation", do you mean the one that depends on a solution to the GR field equations? Those are far from straightforward. Have you solved them for the scenario of the paradox?

Stefano Quattrini

Is t = time?

Can you give a reference for the "version of LT according to Lorentz where vx'/c2 is not a coordinate time"? I didn't know there were versions of LT, I thought the only parameter is v.

Are you saying that whether the string breaks, depends on which version of LT is used?!

I have read Bell's paper. Everyone has their sloppy moments, that was one of Bell's. The other papers in the book are outstanding.

For the umpteenth time there isn't any paradox. In special relativity it just isn't possible to define a greatest spatial distance, because Lorentz transformations don't preserve spatial distances and Lorentz boosts don't define a compact group. Lorentz contraction, itself, is a coordinate artifact. The point that Bell may have had in mind is that (as already mentioned), in special relativity it's not possible to have an extended object of fixed maximal length; so a string that can break, if its length exceeds a certain limit, is a non-relativistic artifact. If it can break in one frame, it can't be unbroken in any frame, related to the first by a global Lorentz transformation. Conversely, if it isn't broken in one frame, it can't be found broken in any frame, related to the first by a global Lorentz transformation.

The reason the string connecting two objects can break is if-and only if-a force is exerted on the objects, that can overcome the (electrostatic) forces that hold the string together. Such a force-which makes the objects accelerate relative to each other-is a tidal force, which means that the objects+string are moving in a curved spacetime. In that framework the geodesic equation provides all the information: Its solution is a standard exercise in general relativity. A constant force isn't sufficient, because a constant force, on each end of the string, doesn't affect the string.

Andrew Dabrowski ,

I don not have a specific reference but I am writing a paper about it

first, you can verify easily that the one below

(1) t'=gamma-1 t - vx'/c2 ; x'=gamma(x-vt)

is LT an equivalent form to the better known classic form

(2) t'=gamma(t - vx/c2) ; x'=gamma(x-vt) found by Einstein by assuming the constancy of SOL for every observer.

(1) is also the form which was found by Lorentz in 1904.

There is a non-trivial difference between the Lorentzian interpretation where it exists a preferred frame and Einstein's SR where all IRF are equivalent .

The difference focuses on the interpretation of the term vx'/c2 = v/c*x'/c

Einstein considered it a coordinate time to allow transformations between equivalent inertial frames. The term keeps the speed of light invariant for every observer.

If we follow LORENTZ's interpretation, it is instead a variation of x'/c, the light time to cross x' at rest, due to motion. You can understand that it has a different physical value and it is there only if Electromagnetic waves are present.

So in that interpretation, the coordinate transformations must be

(3) t'=gamma-1 t ; x'=gamma(x-vt)

the Tangherlini Transformations.

Stam Nicolis ,

to my understanding you are saying that it is not a well-posed problem for SR with flat ST. The solution to the configuration must be found in a curved ST hence GR.

Stam Nicolis

Once again, you seem to be quibbling over the meaning of "cause". You agree that the string breaks, yes? Or do you want to quibble over "break" as well?

Where, in your opinion, did Bell start to go wrong in that article?

One of the reasons I posted this question is that it does not seem to be true that "Its solution is a standard exercise in general relativity."

I have been unable to find a mathematical solution using GR. If you have it, please share it.

Stefano Quattrini

I found Tangherlini's dissertation. Will that help?

Can you answer a more specific question:

Under SR and GR, would the string break or not?

I think your answer is Yes.

Andrew Dabrowski

The tension in the string as a result of the initial acceleration of the first rocket cannot travel faster than the speed of light, so in any frame of reference the tension in the string will reach the second rocket later than it starts. This means that the string will sag near the second rocket. This situation will repeat at any time in the reference frame comoving with the second rocket. Therefore, if the string does not break as a result of the initial tension, then it will not break further.

Wladimir Belayev

Thanks. The tension cannot travel faster than c, because that tension communicates information, correct? Since that applies to any point on the string behind the first rocket, most points of the the string will not move until a nonzero time interval after the first rocket starts.

However, the acceleration can grow as gradually from 0 as desired. So I think means that in your interpretation there are some situations in which the string would not break.

How do you account for the conflict between the constant gap between the two rockets, and the Lorentz contraction acting on the moving string? How could the string survive that?

Andrew Dabrowski

The tension in the rope cannot spread faster because this information is transmitted by material particles.

If the acceleration is constant in its proper reference frames, then the observed acceleration of the 1st rocket will be less than the 2nd and this compensates for the Lorentz contraction. If the accelerations are the same in some inertial frame of reference, then the rope will break.

Wladimir Belayev

The speed limit of c applies to information as well as particles. Moreover, the spread of tension does not necessitate the spread of material particles: the particles don't move much, yet still the tension spreads from one end of the string to the other. So I think in this case it's the spread of information that is primary.

Your statement about the 2nd ship accelerating faster than the first is true in Rindler space, but not under the conditions of Bell's Paradox, which specifically stipulates that the two ships accelerate at the same rate. Of course this means that the two ships have separate inertial reference frames, which complicates the analysis.

Andrew Dabrowski

Lorentz transformations are applicable for uniform motion. Accordingly, the Lorentzian contraction of length cannot be determined during accelerated motion. To determine it, the first rocket must stop accelerating. And at the moment when information about this reaches the second rocket with the help of a light signal, a Lorentz contraction can be considered.

@Wladimir

In principle the contraction can be determined during accelerated motion using GR. In practice that's very difficult - I don't think anyone has succeeded in doing it. That's one reason I asked my question.

But even apart from that, once acceleration has stopped, do you agree that the string will be found to have broken? In that case one question is, Just when did it break?

There's an assumption in relativity called the Clock Hypothesis, which as I understand it says that there is no affect on the performance of a clock due to acceleration aside from the effect of vellocity due to the LT. Is there are similar principle for length contraction?

Andrew Dabrowski

If we consider the problem in Bell's formulation, that is, with the condition that the distance from the observer to the initial position of the rockets is the same, then as the distance moves away, the signal from the first rocket will arrive with a delay by an amount tending to l/c, where l is the distance between the rockets. Accordingly, the distance traveled by both rockets, corresponding to the signals arriving at the same time to the observer, will differ by a(l/c)t, where a is the observed acceleration.. As the speed increases, the difference in distance traveled corresponding to this time will increase. Perhaps this will be enough for the observed distance to decrease no less than the Lorentz contraction without taking into account the direct influence of acceleration on the change in length..

Wladimir Belayev

"the distance from the observer to the initial position of the rockets is the same"

Do you mean the same distance to both rockets?

Why does that matter?

"the distance traveled by both rockets, corresponding to the signals arriving at the same time to the observer, will differ by a(l/c)t"

What is t?

Do you mean one rocket travels farther than the other w.r.t. the rest observer?

The scenario stipulates that both rockets move with the same acceleration "program" in the rest frame, so no change in distance in that frame should occur. In the moving frames, the rockets get farther apart. That's why most believe the string should break.

Andrew Dabrowski

If the distance to the rockets from the observer at the moment of launch is the same, then a Lorentzian reduction in length is obtained. The rope will not break, because in the proper frame of each rocket, the distance between them will not change after the light signal about the start of one rocket reaches the other. t here is the time in the rest frame, in which a reduction in distance will be observed. But this is provided that the acceleration is constant in the rockets’ proper frame.

If the acceleration is constant in the observer’s frame of reference, then the rope will break also because in proper rockets' frame it will correspond to an infinitely increasing acceleration. As rockets approach the speed of light, their acceleration in a rest frame will decrease.

The really rigorously scientific answer to the thread question is given in the SS post on page 3, 5 days ago now. Nonetheless after this post a number of really strange scientifically posts appeared, where the posters, say, write

“…If the distance to the rockets from the observer at the moment of launch is the same, then a Lorentzian reduction in length is obtained. The rope will not break, because in the proper frame of each rocket, the distance between them will not change after the light signal about the start of one rocket reaches the other….”, etc. something like

-in the paradox both ships start simultaneously and further are identically accelerated, so all time have identical speeds, at that any the light signal about anything have no any relation to paradox.

Paradox in the SR is in that according to the SR postulate that in moving frame whole Matter’ space “is contracted”, and so, since both ships are in the same frame, the distance between the ships must be contracted as well, so the string must not break;

- however, from consideration in framework of the SR of the ships’ system in “stationary” frame, the ships started from which, it follows that the distance, L0,[at the start L0=Ls, Ls, is the string’s length], between ships remains always be the same as it was at the ships’ starts, while measured in moving frame distance L is larger than the measured Ls=L0 in Lorentz factor, so the string breaks.

In the stationary frame the measured distance L=L0, but is larger than the measured Ls in Lorentz factor, because of the measured string length contraction, and so in this frame the string breaks as well.

From the last so it follows that at least the “fundamental relativistic property/effect “space contraction”” is wrong.

Though for any normal human in this there is nothing surprising – from the SR postulates that there is no absolute Matter’s spacetime and that all/every inertial reference frame are absolutely completely equivalent and legitimate

[just only provided these postulates are true the really completely illusorily, and fundamentally non-existent, postulated in the SR “fundamental relativistic properties/effects can exist]

- any number of other rigorously senseless consequences completely directly, rigorously, and unambiguously, follow; the Bell paradox is only some example.

In soon 50 years after the Bell’s paradox formulation in mainstream physics rather numerous of “solution” were developed, including in framework of the GR where “in full accordance with the SR, i.e. since space between the ships is contracted, the string doesn’t break”; all of which were/are nothing else than some strange mental constructions.

Including this

“…Once more, there isn't any paradox, just the confusion brought about by sloppy formulations.

If the objects are subject to acceleration, Lorentz contraction is even more irrelevant. What's relevant, in that case, are tidal forces. These, indeed, enter in competition with the electrostatic forces that ensure the existence of the string and it is they that can lead to the string breaking in time. That can be found straightforwardly by solving the geodesic equation(s). ……”

- really is an example. Really of course, FitzGerald-Lorentz contraction of a moving rigid body depends only on the body’s speed at any current time moment, and by no means depends on any acceleration. Any “tidal forces” has no relation to the paradox, since in the paradox accelerations can be arbitrary small, nonetheless the paradox exists. And, though “the electrostatic [more correctly – EM forces between atoms and molecules in the body] forces” indeed ensure the existence of the string, however, again, that has no relation to “tidal forces”, these forces actions depends, again, only on the moving bodies’, including the string’s, speeds.

Really some essential “tidal forces” , since the ships always move with identical speeds, and the distance between ships in “stationary” frame is always constant, simply practically don’t exist, etc.. Including, say, the “tidal forces” don’t break the ships; though, if would act at an acceleration, act on the ships quite equally as act on the string.

More about what is really the Bell’s spaceships paradox and why the string breaks, [and that in this Bell’s scheme there exists the other the SR paradox also, where the “fundamental relativistic property/effect “the relativity of simultaneity”” disappears] see in the SS post page 3, 5 days ago now, and the links in the post.

Cheers

I think my posting this question was well worth, because it is evident there is no real consensus even among physicists whether the string breaks or not. Moreover, even among those who agree on whether it breaks, no two seem to agree on the exact explanation. As an outsider, this seems rather scandalous to me.

Wladimir Belayev

"If the distance to the rockets from the observer at the moment of launch is the same, then a Lorentzian reduction in length is obtained. The rope will not break, because in the proper frame of each rocket, the distance between them will not change after the light signal about the start of one rocket reaches the other."

I can't follow your reasoning. Why does the distance of the rockets the observer matter? Regardless of whether the acceleration is constant or not, if both ships accelerate equally at all times w.r.t. the rest observer, there will be no change in the distance between the rockets w.r.t. that observer.

Sergey Shevchenko

You agree with Bell that the string will break.

You seem to have a somewhat unorthodox view of relativity, that space distortion is not real, even in GR.

Do I have that right?

Andrew Dabrowski

If in the rest frame A, located in the middle between the rockets, they start simultaneously with a constant proper acceleration, then their motion will lead to a standard Lorentzian reduction in the distance between them as they move away from the observer.

If observer B is at the launch site of the 2nd rocket and records a simultaneous launch, then for observer A the 2nd rocket starts with a delay of l/c. Since the rockets are constantly increasing their speed, this delay will lead to an increase in the tension of the rope.

An inertial frame of reference is for all of space-time, not just the local neighborhood. If A and B are in the same inertial frame, that of the rest observer, they will agree on the times of launch, and that both rockets launched simultaneously.

Andrew Dabrowski ,

“…Sergey Shevchenko

You agree with Bell that the string will break….. Do I have that right?….”

- yeah, you have above right. I indeed agree with Bell – and that quite clearly is written in all SS posts above in the thread.

However that

“….You seem to have a somewhat unorthodox view of relativity, that space distortion is not real, even in GR….”

- really is a bit vague passage. Really, of course, any space [and time] distortion is fundamentally not real, since that is fundamentally impossible, including in GR; but that isn’t a somewhat unorthodox view of relativity. That is rigorously true scientific view, while “unorthodox view” is simply the postulated unscientific illusion of the author.

Cheers