Sine-Gordon model for better understanding of special relativity?

[Jarek 31]

TL;DR Summary

Sine-Gordon is relatively trivial model which seems to contain nearly entire SRT - are these proper intuitions? Maybe it is worth to use it in SRT education?

Sine-Gordon model is looking trivial 1D model: just
$$\phi_{tt} = \phi_{xx}-\sin(\phi)$$
which has physical realization as lattice of coupled pendulums, e.g. nice video:

Despite looking so trivial, it e.g.:

• has analogues of massive particles ("kinks") corresponding to complete rotation - there is integer number of them, total number is conserved,
• these particles are created/annihilated in pairs (releasing massless excitations) - from https://en.wikipedia.org/wiki/Topological_defect#Images :
• 
• there is Lorentz contraction for them - traveling kink is narrower exactly as in special relativity, mass/energy scales as in SRT:
  
• There are oscillating solutions ( https://en.wikipedia.org/wiki/Breather ), which slow down while traveling as in SRT time dilation:

I would like to propose a discussion about using it to understand/explain SRT, e.g.
- does it provide proper intuitions?
- which SRT phenomena can/cannot be explained this way?
- maybe it is worth to take it to SRT education (this PDE can be numerically solved) ?

 

[martinbn]

What is the relationship or the analogy? What do the solutions of the equation correspond to?

 

[Jarek 31]

Sine-Gordon is Lorentz-invariant model with massless (shallow waves) and massive particle-like objects (kinks), behaving as in special relativity - with Lorentz contraction, energy/momentum scaling, time dilation for oscillating.

Could you elaborate on your question?

 

[hutchphd]

Summary:: Sine-Gordon is relatively trivial model which seems to contain nearly entire SRT - are these proper intuitions? Maybe it is worth to use it in SRT education?

Sine-Gordon model is looking trivial 1D model: just
$$\phi_{tt} = \phi_{xx}-\sin(\phi)$$
which has physical realization as lattice of coupled pendulums, e.g. nice video:

Despite looking so trivial, it e.g.:

• has analogues of massive particles ("kinks") corresponding to complete rotation - there is integer number of them, total number is conserved,
• these particles are created/annihilated in pairs (releasing massless excitations) - from https://en.wikipedia.org/wiki/Topological_defect#Images :
• 
• there is Lorentz contraction for them - traveling kink is narrower exactly as in special relativity, mass/energy scales as in SRT:View attachment 285306
• There are oscillating solutions ( https://en.wikipedia.org/wiki/Breather ), which slow down while traveling as in SRT time dilation:

View attachment 285307I would like to propose a discussion about using it to understand/explain SRT, e.g.
- does it provide proper intuitions?
- which SRT phenomena can/cannot be explained this way?
- maybe it is worth to take it to SRT education (this PDE can be numerically solved) ?

This is all pretty old physics at this point:

http://scholar.google.com/scholar_u...5Z4_dCLW6EXImfGWLUNUO08eg&nossl=1&oi=scholarr

I know this for a fact because I started my PhD research using the Sine Gordon to model dislocations in crystals, and I am now old. The effort did not produce much useful stuff, and I went a different track for my thesis. It is interesting, but not novel.

 

[Jarek 31]

Sure, it is not supposed to be novel here, only to provide a relatively trivial model for clear intuitions about SRT, difference between massless and massive particles etc.

ps. But generally this field is now quickly developing (e.g. thanks to faster computers) - dozens of talks: http://solitonsatwork.net/?display=archive

 

[hutchphd]

What "intuitions" does it provide for SRT? Is it predictive in some way? I see nothing new over the nearly half-century since my initial encounter, gigabytes or not. Its utility then was partially that it was analytically soluble for solitons
It is perhaps the most interesting of a host of Klein Gordan nonlinrear forms. And it does now exist on youtube...

 

[Jarek 31]

For me, getting e.g. Lorentz contraction and time dilation from such a trivial model was quite surprising.

Sure, we can get them e.g. using "photon bouncing between 2 mirrors" clock, but here it is for model of single particle, already with quantization, pair creation, conserved total charge.
Then we can feel the difference between massive and massless particles, why massive cannot reach propagation speed, that its energy should grow to infinity there.

With entertaining mechanical demonstration like the one from youtube, a kid can feel special relativity and its consequences.
What better way to introduce to SRT do you know?

ps. Regarding recent developments, e.g. they get nuclear models extending to 3+1 D: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.232002

 

[robphy]

For me, getting e.g. Lorentz contraction and time dilation from such a trivial model was quite surprising.

Sure, we can get them e.g. using "photon bouncing between 2 mirrors" clock, but here it is for model of single particle, already with quantization, pair creation, conserved total charge.
Then we can feel the difference between massive and massless particles, why massive cannot reach propagation speed, that its energy should grow to infinity there.

With entertaining mechanical demonstration like the one from youtube, a kid can feel special relativity and its consequences.
What better way to introduce to SRT do you know?

In my opinion, Special Relativity is not so much about length contraction and time dilation
as it is about causality and the spacetime structure. Implicit in this is "the principle of relativity"
where inertial frames are equivalent.

More of a connection (a mapping if you will) to special relativity is needed.

As a mechanical device for showing something possibly resembling features of relativity,
it seems comparable to those curved fabric surfaces on which a ball travels
attempting to explain curved spacetime.

I'm not saying that it's necessarily wrong.
But I think more needs to be done to show that this is a mechanical model of special relativity.

As with many models, one doesn't want to introduce misconceptions.
So, one must be able to compare and contrast.

 

[hutchphd]

For me, getting e.g. Lorentz contraction and time dilation from such a trivial model was quite surprising.

At first glance. But you are starting from a Klein-Gordon equation and adding essentially a unitary "feedback" to it. So it is not shocking to me, but it is certainly interesting.

The paper you reference is essentially the use of the Sine Gordon equation to model magnetic domains in the nucleus. Remarkably similar to my initial (now ancient) work in crystals, only smaller! So there is nothing new under the sun. It is again interesting but not really new news.

As a learning tool for SRT I do not see it at all. What does it teach other than a coincidence? Perhaps as an example to teach nonlinear field theory it has some legs

 

[Jarek 31]

something possibly resembling

This is Klein-Gordon with more complex potential (getting massive particles) - allowing to derive e.g. Lorentz contraction and time dilation.
So what is missing to be able to use it as a proper toymodel of SRT for massive+massless particles?

The paper you reference is essentially the use of the Sine Gordon equation to model magnetic domains in the nucleus. Remarkably similar to my initial (now ancient) work in crystals, only smaller! So there is nothing new under the sun. It is again interesting but not really new news.

I am just listening to talk about this kind of crystals to model neutron star: http://th.if.uj.edu.pl/~wereszcz/sig9.html
Sure, many of these ideas were generally circulating for many decades ... what has changed are e.g. modern computers allowing to practically simulate such nonlinear PDEs in multiple dimensions - this field is now blooming for a few years.

 

[robphy]

This is Klein-Gordon with more complex potential (getting massive particles) - allowing to derive e.g. Lorentz contraction and time dilation.
So what is missing to be able to use it as a proper toymodel of SRT for massive+massless particles?

Other kinematics "effects" like the relativity of simultaneity, doppler effect, velocity transformation, nonrelativistic limit, derivation of the boost transformation, longitudinal light clock, ...
thomas precession.

For dynamics, can one model a collision in numerical agreement with special relativity (allowing for a different, but finite maximum speed of signal propagation)? Do "massless particles" travel with the same invariant speed?

I do note that the Sine-Gordon equation uses the wave-operator.
Are these "relativistic" effects actually coming from the wave-operator?

 

[Jarek 31]

Sure, it only has basic massless and massive particles - could only have effects for which these two are sufficient.
In higher dimensions it is more interesting - some gathered materials: https://www.dropbox.com/s/aj6tu93n04rcgra/soliton.pdf

Regarding Doppler, energy transform is exactly as in SRT, however, interactions are a bit simpler - there are only short-range in 1D (in 2D long-range starts), there was annihilation animation above for opposite charges, for the same charges they bounce instead.
Non-relativistic limit is there - just take small velocity and Taylor will give the same.

It is in 1+1 D, hence has only one type of time dilation - and it is there: traveling beather makes smaller number of "ticks", the boost is given by Lorentz transform:

 

[robphy]

@Jarek 31 While it might appear compelling, the next step is to convincingly demonstrate the correspondence, or at least the limitations of correspondence. Otherwise, in my opinion, it’s just a curiosity and speculation waiting for someone to more fully uncover.A little off topic, but possibly related in spirit.
https://www.tandfonline.com/doi/full/10.1080/09500340.2018.1502826
"Optical analogue between relativistic Thomas effect in special relativity and phase response of the photonic integrated circuits-based all-pass filter" - Journal of Modern Optics , Volume 65, 2018 - Issue 19
...curious... but I'm not sure if there's more of a connection to special relativty.

 

[Nugatory]

For me, getting e.g. Lorentz contraction and time dilation from such a trivial model was quite surprising.

I am deeply skeptical of any model that introduces length contraction and time dilation without relativity of simultaneity.

 

[Jarek 31]

Breather in resting frame of reference has one frequecy, in moving has different frequency - exactly as in SRT time dilation, derived from just phi_tt = phi_xx - sin phi.

Sure this is trivial model missing crucial concepts - but does it disagree within the contained ones: simple massless and massive particles?

It cannot - it is Lorentz invariant model - we can just translate solution from one to another frame of reference with Lorentz transform.