Analogies for relativistic physics

[swansont]

2 hours ago, Killtech said:

The question was, what would happen if we could meaningfully compare the length of a meter and that of a second in between inertial frames. 

No, that wasn’t the question.

Go back and read the exchange; this was about inertial frames vs. preferred frames, and your response was about inertial frames being distinguishable, but your cite was an article about relativity in compact spaces. If we aren’t in a compact space, the article is irrelevant.

[joigus]

3 hours ago, Killtech said:

English isn't my native language, so please help me understand how that only quote of yours in this thread implies there is more then one Lorentz group? I have to admit i still cannot find an interpretation of it that would conform with what you just said. Or was one of your responses maybe deleted?

Neither is mine. And I apologise if any misunderstanding was because of that.

There are infinitely many "Lorentz" groups (quite trivially) because the Lorentz group depends on a particular parameter c, the speed of light in the vacuum, that you could assume to take on a different value, and another, and another.

Would the new set of transformations it be a group? Of course. Would all the structure generated by the Lorentz group be valid? Of course. You would have time-like, space-like, and light-like vectors, the metric, the whole works. The only exception is provided by the choice c=infinity, with which you recover the Galilean group.

But you have to choose what this invariant maximum speed that generates this peculiar geometry (hyperbolically-constrained if you will) is going to be.

Other ideas similar to what you suggest have been and are being tried. Namely: doubly-special relativity. And I'm sure the are others still. Doubly-special relativity has different versions. 

But nothing you say suggests to me that you are trying to go in a similar direction --the interesting one. Rather, you seem to be repeating the known stuff with a different parameter. One such attempt would be totally mathematically consistent. We know, because it was consistent more than a century ago. So the only question is whether it's experimentally true. And we know it to be false.

[Killtech]

2 hours ago, swansont said:

No, that wasn’t the question.

Go back and read the exchange; this was about inertial frames vs. preferred frames, and your response was about inertial frames being distinguishable, but your cite was an article about relativity in compact spaces. If we aren’t in a compact space, the article is irrelevant.

  oh, okay this was merely a sidenote i made as an interesting curiosity. i think you mean this:

On 11/20/2023 at 8:08 PM, Killtech said:

of course i do. In this particular scenario it is possible to find a preferred frame and it is where people age the most and the circumference is minimal.

We can further analyse the frames with a Sagnac detector: send light around the circumference of the torus in both directions at the same time and measure if there is a delay between the in the arriving light signals. In this special topoligy we can use the result to measure the one way speed of light. The preferred frame is a frame where the Sagnac test yields no delay while all other will have one.

If we compare what happens in the same situation with acoustic signals in a Sagnac detector, we will find the exact same result. The preferred frame is the rest frame of the medium while inertial frames that measure delay in the arriving light effectively measure the wind speed.

i fear the bold text may have gone under the bus in our conversation.

But also

On 11/20/2023 at 9:42 PM, Killtech said:

Only globally it makes a difference. it is different then in regular case of SR where the global detection isn't possible.

On 11/20/2023 at 11:15 PM, Killtech said:

the interesting thing about that scenario is that the inertial frames are no longer perfectly equivalent and can be distinguished.

On 11/20/2023 at 11:15 PM, Killtech said:

Indeed the topology discussed in that paper is hypothetical since we never observed the universe to have a limited expanse in any direction so far. However, it that were the case, the existence of preferred frame is a consequence. The issue is that the finite expanse does not allow for the relativity principle to hold globally. 

i am well aware there isn't a preferred frame in general, particularly in a infinite flat Minkowski space. i am sorry if you got the impression i suggested otherwise, but i am not sure what better words i should have chosen instead.

This was the original starting point i meant on this part of a conversation:

On 11/20/2023 at 7:00 PM, Killtech said:

Actually, SR in fact contradicts your assumption. Normally, there is no absolute comparison method available to really check, but there are certain exceptions when this becomes possible.

For example consider the twin paradox. Normally we are not able directly compare the age of the twins since they never meet again, hence the paradox. However, if we assume the world topology is a kind of torus then inertial frames can periodically meet allowing a direct comparison. A torus is special in that it can be a flat space, hence SR still applies.

Logic constrains that it must be uniquely determined which twin is older, yet time dilatation enforces an age difference whenever they meet. In this special scenario SR predicts that there is only one inertial frame where aging proceeds the fastest. Similar, each inertial frame can try to measure the size/circumference of the world. Lorentz contraction and logic now demands that different inertial frames must have different results and there is only one inertial frame where the world has the smallest circumference as it is not a Lorentz invariant quantity.

So no, a clocks do tick differently in inertial frames and lengths change. we are usually unable to make the proper comparison, hence chose to ignore it.

Again, the bold text highlights this procedure is not possible in the normal case of SR.

the inability of direct comparisons between inertial frames allows the relativity principle to not just hold locally, but even apply globally. i have to admit that when i learned about SR, i had some trouble how to interpret time dilation in relativity, whether it was a "real" effect or just a mathematical artifact of the specific representation. So this lead me to look for ways to achieve a direct comparison between inertial frames and figured that such a setup might do the trick. found out there are already papers handling it. while this case is by no means applicable in general, it still helps to build an interpretation, and particularly point out that relativistic effects cannot be considered mere artifacts. Furthermore it is interesting to study this in context to what it actually means for how we define length and time.

[Killtech]

3 hours ago, joigus said:

Neither is mine. And I apologise if any misunderstanding was because of that.

There are infinitely many "Lorentz" groups (quite trivially) because the Lorentz group depends on a particular parameter c, the speed of light in the vacuum, that you could assume to take on a different value, and another, and another.

Would the new set of transformations it be a group? Of course. Would all the structure generated by the Lorentz group be valid? Of course. You would have time-like, space-like, and light-like vectors, the metric, the whole works. The only exception is provided by the choice c=infinity, with which you recover the Galilean group.

But you have to choose what this invariant maximum speed that generates this peculiar geometry (hyperbolically-constrained if you will) is going to be.

Other ideas similar to what you suggest have been and are being tried. Namely: doubly-special relativity. And I'm sure the are others still. Doubly-special relativity has different versions. 

ah, i am more clearly understand your point, but i fear there is still a misunderstanding i need to clean up first.

as a person with a stronger math background, i feel particularly unqualified to make my own physical postulates and instead prefer to start working only with what i am given. this constrains me to do only the most boring well known physics (and technically this subforum too, since everything else would go into speculations?), but there is still quite a few things in the math framework that still allow to get maybe a novel perspective.

3 hours ago, joigus said:

But nothing you say suggests to me that you are trying to go in a similar direction --the interesting one. Rather, you seem to be repeating the known stuff with a different parameter. One such attempt would be totally mathematically consistent. We know, because it was consistent more than a century ago. So the only question is whether it's experimentally true. And we know it to be false.

unlike relativity, i haven't deliberately made a alternative formulation of the relativity principle for sound. i only claim that in my particular scenario the exist one frame such that the well know acoustic wave equation holds. the invariances of that equation are implied by that singular postulate in that frame and math - but they do not really mean much. 

a coordinate can be anything weird like a angle. a coordinate transform like x→2x does not suddenly halve the length of rods. Europe doesn't become smaller then USA in area just because you transform from feet to meters - the difference is merely in numbers, not reality. so if there are just coordinates, that make the equation looks the same, it is at first a mathematical trickery that has no actual impact on physics whatsoever.

and in fact, outside of acoustics itself, in those coordinates not much else of the real world will be (acoustically) Lorentz invariant. instead one has to do some artificial constructions to find anything that is, like i tried here (look for the picture): https://www.scienceforums.net/topic/132777-analogies-for-relativistic-physics/?do=findComment&comment=1254430. the seemingly much lower speed limit is merely a coordinate illusion in the case of acoustics. Except for one thing: an acoustic wave carries energy and momentum and can interact with other objects, yet its physics limit it such that there is no possibility of it accelerating anything past the speed of sound. 

it may be a boring math exercise to use such an illusive framework to still arrive at the same old predictions we have for sound, but it is nevertheless noteworthy that one can make it work (accounting that a real rod is acoustically Lorentz variant). the one thing pointed out by this, is how such a relativity is strictly applies only to all the things directly related to the equation it originates from.

Now if we compare that to electromagnetism, what else is there actually in reality that isn't electromagnetic? is there even a singe weak or strong interaction that does not involve neither light nor charge? And since a real solid state rod is shaped by the electromagnetic force, it has to comply with its invariances, rendering the math of it a reality rather then an illusion of numbers. This seems to be the crucial difference between sound and light.

Then again, considering that the standard model faces some challenges with its predictions, like the muon anomaly or the proton size in muonic atoms, it isn't actually entirely clear to me if we do have enough evidence to fully rule out Lorentz violations by these forces (e.g. what if the muon interaction would have a minor frame dependence?).

 

[joigus]

Seems like you're trying to build an analogical model (based on sound waves) of relativistic electrodynamics...

Is that what you're trying to do?

[studiot]

25 minutes ago, Killtech said:

i only claim that in my particular scenario the exist one frame such that the well know acoustic wave equation holds.

This was not your original claim, you claimed the wave equation to be invariant.

A a person with a "background in maths" you should be aware that holding good in only one frame contravenes this big time.

Invariants appear in both maths and physics in many places and both employ the same meaning for this term, unlike some other common terms like field and vector.

 

30 minutes ago, Killtech said:

unlike relativity, i haven't deliberately made a alternative formulation of the relativity principle for sound. i only claim that in my particular scenario the exist one frame such that the well know acoustic wave equation holds. the invariances of that equation are implied by that singular postulate in that frame and math - but they do not really mean much. 

a coordinate can be anything weird like a angle. a coordinate transform like x→2x does not suddenly halve the length of rods. Europe doesn't become smaller then USA in area just because you transform from feet to meters - the difference is merely in numbers, not reality. so if there are just coordinates, that make the equation looks the same, it is at first a mathematical trickery that has no actual impact on physics whatsoever.

The are physical phenomena associated with both light and sound which depend upon relative velocity, but do not need to be associated with any coordinatem syste at all and can be detected wothout any coordinates.

The physics is there regardless of a the presence or absence of a coordinate system.

[Killtech]

11 hours ago, joigus said:

Seems like you're trying to build an analogical model (based on sound waves) of relativistic electrodynamics...

Is that what you're trying to do?

Yes, sort of. I am trying to understand how a geometric representation works to modelling the medium of a wave equation - as if the medium *was* the underlying spacetime. The idea is to make the math as close as possible to electrodynamics in GR (not the actual physics. that won't change) and find a corresponding interpretation (artificial clocks and rod). When i could achieve that, i would be able to do the reverse for electrodynamics, i.e. find a generalized Lorentz Aether formulation for gravity that however remains equivalent to GR (still boring, still no new physics. though it might be interesting for quantization of gravity).

So far that's only a pure math exercise. Now the actually interesting part: comparing the two i am interested in the special behaviors sound wave can show that light does not: sound in a vortex, i.e. the situation of the medium flowing in a curl. as far i am aware spacetime in GR cannot do that. The question behind it is if extending GR with such new (yet analog) physics could resolve the galaxy rotation curve discrepancy and therefore remove the need for the vast majority of dark matter in the universe. 

The issue: the road there is long and i have to start small... and even the small part turns out hard enough to discuss.

11 hours ago, studiot said:

This was not your original claim, you claimed the wave equation to be invariant.

A a person with a "background in maths" you should be aware that holding good in only one frame contravenes this big time.

Invariants appear in both maths and physics in many places and both employ the same meaning for this term, unlike some other common terms like field and vector.

That is the only claim i used. you do not need anything more that an equation in a single frame and its coordinates to evaluate the structure and invariances of it. all coordinates are defined relative to that one base frame/coordinates where the equation holds. if you change coordinates only, you do not automatically change the frame. energies and momentum stay as they were in the base frame (as in you have to rewrite energy in the base frame according to the new coordinates). I have made the mistake of taking that for granted, and probably made the mistake myself that i may have called a change of coordinates a change of frames, which are two different things. 

it takes a change of the metric (geometry) to do more - which requires the introduction of artificial acoustic clocks and rods. those new clocks and rods would then be needed for all measurements to ensure experiments still agree with the model. i realized that adding that step would confuse people even more so i tried to constrained the discussion to coordinates only after the initial posts.

One way or another, you do need nothing more then a quite weak physical postulate (wave equation in one frame) and the coordinate invariance follows. This is of course only half way to constructing something like a relativity principle analog for sound.

 

[studiot]

1 hour ago, Killtech said:

That is the only claim i used. you do not need anything more that an equation in a single frame and its coordinates to evaluate the structure and invariances of it. all coordinates are defined relative to that one base frame/coordinates where the equation holds. if you change coordinates only, you do not automatically change the frame. energies and momentum stay as they were in the base frame (as in you have to rewrite energy in the base frame according to the new coordinates). I have made the mistake of taking that for granted, and probably made the mistake myself that i may have called a change of coordinates a change of frames, which are two different things. 

it takes a change of the metric (geometry) to do more - which requires the introduction of artificial acoustic clocks and rods. those new clocks and rods would then be needed for all measurements to ensure experiments still agree with the model. i realized that adding that step would confuse people even more so i tried to constrained the discussion to coordinates only after the initial posts.

One way or another, you do need nothing more then a quite weak physical postulate (wave equation in one frame) and the coordinate invariance follows. This is of course only half way to constructing something like a relativity principle analog for sound.

 

I see you totally ignored my second comment in the post you replied to.

Why was that ?

 

The equations of Physics are required by the Physics Principle of Relativity to be form invariant.

Do you understand form invariance ?

 

Do you realise that Newton's 'Laws' are not form invariant as commonly formulated ?

This was the big issue, that was well understood before SR, and that special relativity addressed.

[Killtech]

32 minutes ago, studiot said:

I see you totally ignored my second comment in the post you replied to.

Why was that ?

what you refer to is the geometric formulation using the metric. i covered that in the response above and saw no need to repeat.

32 minutes ago, studiot said:

Do you realise that Newton's 'Laws' are not form invariant as commonly formulated ?

This was the big issue, that was well understood before SR, and that special relativity addressed.

Classical mechanics are invariant to translations, uniform motion and rotations, the Galilean group. Noether uses that to deduct the corresponding conservation these invariances imply. 

Edited November 26, 2023 by Killtech

[studiot]

1 hour ago, Killtech said:

what you refer to is the geometric formulation using the metric. i covered that in the response above and saw no need to repeat.

I see no connection whatsoever with my statement that coordinate systems are sometimes unecessary in Physics.

 

1 hour ago, Killtech said:

Classical mechanics are invariant to translations, uniform motion and rotations, the Galilean group. Noether uses that to deduct the corresponding conservation these invariances imply. 

That is a bold claim, because my university textbok (and many others) say otherwise.

So can you prove it.

I can definitely prove otherwise.

 

[Killtech]

4 minutes ago, studiot said:

I see no connection whatsoever with my statement that coordinate systems are sometimes unecessary in Physics.

coordinates are needed to under the hood almost everywhere in physics. this is because in order to define differentials they are the minimum requirement. and even the coordinate free formulation of equations in geometry isn't entirely free of coordinates, because it still requires smooth manifolds to define them which in turn use atlases of coordinate maps for their definition.

15 hours ago, studiot said:

The are physical phenomena associated with both light and sound which depend upon relative velocity, but do not need to be associated with any coordinatem syste at all and can be detected wothout any coordinates.

i am not sure i understand what you are trying to imply here. there is absolutely no physical phenomenon which depends in any way on your choice of coordinates. there cannot be, coordinates only affect the calculus and are entirely independent of physical predictions. the only thing that depends on the choice of coordinates is the length of your calculation needed to make a prediction - the result does not.

4 minutes ago, studiot said:

That is a bold claim, because my university textbok (and many others) say otherwise.

So can you prove it.

I can definitely prove otherwise.

Look up the Noether theorem. it's pretty standard. Specifically look up how (classical) energy conservation is derived in classical mechanics.

the translation symmetry is needed to deduct the momentum conservation, while rotation symmetry gives you conservation of angular momentum. i hope you do not question those in classical mechanics?

Show me what your textbooks says, so we can clear up the misunderstanding.

[studiot]

19 minutes ago, Killtech said:

i am not sure i understand what you are trying to imply here. there is absolutely no physical phenomenon which depends in any way on your choice of coordinates. there cannot be, coordinates only affect the calculus and are entirely independent of physical predictions. the only thing that depends on the choice of coordinates is the length of your calculation needed to make a prediction - the result does not.

This makes it very clear that you have not understood what I said,

since you have just stated very nearly the exact opposite.

So let's take it one step at a time.

 

16 hours ago, studiot said:

The are physical phenomena associated with both light and sound which depend upon relative velocity, but do not need to be associated with any coordinatem syste at all and can be detected wothout any coordinates.

The physics is there regardless of a the presence or absence of a coordinate system.

The physics is there regardless of the presence or absence of a coordinate system.

In what way does this contradict your statement ?

24 minutes ago, Killtech said:

there is absolutely no physical phenomenon which depends in any way on your choice of coordinates.

So we appear to be agreed on this.

 

However your statement differs from mine in that it seems to imply t5hat a coordinate system is necessary for all calculations in Physics.

Whereas my statement allows for the possibility that the is no coordinate system in use for some calculations.

Note this does not say that you cannot use coordinates, if you want, only that you do not need to.

Off the top of my head, examples are

Iin optics are the magnifying power of an optical instrument

In mechanics, the velocity ratio, the mechanical advantage, and the efficiency.

and of course the wave example I originally cited, which you don't seem interested in.

 

35 minutes ago, Killtech said:

Look up the Noether theorem. it's pretty standard. Specifically look up how (classical) energy conservation is derived in classical mechanics.

the translation symmetry is needed to deduct the momentum conservation, while rotation symmetry gives you conservation of angular momentum. i hope you do not question those in classical mechanics?

Show me what your textbooks says, so we can clear up the misunderstanding.

It is very easy to show that Newton's Second Law

[math]F = m\frac{{{d^2}x}}{{d{t^2}}}[/math]

 

does not satisfy the Principle of Relativity.

I look forward to your proof that it does.

[Killtech]

26 minutes ago, studiot said:

It is very easy to show that Newton's Second Law

F=md2xdt2

 

does not satisfy the Principle of Relativity.

I look forward to your proof that it does.

lets take a combination of a uniform motion plus a translation like

x′=x+vt+x0
t′=t+t0

then \(\frac{dx'}{dt'}=\frac{dx}{dt}+v\) and \(\frac{d^{2}x'}{dt'^{2}}=\frac{d^{2}x}{dt^{2}}\)

therefore \(F'=m\frac{d^{2}x'}{dt'^{2}}\)

you can go through the other elements of the Galilei group and check it remails invariant.

EDIT: shouldn't have bother to write it down myself. here you have it on wikipedia, the Galiean relativity . also latex \frac breaks on editing

Edited November 26, 2023 by Killtech

[studiot]

42 minutes ago, Killtech said:

EDIT: shouldn't have bother to write it down myself. here you have it on wikipedia, the Galiean relativity . also latex \frac breaks on editing

Where do F and F' magically come from in mechanics ?

This has nothing to do with Group theory or Noether

This simply require properly substituting for every force acting in two frames and comparing the results

You need two particles to consider this properly.

 

Consider two particles acting through a force F (x1, x2) where x1 and x2 are the x coordinates of particles 1 and 2 respectively and m1 and m2 are their masses.

We have due to the force of interaction by Newton's third Law.

[math]F\left( {{x_1},{x_2}} \right) = {m_1}\frac{{{d^2}{x_1}}}{{d{t^2}}}[/math]

and

[math] - F\left( {{x_1},{x_2}} \right) = {m_2}\frac{{{d^2}{x_2}}}{{d{t^2}}}[/math]

 

now imagine a second frame (denoted by dashes or primes) translated so that its origin is at x₀ in the original frame

 

We have

[math]{x_1} = {x_1}' + {x_0}[/math]

and

[math]{x_2} = {x_2}' + {x_0}[/math]

 

Substituting the new parameters into out master equation we have

[math]F\left( {x{'_1} + {x_0},x{'_2} + {x_0}} \right) = {m_1}\frac{{{d^2}x{'_1}}}{{d{t^2}}}[/math]

and

[math]F\left( {x{'_1} + {x_0},x{'_2} + {x_0}} \right) = {m_2}\frac{{{d^2}x{'_2}}}{{d{t^2}}}[/math]

 

 

Now please explain why you think there is form invariance between the x and x' frames, when the form of the equations in the x' frame is so clearly different from that of the x frame ?

Further the equation depends upon the origin of the x' frame, which the original does not.

 

Edited November 26, 2023 by studiot

[Killtech]

11 minutes ago, studiot said:

Where do F and F' magically come from in mechanics ?

F is a function of coordinates x and t, i.e. F(x,t) whereas F' is a function of coordinates x',t' i.e. F'(x',t')

please look up how a change of variables works in math: https://en.wikipedia.org/wiki/Change_of_variables_(PDE) . the wiki article on Galilean invariance also explains it for the specific case of Galilean transformation. i really recommend you to read and understand it first.

what you have written is a mixed picture using functions of old and new coordinates. you did not correctly/fully exchanged the variables. 

[joigus]

19 minutes ago, studiot said:

Where do F and F' magically come from in mechanics ?

This has nothing to do with Group theory or Noether

This simply require properly substituting for every force acting in two frames and comparing the results

You need two particles to consider this properly.

 

Consider two particles acting through a force F (x1, x2) where x1 and x2 are the x coordinates of particles 1 and 2 respectively and m1 and m2 are their masses.

We have due to the force of interaction by Newton's third Law.

F(x1,x2)=m1d2x1dt2

and

−F(x1,x2)=m2d2x2dt2

 

now imagine a second frame (denoted by dashes or primes) translated so that its origin is at x₀ in the original frame

 

We have

x1=x1′+x0

and

x2=x2′+x0

 

Substituting the new parameters into out master equation we have

F(x′1+x0,x′2+x0)=m1d2x′1dt2

and

F(x′1+x0,x′2+x0)=m2d2x′2dt2

 

 

Now please explain why you think there is form invariance between the x and x' frames, when the form of the equations in the x' frame is so clearly different from that of the x frame ?

Further the equation depends upon the origin of the x' frame, which the original does not.

 

I agree. There is nothing forcing F=F' a priori. We must be told how the force changes under the transformations. It is no coincidence that all basic forces depend on the differences of the coordinates (invariance under translations), they do it through the gradient of a potential-energy which is a scalar under rotations (so F rotates covariantly as an SO(3) vector) and the force law does not depend on higher time derivatives of the coordinates.

In fact, if you study in depth what kind of velocity dependence you can have in the law of force, curiously enough, the answer is the Lorentz force: vxB, where B is a vector field.

Only a small class of force laws gives you consistency with Galilean relativity.

Edited November 26, 2023 by joigus
some formatting

[joigus]

Ultimately, either Galilean or Lorentz invariance is a salient fact from experiment, of course.