Killtech

Mass is a thing of reality, while energy is an abstract concept of our description of nature, that reality does not need to care about. the concept of energy arises conservation laws which are a consequence of symmetries, which again are arise from how we define time and space relative to natures laws. different definitions of the latter lead to different deductions of the former. Einsteins relativistic notion of space time (specifically his clocks) leads to the relation you mentioned, but if you were to use an absolute space time defined by a coordinate time and space and took the symmetries it is subject to, you get a coordinate energy unit with different relations. energy is an extremely useful tool for the calculus, but it is not real in the same sense as mass, space, time or change, which all come with their own unique units expanding the degrees of freedom of the universe, while the deducted quantities like energy, (angular) momentum all have units assembled from those fundamental ones and describe the constrains of the world.

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.

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

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. 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. 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.

what you refer to is the geometric formulation using the metric. i covered that in the response above and saw no need to repeat. Classical mechanics are invariant to translations, uniform motion and rotations, the Galilean group. Noether uses that to deduct the corresponding conservation these invariances imply.

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. 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.

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. 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?).

oh, okay this was merely a sidenote i made as an interesting curiosity. i think you mean this: i fear the bold text may have gone under the bus in our conversation. But also 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: 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.

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.

Is it a neccessary condition? Actually, aren't some big bang models of the universe compact (honestly i don't know)? what you need for comparisons are geodesics that repeatedly meet.

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? Also where did you get the impression that i mixed up the Lorentz groups? i have tried to point out quite a lot that this is an analogy, i.e. not the same thing, and i tried to stress that out a lot by prefixing the concepts with the word "analog" to make it apparent it is not the same as in relativity, but rather a mathematical analog with the constant replaced by another.

If you cared to read the conversation you were responding to, you would know you don't add anything already said. But if you put it that way, the "general" case does not apply so generally, if there are counterexamples against it. note that the compact space case is just the easiest example to illustrate the issue, so do not make the fallacy to assume its a necessary a condition. oh, sorry it didn't cross my mind that anyone on these forums wouldn't know these, so i didn't understand your question. Yeah, you misread the original post so much, i didn't even bother to try to correct you. But fair enough, especially for physicists, i suppose it is very easy to forget that there is not just one Lorentz group since only the one specific to the speed of light is used in physics. You forgot that if you take the Lorentz group and replace c by any other constant, you get another group with identical structure - and it is also still commonly referred to Loretz group, because in math c play no special role. However, the group acts quite differently on the space, and in particular combining elements of different Lorentz groups end up being elements of neither. So in this case this creates a little ambiguity and i guess i cannot entirely blame someone for stopping reading at "sound wave equation is Lorentz-invariant" and therefore missing out that a different Lorentz group was meant, which should have become clearer when reading further. Since you mentioned the d'Alembert operator, lets go back to the linear acoustic wave equation and notice how we are talking about the same structure - where only the constant is a different one. One could just write \(\Box p=0\) instead but that would omit the material specific propagation constant within, which just fuels the confusion you fell for. But since i apparently do not understand the concepts of relativity, would you please be so kind to enlighten me and the other people in the conversation on the invariances of the d'Alembert operator (or any other operator with the identical structure) and name the 3 subgroups of transformations of the Poincaré group it is invariant under?

it's this one here: https://en.wikipedia.org/wiki/Acoustic_wave_equation#/media/File:Derivation_of_acoustic_wave_equation.png the time derivative is also 2nd order but one character is missing. without a preview function i had some trouble with the latex. As one can easily see the solutions to that equation are normal travelling waves. remember you posted a more general book on the topic and you can find the equation in your link as well (https://www.scienceforums.net/topic/132856-wave-equation-in-a-medium-with-smooth-nx-refractive-index/#comment-1253798) where travelling waves are studied. i am not sure i understand that remark on wiki mentioning the standing wavefield. Here is also the slightly more general case used for traveling waves in a medium with refractive index of which this isa special case: https://wiki.seg.org/wiki/The_eikonal_equation 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. the interesting thing about that scenario is that the inertial frames are no longer perfectly equivalent and can be distinguished. in other words the relativity principle cannot not hold globally any longer and instead it defines an unique preferred inertial frame. That is the crux of the matter and it is discussed there e.g. here: https://arxiv.org/abs/gr-qc/0101014 you can find other papers on the topic since it is a known case. the wiki article has a section about the special meaning of the preferred frame in aether theories and that is the meaning here as well.

https://en.wikipedia.org/wiki/Preferred_frame that's the meaning and that is the case in that special topology.

I made that typo in my original post, and i corrected it in the repost. the value i used for \(\gamma_s^2\) is correct though and the calculation that follows is too. thanks for finally reading it. No, this is different. Each person has the means to determine if it is in the preferred frame or not. Locally the preferred frame changes nothing, so you cannot detect it by local means if your frame is in fact the preferred one or not. Only globally it makes a difference. it is different then in regular case of SR where the global detection isn't possible.