Killtech

That i do not. Of course a SI meter won't behave like that. However, i am mathematician and i will immediately look up the definition of a meter and recognize how it is constructed from the relativity principle and that the analog definition using sound signals instead of light actually ensures the contraction in the wind frame. Anyhow, coordinates don't care at all about such matters and can contract irrespective of what an actual rod may do. coordinates are instead just a question of preference and we can chose any we like. The ones using a principle of relativity for acoustic do show they achieve the correct result using an analog of relativistic calculus. A real solid state rod is electromagnetic in origin - its size ist mostly governed by the interaction of the positively charged nucleus with its electron shells via the electromagnetic force which core characteristic is the speed of light. naturally it will therefore conform to the relativity principle of light and not acoustics. But definitions are just that - convention we choose. and we can construct something which size is governed by acoustics instead of electromagnetism and therefore conform to the relativity principle of acoustics and contracts accordingly. Admittedly such constructs do not appear very intuitive (except maybe for a bat which perceives the world around it via acoustics) but mathematically they are a perfect analogy. The issue is that we live in a world almost entirely governed by the electromagnetic force, hence we become blind to anything past that perspective. building up the math that shows how sound can be treated very much in the same way if we chose to adjust our standard definitions, is quite important to get a different perspective on what we are actually dealing with.

Nor is SR able to correctly predict the bending of light due to massive bodies. It's absurd to apply physical equation to situations that they are not suited for/do not try to model just to claim they are invalid. What you are doing here is called a strawman argument :( For the linearized acoustic wave equation, we know quite well the domain it is valid for. So we have to limit our discussion to the case of a static perfect medium as long as we are discussing this particular case. Only there it does agree with the experiment and this is where the analogy with light also works. The generalization to a locally inhomogenes medium is more involved and requires different equations. However, similar to SR, the core framework and its terminology is setup in the easiest possible situation first and generalized to other cases later and i intend to do it the same way.

The assumptions of the acoustic wave equation is a homogeneous static medium, in your case the medium has two layers moving relative to each other, so you are discussing a different scenario. Coordinates don't have anything to do with that. An analogy to SR requires to have analogous conditions, hence why for the start we assume a simple medium setup, same as LAT and SR does. In relativity the analogue of your example however already requires the use of an Alcubierre metric to achieve an analoge result with light. So this requires quite a bit of GR and a lot of more involved math. Let's go slow and first discuss the simple situations before we go there. yes, and now think it through. the meter and second at rest in one frame are the same as they are at rest but another frame, right? how would you know they are the same? if you put them next to each other for comparison (but still with their relative movement) they actually won't be - length contraction, remember? so you measure the proper length of each, i.e. you measure the meter in meters in its rest frame. you do that with the other meter, too. you found out that relative to itself the meter did not change... but how could it? as you can see, the trick it to define a compare relation that makes it so.

The linear acoustic wave equation works and of course it is confirmed by experiment. There is a reason we use it after all. but if you are tying to imply that physical predictions change depending on your choice of coordinates, oh boy, i want you to try to prove that. You can then surely explain why different coordinates still produce identical predictions in this case here: https://www.scienceforums.net/topic/132777-analogies-for-relativistic-physics/#comment-1253247 (note: there is one typo in that post towards the end, where i wrote \(\gamma_s = 5/3\) instead of \(5/4\))

Coordinates cannot change anything about physics. if an equation is considered adequate in one coordinates, then it works in every possible coordinate. Coordinates don't care about velocities or any such terminology. there are coordinates which agree with the metric and therefore for these choices a ratio of two coordinate differences may be interpreted as a velocity, but in general they can be very abstract things. Think for example generalized coordinates in Hamilton Jacobi equation. The coordinate invariance of an equation is therefore a quite abstract idea to begin with. But the analog principle of relativity does work for sound. I did the math here in this example: https://www.scienceforums.net/topic/132777-analogies-for-relativistic-physics/#comment-1253247 if you want to challenge it, then you should be able to challenge the simple math in that post. it is a very simple example to dissect and analyze the question. math is more convincing argument then hand-waving. Besides, no, in Einsteins SR coordinates in different inertial frame do not mean the same thing!! The term "relativity" implies there is a dependence, i.e. things change and are explicitly not the same. 1s and 1m means something else in each frame - it goes by the name of time dilatation and length contraction. even a concept like what events are simultaneous becomes relative on the frame. It is a direct results of building a metric from coordinates that mix time and space - and there is no logical contradiction in doing so, neither for light nor for sound. in Newtonian physics these concepts are absolutes, that is the same irrelevant of the frame and location, while SR has a frame dependence of time and length and GR adds a dependence of location, that is the gist of relativity.

One can start by making an alternative postulate of relativity that you named after me. As you say, such a principle all by itself is all nice and self consistent, but it doesn't lead anywhere without a proper foundation in reality. And indeed it isn't immediately clear for what clocks and rods this principle actually holds true, apart from that these cannot be those we usually use - i.e. the SI standards. Einsteins theory and gedankenexperiments talk a lot about clocks and rods, yet there is nowhere a postulate/definition of what these devices are supposed to be in reality. And nature offers a wide range of possible oscillators which we can take as a basis for time measurement, yet they won't all produce an equivalent definition of time. So instead, physicists found that the clocks and rods needed for Einsteins principle can be deducted right from the relativity principle itself. The geodesic clocks is an example of that. The problem with such definitions is that they are constructed in just such a way that they guarantee the principle to work. The definition of the SI meter makes it plain obvious for example: there is no logical way left how the speed of light could possibly deviate - irrespective of what nature does. And these definitions/concepts are in no way exclusive to light signals but can be adapted to audio signals as well. Therefore you can construct acoustics clocks and rods that measure time and space in reality consistent with the alternative acoustics relativity principle - you can measure nature in such a way that it behaves according to it. The thing about the principle of relativity is that it is mostly math based on different references of measurement, a different representation of physics, that in some cases has its advantages over the alternative. But as predictions go, it does not make any that differ from Lorentz aether theory. It is just a question of practicability if its easier to handle the medium explicitly as a field or alternatively implicitly via coordinates or more refined via the geometry.

i used the well known linearized acoustic wave equation as a starting point because the equation is invariant under Lorentz transformation. The rest is going back in history, looking how special relativity developed from the Lorentz aether, back when light was still modelled as having a medium (with a possible wind effect) and replaying the same game but for acoustics instead - simply because it works just the same. And since all that is just playing around with a different representation of the original equation it is guaranteed not to be wrong. And that new representation suddenly made the question of the motion of the medium irrelevant also annoyed Lorentz. The same irritation seems to pop up when people are demonstrated that the math works the same for sound. Here is the post where i did it explicitly: Analogies for relativistic physics - Relativity - Science Forums - so if you have any question how that is supposed to work, the simple math is there. Imagine how much trouble it was back in the day for physicists when Lorentz coordinates were introduced the first time. Interpreting them without a precedence was a lot harder. But coordinates are coordinates, so anything goes. Let's do baby steps and confirm that they work first. invariance of a given equation under a group of transformation is well defined. Any equation has a larger group it is invariant under and for the linearized acoustic wave equation, this happens to be the hyperbolic rotations of spacetime based around \(v_s\). This is a different group then the Lorentz group of SR obviously, but it has the identical structure since the only difference is that the limit speed those trafos use, that is \(v_s\) instead of \(c\). i posted an example in this thread where i did it step by step with all the math included. This is about the mathematical formalism of relativity. It developed from the Lorentz aether theory, where light waves were analog to sound waves and were modelled as having a medium (the luminuferious aether). We can try to do all the historic development of relativity but applying it to sound instead and see how far we can get. The question is simply if the formalism allows to hide all aspects of the medium entirely through the use of tricky coordinates. and later geometry.

Help me out a bit. The post above demonstrates how one can remove the wind from the equation using the Lorentz formalism. Going further to the case of a medium with a refractive index, that is if we have a sound equation like \(\partial_{x}^{2}p-c(x)^{-2}\partial_{t}^{2}p=0\) with non-constant speed of sound \(c(x)\). I want to find coordinates such that it reverts back to the previous case where c was constant. Looking at the known solutions for that case i figured a coordinate trafo like \(t'\rightarrow t+T(x)\) will do the trick where the additional term fulfils \((\nabla T)^{2}=n(x)^{-2}\) the eikonal equation with \(n(x)=\frac{c(x)}{c_{0}}\) the refractive index. And it almost works but doing the change of variables i get an additional term \(\partial_{x}^{2}T\partial_{x}p\) when doing the second derivative of \(\frac{\partial p}{\partial x}=\frac{\partial t'}{\partial x}\frac{\partial p}{\partial t'}\) due to the product rule. So instead i was thinking to define the spatial coordinate \(x'\) in term of the rays \(x'(s)\) implied the the eikonal equation, that is \(\frac{d}{ds}n\frac{dx'}{ds}=\nabla n\). Such coordinates will only work locally, since due to the possibility of lensing effects initially parallel rays may intersect. The idea is that \(x'\) resembles the shape of null-geodesic in GR, i.e. it is intended as an analog to geodesic coordinate. Anyhow, written in terms of coordinates that follow the rays of the wavefronts, i don't see how the wave equation itself could look any different then in the trivial case.

found what i was looking for here: https://wiki.seg.org/wiki/The_eikonal_equation apart from the eikonal equation (4), there is also the corresponding wave equation (1) and it is indeed just the regular standard 2nd order wave PDE but with a non-constant \(c(x)\). (1) produces just the wavy solutions i was looking for.

same as in the article x is meant to be the coordinates \(\boldsymbol{x}\) with 3 dimensions (i am struggling with using latex in the forums without an editor with better support). but they also mix it up and sometimes it just means the first component, specifically when they use \(\partial_x\). hmm, does in that article \(c(x)\) vary only along one dimension? as far as i understand it, the eikonal equation (the one in the wiki article) can be interpreted as the path a wavefront takes through the medium, but it is not the actual equation of the wave itself. so yes, it is very closely related to what i am looking for, but not exactly it.

okay, nah, there is something wrong in the article. i don't see their factorization of the equation into 1st order PDEs to work, because \(c(x)\partial_{x}(c(x)\partial_{x})\neq c(x)^{2}\partial_{x}^{2}\) unless \(c(x)\) is constant. a term \((\nabla c)\nabla\) would creep into the wave equation.

Thanks. They suggest an equation with the simple form \(\partial_{t}^{2}\phi-c{}^{2}\partial_{x}^{2}\phi=0\) where \(\phi=Es(x)\) and \(c=c(x)=n(x)c_{0}\) i suppose, okay. That was my first guess, too. But i stopped there because simply replacing a constant \(c\) by a locally dependent one seemed a little too easy, specifically for a plane wave the the spatial dimensions can be easily treated independently. Am i just blind and missing something or is this equation one approximation too many such that the smooth refractive index here won't produce any lensing effects?

how is such an equation called? Im looking for an simplest wave equation for a non-homogenous static medium with a smooth refractive index n(x). i am more interested into the case for acoustics, though i guess it will be quite the same for optics. i am failing to google the right thing, so i though i just ask people that can answer me right away. I know the eikonal equation is related, but i am looking for the equation of the actual wave.

The speed of light is defined, not measured. Consequently there is no logic way it can deviate, because the concept of length is defined via the (local) speed of light in vacuum. Or to rephrase it, how is the (local) speed of light measured in units of the (local) speed of light supposed to deviate? Natural units express that even better by simply using \(c=1\). The value we chose for the speed of light is mostly due to downwards compatibility with older data and measurements, but in principle it can be be set to whatever value. Not exactly. You can measure it to check if you implemented the specification of the SI system correctly but other then that it bears no physical meaning. the definition of the SI meter cancels out all physical aspects of the constant and makes it a pure mathematical convention.

the constantly or isotropy of the speed of light always refers to the constant \(c\) as it appears in the Maxwell equations for vacuum and it is strictly constant by postulate. the speed of light in a medium is not - and most text name it as such so not to mistake it with the (vacuum) speed of light. Due to its specific context to the vacuum Maxwell equations, the constancy or isotropy of it in all frames almost uniquely fixes the form of that equation in every frame to the same shape (you could argue that hypothetically the ratio between \(\epsilon_0\) and \(\mu_0\) could change in the equations... but no), which practically implies it has to be invariant. In reverse, the invariance assures that all constants that appear in the equation stay the same. So for constancy of \(c\) and invariance of vacuum Maxwell are almost equivalent in SR. \(c\) is different from a spring constant which definition isn't that strict, such that it may change with rising temperatures. More generally in order to be able to speak about constancy, a quantity must be represented as a value or some other mathematical structure for which such a relation is even defined. objects in reality aren't made out of number and it is only measurement that associates them with numbers. Not all measurement methods are per se guaranteed to be consistent to each other, i.e. distances can be measured by in units of a rod or the time light in vacuum takes to travel that distance, or they could be just given as a difference of coordinate - one may find two distances to have same length with one method but mismatch with another. physical frameworks usually define everything clearly enough, including the valid methods of measurement leaving no ambiguity and therefore within such a framework it is clear if a physical entity is a constant or not. what defines such a framework is not just the laws of physics, but also a lot of technical definitions and conventions. Therefore, it may be a question of representation rather then physics if a quantity is constant or not.

from these answers i got here, i think i have a better idea how to rephrase my initial post to make the concept much clearer. It will require some knowledge of the historic development of SR and its predecessor LAT. of course i meant the vacuum case, where light does not have a medium. However, i was applying the analogy of relativity to sound in the same way as the historic predecessor to SR did, the Lorentz Aether theory, when light was still described as having a medium in vacuum, the luminoferious aether. Below is the discussion what acoustics are invariant to. So let's rephrase the whole concept: Let's start again and proceed by baby steps at the risk of stating the obvious, but whatever. The linear approximated acoustic wave equation is \(\partial_{x}^{2}p-c^{-2}\partial_{t}^{2}p=0\) where \(c_{s}=\frac{1}{3}\frac{km}{s}\) is the speed of sound in the medium we use (close enough to air). This is of course the version with only one spatial dimension, but for simplicity it is enough for now. The equation should hold in the rest frame of the sonic medium. Let's compare this model with the historic start of SR, when light in vacuum was still modelled as traveling through a luminoferious aether medium. Apart from having an equation for a transversal wave instead of a longitudinal one, that model is quite similar to the acoustic case. And indeed a wave in a medium was what inspired Lorentz approach. However, the peeps of old found out quickly, that these type of equations are invariant under a special type of coordinate transformations, the Lorentz trafos. Coordinates are just a math tool and have not much to do with physics per se, so let's ask if we can play an analogue trick for acoustics. in fact a trafo like \(t'=\gamma_{s}(t-vc_{s}^{-2}x)\) and \(x'=\gamma(x-vt)\) with \(\gamma_{s}^{-2}=(1-\beta_{s}^{2})\) and \(\beta_{s}=vc_{s}^{-1}\) (note that here cs is the speed of sound!!) transforms the wave equation into \(\partial_{x'}^{2}p(x',t')-c^{-2}\partial_{t'}^{2}p(x',t')=0\) , (i don't think i need show this explicitly, since the change in variables works exactly the same as in relativity theory) i.e. sound remains invariant under these kind of coordinate changes. The \(c_s\) in that equation however isn't the real speed of sound anymore but an artificial coordinate speed of sound. More on that later. Let's call these acoustic Lorentz transformations, short A'rentz trafos. In order to give a little more life to this math curiosity, let's consider @studiot simple experiment where we have one observer and a wall on the ground 1km away from each other. The observer emits a sound wave, it is reflected by the wall and an echo returns to the observer. The time is measured how long it takes to go there and back again... not the hobbit, just his sound. This setup will be discussed under two situations, in calm weather with no wind speed and in windy conditions with a speed of \(v=\frac{1}{5}\frac{km}{s}\) For reference, let's do it classically first, that is in calm weather it takes the echo \(\Delta t=c_{s}^{-1}\cdot(1km+1km)=6s\) to get back. When there is wind it's \(\Delta t=(c_{s}+v)^{-1}1km+(c_{s}-v)^{-1}1km)=(\frac{15}{8}+\frac{15}{2})s=\frac{15+60}{8}=9.375s\) Now let's do the wind case differently. Exploiting the A'rentz invariance of sound, we can start in the wind frame (rest frame of the medium) where we have the wave equation for and just transform to the ground/observer frame, right? For that we get \(\beta_{s}=vc_{s}^{-1}=\frac{3}{5}\) and \(\gamma^{-2}=(1-\frac{9}{25})=\frac{16}{\text{25}}\), thus \(\gamma_{s}=\frac{5}{3}\). Appling that ensures the equation keeps its form same as in the calm weather case. Unfortunately, there is a bit more to do, because if the distance between observer and wall in the wind frame was \(\Delta x=1km\) and moving with \(v\), then the new coordinates, which are its rest frame, will therefore undo an A'rentz length contraction, so \(\Delta x'=1\gamma_{s}=\frac{5}{3}\) and hence the signal will need a travel time of \(\Delta t'=2\Delta x'c_{s}^{-1}\). However, notice that the time is given in the coordinate time \(t'\) and not \(t\) and therefore we have to transform it back, thus \(\Delta t=\gamma_{s}\Delta t'=2\gamma_{s}^{2}c_{s}^{-1}=\frac{2\cdot 25\cdot 3}{16}=\frac{75}{8}=9.375s\). I think I best to stop at this simple point for now, so we can check that we all agree that these calculations are correct, and importantly that both the classic calculation and the acoustic analog of the relativistic formalism do yield the same result. Furthermore let's observe that the Lorentz invariance allows to remove the rest frame of the medium from the calculation by a proper choice of coordinates for any simple wave equations, not just for light and its aether. More on that later. ps if i gonna do the latex here, i need a working preview of the formulas i write. HOW DO I GET THAT?!?

Sorry. Bad habit of a mathematician not to care about actual numbers, but rather how we can get to them. Let's first acknowledge that of course that treatment you quoted is entirely correct and so is the result. However, if we are willing to use a different concept of space time, there is an alternative approach to the same problem. i wrote it down explicitly in this post (i hope the latex expressions do survive the quoting. if not, please go to my quoted post [Edit: the quoting broke the "\frac" latex command]):

1000 / (333+50) = 2.611 1000 / (333-50) = 3.533 i don't see the value in using odd numbers for this particular example, so i rounded them for convenience. i did this also for the speed of sound by assuming a medium that has exactly that speed so the back and forth time is exactly 6. and yes, i slipped up at one digit when adding up the numbers, sorry.

Then let's account the version of Lorentz Aether Theory as corrected by Poincaré and which in that final form is equivalent to SR - which was developed from it. In that scenario, light in vacuum is assumed to move through a medium, the aether. Therefore it is almost the same as for acoustic waves. In that theory, there can be an aether wind in vacuum just as in our acoustic case. It still turns our that it does not matter because using a special type of coordinates, the wind can be transformed away, just as it can be done for acoustics. However, that invariance to the wind holds only in specially selected coordinates of time and space. Lorentz and Poincaré have shown that even if light was a wave propagating through a luminuferious aether, it would still have the exact same physical behavior that it has in SR. The Michelson Morley experiment does yield a null result in LAT just the same. The big jump from LAT to Einsteins SR is the declaration that these special coordinate times and spatial distances are not just an obscure mathematical transformation but the actual time observed by some clocks and actual distances observed by some rods. Einstein postulates of relativity translate into just that. Maybe Poincarés thoughts on the subject can help you understand what Einstein postulates do: https://en.wikipedia.org/wiki/Lorentz_ether_theory#Principles_and_conventions if you do want to argue that the wind cannot be transformed away in acoustics by proper coordinates, i would encourage you to try to show that for the transformation i used in my post before.

For me it would make more sense to stick to the frame of the flags in either case, but we can also switch to the wind frame. Now let's make things a bit simpler and choose a medium such that the speed of sound is \(c_{s}=\frac{1\text{000}}{3}\frac{m}{s} \) in SI. Let's assume the wind speed in the scenario 2 is at \( v=\frac{150}{3}\frac{m}{s} \). Let's remember the linear acoustic wave equations is \( \triangle\boldsymbol{p}-c_{s}^{-2}\frac{\partial\boldsymbol{p}}{\partial t^{2}}=0 \) in the rest frame of the medium. Let us now do an acoustic Lorentz transformation of that equation into a frame that moves with \(v\) relative to the medium (i.e. where we would feel the wind). so therefore we have \( \beta=\frac{v}{c_{s}}=\frac{150}{1000}=0.6 \) and \(\gamma=1.25 \). It may be intuitive to use Galilean transformed coordinates for such low speeds, but let's do choose the crazy acoustic Lorentz coordinates instead with \(x'=\gamma(x-vt)\) and \(t'=\gamma(t-\frac{vx}{c_{s}^{2}}) \). Now rewriting the acoustic wave equation into the new exotic acoustic coordinates yields \( \triangle'\boldsymbol{p}(\boldsymbol{x}',t')-c_{s}^{-2}\frac{\partial\boldsymbol{p}(\boldsymbol{x}',t')}{\partial t^{'2}}=0 \). So in fact we find that if we use these type of coordinates, it becomes entirely irrelevant if there is any wind present or not. That's the whole point of Lorentz trafos as back in the day Poincaré and others studying Lorentz aether theory observed. However, we changed to a frame where now the flags are moving relative to our frame, therefore we have to account for that and within the new coordinates the (coordinate) distance between them contracts by the factor \( \gamma^{-1} = 0.8\). In the 2nd scenario we can assume that we were actually using exotic acoustic Lorentz coordinates for the flag frame, since those assure us that we can use the wave equations just so as if there was no wind present. These are very different from what we would classically use to discuss the problem, specifically mixing time and space at very low velocities already. Transforming to the wind frame by an acoustic Lorentz trafo will just reverse the trafo we had to use to get into the flags rest frame, so it brings us back to the normal classical coordinates of the problem. When there is no wind and we are in the rest frame of the medium the coordinates agree with our usual choice and the waves take \(\Delta t=\frac{2\cdot 1000}{c_{s}}=6\) where \(\Delta t\) is in coordinate time \(t\). If there is wind we can do the same and yield the same result, however \(\Delta t'\) is now provided in the quite different coordinate time \(t'\) compared to the calm weather case. Keep in mind that even if we are in the same frame, we use different coordinates to remove the wind from the equation. Now let's consider transforming to the frame of the wind. While the distance between the flags are contracted, the flags are still moving so the signal has to cover a distance of \(L\gamma^{-1}+v\Delta t_{1}\) in one direction and \(L\gamma^{-1}-v\Delta t_{2}\) in the other. So these new coordinates make it quite complicated to do the calculation in the rest frame of the medium.

i gave you the task to apply Lorentz trafos to the raw equations and verify that it stays invariant. Lorentz trafos are merely coordinate trafos and there is no physics involved in applying them - its just pure and simple math. you are using SI lengths and therefore a different geometry. The analogy works only if you construct the units of time and length in an analogue way via acoustic signals. Now we have to translate everything into a different geometry first. the distance between the flag poles has to be measured first and more importantly sonic means only. So we measure how much time sound takes to go from one flag post to another and back. In calm weather that will take roughly 6 seconds. in windy conditions it's 2.61s in one direction and 3.53 in the other, so 6.04s in total - so it would seem that flags have moved away from each other. However, we still measured time with an SI clock and have to account that a geodesic sonic clock tick rate will be slightly slowed down by the windy conditions. In fact, a geodesic clocks works quite the same setup as your flags setup and thus we can calculate that it is slowed by a factor of 1.0067. Appling that to the sonic distance, we yield that in fact the flag poles did not move away from each other in soncic-meters as well. If we now also translate the SI seconds into acoustic seconds in either case, we yield that in both times it took the acoustic signal 6 sonic-seconds to go back and forth, independently of the wind. So in terms of sonic proper time, there effect of the wind is irrelevant. Tada, the magic of frame and location dependent units in relativity. we can also reverse the situation and make a similar experiment with light. instead of the calm weather take put one flag experiment into the rest frame of the barycentric coordinates, and the other in a frame moving with a given speed relative to that. Now we are stubborn and instead of providing the results of the experiments in time in proper time of the specific frame, we do it as IAU would do and give the result in TDB coordinate time. We now realize that in that time reference it took light a different amount of time to get back and forth.

Oh, then i leave it open to you to prove how the linearized 2nd order PDE acoustic wave equations is mathematically not Lorentz invariant to the corresponding Lorentz group. You have to read Einsteins postulates more carefully and notice how there is no postulate about what clocks and rods have to be used. In fact that is left entirely open and instead those are constructed right from the postulates themselves. The construction i quoted does exactly that and as one can see, it works quite the same when applied to another wave signal. The reason is that Einsteins first postulate can be safely assumed for any simple physical model of waves when there is no other physics (particularly no other wave equations with a different propagation speed) that serves as a reference. That is why Einstein postulates in fact work for a much wider range of various wave equations when they are treated in isolation from other physics and if you construct the clocks and rods implied by treating the wave signals as null geodesics of some wave-specific geometry. Through the construction of proper clocks and rods, any wave equation becomes Lorentz-invariant even in general non-linear case. You can just always find a geometry which assures that.

The concept of relativity was quite the revolution for physics back in the day. But is it really something entirely new without any classical analoge? What i am struggling with is that Lorentz invariance is not particularly special to light/Maxwell, but is a rather a basic property of any linear wave equations. Let's consider a simple old classical model of an ideal uniformly distributed gas at rest in the lab frame and let's have a look at the acoustic wave equations we have for that case. Within this approximation the equation is a linear 2nd order PDE. Let's keep our minimal physical model limited to acoustics alone for now. Formally looking at the equation it becomes immediate clear that it must technically be also invariant under Lorentz group of \(c_s\) - though in this case it is the speed of sound and not of light. So if we were to use \(c_s\)-Lorentz transformed coordinates, the equation will always maintain it's form. While that may be unintuitive to mix time and space coordinates in such a scenario, we can embrace it and see where that leads us to. Note that any physical theory needs an interpretation, that translates between objects of the theory and real objects so it can be tested in experiments. Locations of events and distances in between them are a fundament of that and units are the means to express these. Since we restricted ourselves to acoustic physics only, it actually significantly limits our options to come up with a way to measure time. Remember that in order to build any kind of clock at all, we need some kind of physical oscillator as a reference available both in theory and reality. Let's look at the specification of a geodesic clock: The concept is based on using bouncing light signals to construct an oscillator that serves as a clock. This concept is fully adaptable to use acoustic signals/waves. In reality it might be somewhat difficult to find a structure that reflects sound but allows the medium to pass through it unhindered, however we can practically calculate accurately what that ideal clock will measure and simulate it by means that are easier to implement. (we don't use geodesic clocks for light either but also find clocks that behave equivalently to the theoretical ideal). Let's now turn to define a measure of spatial distance. Analoge to the ideas of the SI system and considering that we have defined a unit of time, we can now use that to define the unit length as the distance an acoustic signal travels in one unit of time (times the \(c_s\) constant). Apparently those definitions now mathematically guarantee that the speed of sound must be perfectly constant, no matter what. In fact, analogue to SI system we can just define its value at will as it cannot logically even have a measurement uncertainty when expressed in those new units - as implied by construction. Yet of course the speed of sound isn't really constant, not even in an ideal gas, but be aware that this constancy is just a representation trick from a special kind of transformation and not a physical effect: the clocks and lengths we are using to express it behave very differently from their SI counterparts. As this point, we have a minimalistic \(c_s\) Lorentz invariant model of acoustic physics. It won't do any different predictions to the original classic model, if we correctly transform from the new units of time and length to their correspondence in the SI system - and note that it isn't just a simple unit trafo but a far more involved local transformation because with the exchange of the metric we have also exchanged the physical geometry from Euclidean to Minkowski which is accompanied by a reshaping of the laws of physics. In this very simple model, we can now observe that most of Einstein's gedankenexperiments conducted using light signals, work quite analoge with acoustic signals, if we chose to represent them in a structurally similar framework. In fact, if we take the perspective of a bat, a creature perceiving its environment through acoustics rather then optics, it becomes somewhat natural. Now, let's consider that the speed of sound is not actually constant because even for an ideal gas it depends on the gas pressure. This will modify the wave equations by adding the refractive index \(c_s(t,x)\) to the medium that will differ locally. For example gravity induces a gradient onto the gas pressure and consequently this will cause acoustic waves to be be bent by it. However, with the definitions of the new units we won't be able to observe the change in the speed of sound directly. Instead our definitions have made the path an acoustic signal takes a null geodesic of the new geometry. So consequently, the induced pressure gradient causes instead the new metric to deviate from the flat Minkowski space and curvature appears while the wave equation maintains its original form. The predicted effect how an wave is bent by gravity pressure will be the same in either description. Finally, one can mention that the acoustic waves discussed here are pressure waves, and because the speed of sounds dependence on the local pressure, two sound waves passing through each other actually weakly interact. Therefore a more accurate model of acoustic waves is a non-linear PDE. The point of this lengthy post is put up the question, how much of relativity is indeed special and what part of it comes from physics and which is mathematics and representation. Of course i selected only very specific aspects which show similarities and there are obviously very significant differences between acoustics and light signals in GR in general, yet we should be careful to account every difference to physics alone.

I am quite familiar with non-linear equations. i work in finance and we have quite a bit of non-linear models to tackle as well. In terms of physics i have studied a bit soliton solutions which only appear in non-linear equations. But i have to admit that i am quite new to GR. I mean i knew that Maxwell equations are non-linear in GR through their contribution to energy and therefore to curvature, but for gravity i haven't somehow realized it by looking at the equations. The terms hidden in the curvature tensor easily slip ones attention. But again, the issue with non-linearity is that it is in fact way more complex not just in terms of solutions, but how many different kinds of non-linearities there may be. In comparison linear models are structurally almost uniquely determined. So having a lot more possibilities allows to fit any data at the cost of a significantly more complex parameterizations to calibrate - success of AI tells a story about that. But if we were to train an AI model to predict the time evolution of physical gravity systems based on our observation data, i bet it will come up with a quite different model which will deviate from Einstein's GR mostly in the extrapolating regime where we lack sufficient observations. Non-linear models are usually quite terrible at extrapolating. So there is a good reason to not blindly trust such models and thoroughly tests them through out all regimes. This is why i want to understand how well we have tested the particular shape of the non-linear contributions. Hmm, sounds familiar, a bit like perturbation theory up to the first order. Thanks for the explanation. The linear field equation is admittedly much easier to visualize and understand the core of the theory, so i suppose I should have started with that. Ah well.

That's what i have intuitively assumed. That scenario however allows for very accurate tests though, since the proximity allows to minimize the influence of unknown factors. Of course by "experimentally", i meant to include observational tests as is the case of Mercury's perihelion precession. our ability to detect gravitational waves directly is kind of new though, so i am a bit surprised to hear that they already detailed enough to allow such analysis. Or is that more based on indirect observational data for which we have more history? The issue with non-linearity in a purely empiric model context is that it requires far more detailed and precise data taken from different regimes to determine its exact form because unlike in the linear case, there are way more degrees of freedom/parameters the non-linearities can express themselves. Einsteins postulates are made based on experiences in a weak field regime and produce a very particular kind of non-linearity, but there is no immediate guarantee they still hold everywhere through the strong field regimes. Hence, it is important to test their exact predictions with observation data and look for deviations. Given how much dark matter GR needs to be somewhat consistent with observation and the singularities it produces, i am struggling to understand how well verified the theory is throughout all the regimes. It is of course already important to know that gravity cannot be fully linear, but such an observation/distinction alone does not reveal that much about the detailed nature of the non-linearities in reality. Anyhow i didn't know there is a linearized model of GR. Guess, it makes sense to have that as a computationally much easier proxy. Have to look how that works, thanks.