Killtech

https://arxiv.org/abs/gr-qc/0101014 if the math agrees with the experiment, what more do you want? https://www.scienceforums.net/topic/132777-analogies-for-relativistic-physics/#comment-1253247

That post was the argument. How am i supposed to convince you when you will simply skip any tangible argumentation with math? And the other posts you respond in a way making it obvious you haven't even read them properly. Why waste both of our time? Either you are interested in the discussion or not. It's impolite to continue a discussion disrespecting your counterpart by not bothering to listen. You do the math to make a prediction for an experiment. Since the scenario is chosen for a well know case where we already known the outcome of the experiment, we can check if the prediction is able to reproduce the know result using a different framework. Apparently the flaw in your argument is that you haven't read the post in question. or is it possible that latex expressions are bugged and not displayed for some members? i have a few technical issues with these forums myself.

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.

Please read the whole post before responding...

And it does agree with the experiment. I did the math and pointed it out to you a few times that it makes the correct predictions. Since you didn't/weren't able to pointed out any flaw in it, it stands that it is correct. In some situations a sonar puls is the best available method to measure distances, for example under water or when you are a bat. Similarly seismic waves can be used for locating interesting features within earths mantle and core. However this method does not measure a distance in meters but rather acoustic propagation time. You do not seem to understand the difference between meter distance and an acoustic distance. They are just not the same thing! However, in mathematics the concept of a metric space / distance is significantly more general and both concepts are mathematically valid, so we can build a physical model on either of them. A conversion from one to the other is not always possible if detailed information about the medium along the path is missing. Now note what happens to sonar ranging in when used in a strong current (i.e. by a torpedo): the signal travel time will be affected resulting in a change of measured distances. A rod (or a torpedo) which size is measured by sonar will therefore contract, because its acoustic length changes while its metric length remains the same. The relativistic theory just happens to describe this phenomenon as it has mathematically the same origin. Therefore the experiment will indeed observe a size contraction, but size does not equal size.

I can understand your both hesitance, because indeed this starts off as a very physically unmotivated mathematical exercise. First and foremost, i need to establish that the math of relativity in light has can in principle be also applied to sound in an analoge way, if we were to use some unintuitive artificial measurement definitions. The idea of this analogy is not intended to provide novel physics for sound nor make it easier to calculate. However, if we could somehow achieve it even by very artificial means, then we could potentially use sound as much easier experimentally accessible model for relativistic effects. The ultimate use case would be to do a lot of math exercise to bent sound physics artificially look like GR, just so we can study a GR solution like Alcubierre metric in the lab and get an understanding where the negative energy (in such a geometry) comes from so we get a better idea what to look for when we try to build the same but for GR. With the math setup, we can now start to build a rod, i.e. an experimentally viable model, that actually contracts around the speed of sound. The first challenge is to find a construct that is analog to an atom, but instead of using the electromagnetic force to define its size, it ususes sound instead. So consider some small devices that constantly emit a loud sound. Now, we know that sound waves do interact with objects and can exchange momentum and energy with them, so in principle, if the emitters have low enough mass, their constant emission will cause them to noticeably repulse each other, and the repulsion grows stronger the closer they are. Let's now take a large amount of these and use some external force to make them clump together in an area of space. After some time we expect an equilibrium to set in with the forces at balance and it should look like this: i.e. we expect a lattice-like structure to form, a very artificial model of a solid state using the "force of sound" and a crude analogy for a rod build via acoustics. Now what does happen to this grid when a wind appears? (assuming the emitters are prevented from being blown away by the wind and we use such a medium and setup where no turbulences appear). The balance of forces will change, mainly the repulsion parallel to the direction of the wind will be weakened, hence the rod will start to contract. In fact, the contraction formula will be exactly analog to relativity - but with the speed of sound replacing light. In fact, we can now fully make use of the invariance of sound wave equation here to simplify the handling and in particular, the otherwise weird choice of coordinates now provides a description of the grid that does not change depending on the wind.

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.