Killtech

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.

How well is the self coupling experimentally verified and understood? Though if we go away from a linear field like thinking, it makes intuitively sense, i would still like to see what evidence we have for that. Not looking for a general test of GR though. i admit that singling out an aspect of a theory without a proper competing model that differs only there may be hard to do. Maybe it is just enough to have a theory with a limited propagation speed and therefore replacing Newtons gravity with Maxwell-like equations for gravity would do the trick? What would that predict for the rotation of Mercury's perihelion?

ideally the latter is the quantized version of the former and in particular in the classical limit we should yield one from the other. But since the latter does not exist yet, we have to study the specialties of the former to understand where the trouble comes from. In a closed physical system this cannot happen and it doesn't actually matter if the black hole is made of mass or something else - the curvature is caused by the energy stress tensor after all. Therefore the back hole could be made up entirely of light of sufficient intensity in principle. Clearly the gedankenexperiment violates energy conservation in Newtons and Einsteins physics all the same, so let's just assume it's an open system and treat it analoge to a driven oscillator. The physicists that formulated that old gedankenexperiment didn't intend for it to be a realistic situation but wanted to illustrate a particular question about propagation of gravity. Just as it was valid to show how Newton's gravity is instantaneous compared to Einstein's, it is still an interesting case to drill down the properties of gravity to some of its essential aspects.

But does gravity actually slow down gravity? Let's consider the very old gedankenexperiment: what would happen is a black hole suddenly disappeared, how long would it take for different observers around it to notice it? In particular, if we say gravity travels at \(c\) and for the outside observer clocks within a gravity field are massively slowed, then the speed measured in term of the outside observers clock should appear much slower. Or let's rephrase it: given a massive object with a time dependent mass/energy of \( m(t)=1+\sin (\omega t) \), i.e. producing a field resembling a longitudinal wave - how fast would the curvature changes it produces propagate? what wavelength would that curvature field have locally? going back to the graviton, or in fact any virtual particle. They appear in Feynmen diagrams as part of the integral kernel. But that kernel still contains these paths by physics, which can be interpreted as allowing only somewhat possible paths - that is paths which require more then infinite energy have a contribution/probability density of 0. So if virtual particles are constrains by the same geometry, how are they able to contribute to the amplitude starting at a point within \( r_s \) to a point outside? Well. of course we do not have a theory of quantum gravity or in fact any quantum theory that can deal with curvature, so i guess the question probably does not have a good answer as of now?

test sentence \[ \rho^2 \] ... do i have to post to get a preview? 😮 [math]x^{2}[/math] man make it work! inline test \( \sqrt{2} \)

Maybe let's first settle much simpler question about just GR: is a gravitational wave subject to gravitational lensing? The question is two-fold: what does the theory say and what the experiment. the latter is more relevant but probably still unclear, since it is not that long ago that we were able to even make a first detection. a gravitinon would have to follow the same propagational behavior as the wave.

Hmm, what an inkonvenient technical pause to the conversation. Sure, so you reduce time to the instructions to construct it and in the case of SI, we have very specific instructions. That definition is well defined, sure, but it is decided upon by a committee of people and not actually nature. If you look into the details, you will find a lot of instructions how to correct the Caesium atoms readings for specific effects and you will additionally find a passage explicitly stating that effects of gravity must not be corrected. These seemingly arbitrary specifications make it clear that it is a convention we come up with, same as Einstein's synchronization is and that raises further questions. A mathematicians first natural reflex here is to ask, what other choices of such instructions could we use instead that lead us to a well defined time? are all of those equivalent? And really, thinking a bit about it, it turns out that geometry rises such question and has figured out the answers long ago. It turns out that a metrizable topological space with a differential structure allows for way more one Riemann manifold to construct on it. So we know that there is a large set of possible alternative concepts of time that are not isometric to the one we use. We can deduct how clocks of the different definitions of time relate to each other, and we can formulate how the laws of physics and their symmetries look from the perspective of other alternative clocks. We cannot go wrong when we change conventions our theories work with, can we?

And you are not entirely wrong here. However, math is a mean bastard when you go deep into some seeming trivial details. There is just no one singular way to represent and model physics because it turns out you will requiring quite a few of additional assumptions that cannot be experimentally verified. Those are technically conventions. The choice of those will however have an impact on the resulting invariances and therefore laws of physics. Uff, that isn't so easy to answer. But how do you define a clock or a time measurement in general? let's say we are in a different universe then this one with other laws an all (or just in a computer simulated reality like in the matrix films). How do we define in a general abstract case? Usually it helps by asking what do we need time measurements for to solidify which axioms those measurements have to adhere to in order to fulfill that purpose. This is maybe a mathematical approach physicist don't often consider. A key aspect of measurements is their ability to compare results and translate real world relations into numeric values we can do calculus on. The mathematical concept of a metric very accurately reflect this fundamental of measurements. But given one metrizible topology, we know from mathematics there is way more then one possible metric. Note that nature does not actually need any numbers to work. But we do to model nature. So for nature a smooth topology is enough and it is us that adds a numeric structure of a metric with its comparison relation. in doing so we introduce a lot of untestable assumptions (that implicitly define the metric we use) that mix with the laws of nature into their familiar representation. As a somewhat analog example consider how different symmetries look from the perspective of different coordinates. The same is true for geometries / metrics. But your initial believe holds in a sense as long as the model suffices to Noethers prerequisites: if we choose a suitable metric, it will still have invariances and well handable laws of physics - those will depend on chosen metric / geometry though. Technically you could however work with violated Noether assumptions... but that will be very annoying to handle a system where the total energy isn't conserved but evolves by a deterministic function which means the laws of physics will have some nasty absolute time dependence. I did pick the TDB time coordinate as a time metric explicitly because it guarantees a Galilean invariance with a corresponding energy conservation but requires altered laws of physics that fit those while still reproducing the same relativistic physics.

I have written that a few times, but you are seem keen on jumping over it. the problem is the galaxy rotation curves as indicated by the mass-to-light ratios anomalies since we have no solution to this problem. If you cite dark matter as a solution, then note how that hypothesis works: given a deep mismatch between the GR model and observation we introduce a completely unknown field-like degree of freedom to GR, just so generic and flexible that it can fit almost any deviation. But that flexibility renders it no actual quantifiable theory but free empiric parameters that must be fit experimentally. And experimentally, our means are not good enough to do even that to a satisfactory degree. Sure there have been a lot of proposals to what dark matter may be in order to somewhat constrain these parameters but let's be honest, so far we haven't come far understanding it at all. Adding torsion to GR is just another alternative to fix the same problem and it is a hypothesis similarly generic to dark matter with its own degrees of freedom, meaning it opens up a large class of possible models with wider range of predictions - same as dark matter does. And since the concept is similarly flexible, it can potentially make most of the dark matter obsolete. Unless you consider the question of dark matter resolved and well understood, there is still a big problem to solve. Look how much we try to learn about dark matter from simulations. For these the concept is very feasible - albeit admittedly just as satisfactory as the dark matter explanation. In terms of actual observations i was thinking that one could potentially try to do a Segnac experiment in a much smaller system, like around earths orbit via satelites. But of course here the effect might be much less prominent and thus harder to detect.

fair enough, bad wording. an interferometer for the purpose would have use a pretty extreme wavelength. in my mind the plastic model of a spiral galaxy was shaped more like an airscrew. but fair enough, a perfect flat disc wouldn't be able to move a medium without friction. you got the point anyway. I know the prediction GR makes. But i also I don't know any kind of experiments which would already reliably disprove this possibility, though i may be wrong - so correct me if i am. For this scenario the theory does not seem to be well tested and its predictions are therefore in an extrapolating regime. But of course that does not disprove it in any way. As i stated in my opening post, the purpose is to formulate/sketch the idea for an experiment that might be looking for physics beyond the currently known. After all the purpose of experiments is to test a theory - and that's especially interesting for things it hasn't been tested for. well, look up the galaxy rotation curve issue which shows a big discrepancy between model if we only consider the visible matter. The introduction of large quantities of dark matter outside the galaxy is the preferred hypothesis to mend the discrepancy between the model and the observation. MOND is another approach. however, both are merely assumptions, neither is considered experimentally established. A rotating aether or equivalently introducing torsion degrees of freedom to the affine connection GR uses might be another alternative. Anyhow, the issue is that a galaxy spanning ring is not particularly ideal to build for an experiment - yet this is the case where the discrepancies with observation are the largest. Is there a better way to test such a hypothesis?

Off topic, but you rise an very important point: how do i use LaTeX on this forums?

Yes, the gedankenexperiment is to place a Sagnac sensor (thanks for the proper name) moving along a galaxies disk (i.e. no relative rotation) and use it to measure the angular velocity at that radius to compare it with usual methods of determining a galaxies rotation curve from afar. Our current physical model would expect these methods yield agreeing results. However, i suggest to test if this assumption holds in such circumstance. For example one possible outcome could be that the Sagnac interferometer would yield an angular velocity close to our current model of gravity without the presence of dark matter, which is much lower to what the outside observer measures. The idea is merely based on the cylinder case where also a Sagnac test is used to determine the difference in the one way speed of light. I only mentioned SR because it allows to study this surprising case and its implications - particularly the concept of a preferred frame in relativity. Maybe consider doing a similar experiment in other area of physics to understand the motivation better: Let's take a miniature plastic model of a galaxy and let it spin under water (or better, a superfluid). The spinning galaxy will also cause the surrounding medium to partially flow along the rotation in a curl flow, a vortex will form. We conduct an analoge of a Sagnac sensor but instead of using light, we use acoustic signals which propagate via the medium. Calculating the delay between the signals works similar to light except that the medium defines the one way speed of sound along the signals path. In this case the sensor therefore only measure the angular velocity of the galaxy relative to the rotating medium around it. Because of this the measured result will deviate from the actual angular velocity observed from afar.

It is known that the one way speed of light cannot be measured under normal circumstance, however there are a few special interesting cases. Let's consider the situation of the twin paraoxon, except we assume the world is shaped like a cylinder thus having a finite length in one direction. In this unique settings we can still discuss the twin paradox within special relativity where the two twins travel exclusively in inertial frames, yet are able to periodically meet each other since geodesics around the circumference of the cylinder form closed loops. These meetings allow to make age comparisons locally and logic requires that there must be a uniquely determined, frame independent answer which twin is older. So globally inertial frames are not strictly equivalent in such a case and indeed there can only exist one frame where aging will happen the fastest. In the same situation each twin could also send two light signals (instead of his twins) in opposed directions around the circumference, wait for their return and determine the delay between them. This delay is the difference in the one way speed of light along those two directions. There is only one frame where both signals arrive simultaneously and which is also the same where clock tick the fastest. So we have a case in SR where a preferred frame exists that acts a bit as absolute rest frame. This is a pure theoretical scenario since we have no experimental indication for a topology with nontrivial homotopy class (i think). But I was contemplating if we can make use of these considerations in a normal situation. After all we still can send light along closed loops in two directions and determine the delay, for example along a planet's orbit. But i figured that will merely measure which angular velocity the light ring has. In general, if we consider the difference in the one way speed of light to be a vector field, we find that a closed loop only measures its average difference along the loops tangent vector, hence it will always yield a 0 result for constant or conservative vector fields. But if the field has a curl, it will not. So... what would happen if we place such a light-signal ring along the outer rim of a galaxy? It is an open issue that our current physical models find the observed galaxy rotation curve to be quite abnormal. Is it thinkable that the angular velocity obtained from observing the delay in light signals measured by the ring will differ from the velocity obtained by other means? This is another approach to Mach's question of absolute rotation. Because the possibility of a curl in the one way speed of light could be practically considered as space itself being partially dragged along a rotating object, therefore the object will perceive a lower angular momentum then a distant observer would visually assume. Such an additional degree of freedom would allow a galaxy to match the modelled angular momentum curve and with the angular velocity of space added on top still obtain the observed behavior. This is of course pure hypothetical/speculative, but the question whether that makes sense to look for experimentally is not. finding things that may be of interest for measurement is always looking for possibilities beyond the established - hence why i posted this here rather then the speculation forum section. As far as i understand GR, it isn't able to account such a possibility as that would at least require to introduce a torsion degree of freedom to its connection - thus this is looking for new physics. I don't know of any experiments that would have discounted such a possibility already, or does someone know any?

Indeed me too. I am still figuring if such an approach is viable in general and if there are already similar concepts people worked on that may help here. As for the details, i am still figuring out how the correction function applied to clocks in TDB looks like when generalized to any possible theoretical situation, i.e. what the metric that TDB units imply looks like relative to the usual metric of GR - because that's what mostly defines the transition between geometries. It doesn't work like that. A such a big change in geometry also is accompanied by a change of some symmetries. Particularly having a locally dependent speed of light does not work too well with Lorentz invariance. Other geometries means other laws of physics, and in this special case the laws of physics using a time (metric) that is shared between all frames and locations (effectively is an absolute time by Newtons definition), their invariance can at best be only Galilean. This is the form where the equations can start to resemble those of fluid dynamics.

You have to be careful with such statements. c is a very different constant from all else given its connection to the definitions of time and length. As Poincaré nicely demonstrates by the example of how Astronomers determined that c was constant, that in fact they had to assume how light moves through the vacuum beforehand in order to make measurements at all, hence they showed that if c is const then c ist const. Therefore if you account how the interpretation ties into the model, the experiment actually measured the function c(c). His example is a good point of study for the general unresolvable interdependence. Besides, with the current definition of the SI meter, it is logically impossible that c can vary in any way. i am aware that this definition was chosen much later and for a reason. But let's view it the other way around: if we simply ignore what c "really" does, define its behavior ourself instead and make all experiments maintain that convention (SI system), would we be able to find out that we are "wrong"? For as long as a meter defined liked this provides a well defined measure for length (a rod that has some hysteresis when moved around won't suffice the axioms of a length measure), then no, because all experiments will just provide some results and we can always will find some model that reproduces them. The question is a chicken or the egg causality dilemma between definitions of units and laws of physics. if one assumes a constancy the other inherits it. In reality we can only observe how physical entities change relatively to each other but never how they change absolutely. So what we can do is compare two physical processes where c is involved against each other and compare that c obtained from one is same as the other. A deviation would be interpreted as our model/understanding of one the processes is wrong. I know experiments were conducted to check how stable constant were, but we have to be a lot more careful interpreting the results. Well, one can package all components of Maxwell into a rank-2 energy-stress tensor and the field equation of GR provide the its time evolution. The analogue could work for gravity... maybe it would just effectively replace the Einstein Tensor with another and reshape the remaining Maxwell energy stress tensor into a trivial geometry. With the metric trivialized to globally Euclidean and the two tensors becoming analogue in interpretation, one could combine them into one thingy. So i am not convinced this must lead to a less simple formalism. There are a lot of reasons to ask questions. We still haven't solved the issue of quantum gravity. Looking on a problem from another perspective may help, specifically in a flat Euclidean geometry quantization might be easier. Also reshaping spacetime like this allows comparison to familiar classical fluids and their equations. I would be interested to study how light inside a warp-bubble solution compares to the situation of sound waves inside a cockpit of a supersonic jet. Maybe if we can bring sound and light into a comparable metric, an analogy which might help us know where to look to find solutions to circumvent certain speed limits. There is also an unresolved issue with galaxy rotation, which i have a hypothesis i an interested to test experimentally. I wanted post on that later, but since it has connection to this concept here, i decided to post this first.

yes and no. you are right that i can do a lot of things already with the coordinates but not all of it. but coordinates are one thing, units another. energy for example does not depend on the choice of coordinates (apart from the frame), yet its unit is made up of length and time units. using a locally different time unit as a basis for energy defines a very different physical entity that actually belongs to a different geometry. Noether guarantees us that as long as we can find the symmetries in the new geometry, there will be a alternative concept of energy which will be preserved. Coordinates cannot help with the energy question. Consider Euclidean clocks which behave differently between frames compared to proper time, particularly lacking the singularity at c - if we insist on those to measure an alternative energy, we get diverging results. It requires very different laws of physics to make that new energy (and its action) produce the same outcomes as the relativistic Minkowski geometry does. That is what i am aiming to look for by changing of the metric and particularly hope it can provide a direct translation mechanism in between these different concepts of energy and geometry. I think it is also the metric which carries units while coordinates are usually treated as dimensionless. of course we often use conventions like c=1 to hide and simplify the equations. It is worthwhile to spend some time reading Henry Poincaré's notes on measuring time and what that means for the speed of light. So how would be even notice c to vary, if even Poincaré's corrected Lorentz Aether Theory concludes the same result for the Michelson interferometer? Any attempt to measure c requires us to be able to measure time and length or at least assure we can maintain intervals of constant lengths for the measurement. Yet all the definitions of length and time we used were purely electromagnetic in origin. And it is precisely there where we go the full circle. if our concepts of length and time are implicitly based on c and use it as a reference rendering it constant, then all our measurement will show exactly this and none will be able to record any deviations unless it breaks from the specifications of the SI system. Look at the definition of a geodesic clock @Genady posted earlier and consider how it is affected by a locally varying c(x). it demonstrates how the assumptions on the speed of light is tight to definition of clocks and that it will work with any assumption you put into it. Now consider using that clock in reverse to measure c - those are two side of the same coin. We do know from experiments that clocks run out of synch depending how close they are to a gravity well. We can interpret it as usual, or we can assume that our reference oscillator for time is affected by some local effect and needs correction - same as we had to correct for thermal expansion of the original meter bar and same as the official SI definition of second via Caesium lists required corrections singling out gravity as the only local influence that must not be corrected. If we do that however, we speed up time locally at the immediate consequence that c(x) gets a local dependence. isotropy of light is incompatible with isotropy of clocks. I am open to the result of whatever the metric transition will require. i highly doubt it will be however a scalar field theory, after all the existing degrees of freedom gravity has in GR embedded into the geometry have to go somewhere. I do however think there will be at least one dominant scalar field, especially in the Maxwell equation: a refractive index c(x). But if gravity travels at c and has transversal waves, it likely needs quite a few field equations... actually i would think it might look analogue to Maxwell with two force fields, a vector current and a scalar density. each dimension reflecting one degree of freedom of the GR metric tensor. actually i stumbled on this recently: https://arxiv.org/pdf/gr-qc/0205035.pd . haven't yet time to go through it, but it goes in a similar direction albeit with a different starting point.

Yes, this is indeed where i want to start from. As I understand it, the metric is the important bridge between model and experiment and one finds almost all measurements contain the units of lengths and time. The interpretation isn't actually trivial, because looking deeper at the definitions, one has to make quite a few implicit assumptions in order to define any kind of unit. So the question arises what happens, if we changed some of those assumptions/definitions? We could for example assume a real physical oscillator, on which frequency we base our unit on, is influenced by certain local conditions and consequently we want to apply location and frame specific correction factors to counter these effects. But that interests me for another reason: In science we want to test the assumptions of our models against experiments and in the process it's sometimes easier to formulate counter-hypothesis and check those instead. For some postulates that runs into logical problems: e.g. the isotropy of the speed of light. If we assume a model where it isn't a constant, we run into contradictions with how measurement in experiments works which still implicitly assumes otherwise. If we however account those new assumptions in measurement, the required corrections will yield different measurements as we practically use a different metric (implicitly also geometry). In that case, we may end up doing what i want to discuss. If physics can be reformulated into a different geometry with different laws of physics such that it yields identical predictions, then a counter-hypothesis may prove physically equivalent to the base postulate. In that case i would consider such postulates untestable and treat them as conventions. Before i can go deeper exploring various approaches for physical models, i want to first get a good understanding of the the fundamental relations between a model, its interpretation, measurement and experimental testing. Furthermore, what exact role does the metric plays in this and is the way i think about it correct? I have looked tiny bit into GTG and it does sound quite interesting, though i have not yet understood how its interpretation works. I will have to look better into the techniques applied, though the idea to start a theory from the action and deduct the model from there is inconvenient for my case, because my starting point is indeed the metric and i know too little about what the resulting model may be. Also, i'm not sure if a flat Minkowski space is a good basis for a formulation where c(x) is deliberately made non constant. During the week i usually don't have too much time to focus on such topics. For now you gave me plenty of stuff to read

Okay, i totally failed explaining what i mean by "changing the metric". In my defense i don't know any appropriate terminology for that particular procedure and googling didn't help... so i turn to the forums. Maybe let me try to rephrase it: Given a Riemann manifold (X,g) we can also consider it a simple metric space (ignoring its differential structure for the start). Now let's consider the identity map id of X to itself. I want to introduce an alternative metric structure on X to make it a different metric space (X,f). In that scenario id also becomes a map between two metric spaces and the intention of the choice of f is that id won't be an isometry. Now accounting that X is a smooth manifold we have two distinct Riemann manifolds, each with its own LC connection and they must consequently fail the Cartan-Kalhede test. I started reading on teleparallelism and it goes quite along what i am interested in. The tetrad field in my case would be build from the unit vectors of the TDB and BCRS coordinates. I am just not sure i understand the choice of metric and connection in that case yet. gimme some time. But choice of metric indeed also tries to study a case of a flat geometry, but i intend to stay within the context Riemann geometry. The major difference is that i do not want to postulate any new physical laws on my own but rather would like to deduct the laws in the new geometry from the starting theory using a transformation like Steven posted. In particular, i want to move all influence of gravity from the geometry and torsion (rendering it trivial) and instead separate it out into its own fields: in terms of the transition to the new equation of motion of a particle, the remaining difference between the new and the old Christoffel symbols needs to be interpreted as physical fields representing gravity.

Yes, and that connection is always given by the metric of the Riemann manifold via Levi-Civita. This is why the definition skips out out to mention it. In the special case of Riemann geometry, the metric uniquely dictates the connection and gives it a special name, the LC connection. But don't misunderstand me, i am not insisting that in general a connection requires a metric for its definition. It does not. In that sense it is indeed an entirely independent object around which there is a separate field of study. But we are not discussing the connection itself but geometry, and that is another matter. A some geometric properties have redundant definitions as they can be defined via different concepts, like e.g. geodesics. Besides of what the name geometry already implies, as you can see from the field of pure metric geometry, main geometric definitions don't need to have a connection at all. This is where it is important that when different concepts are available at the same time, their compatibility must be ensured. It is weird to work with a metric and a connection that contradict each other showing two very different geometries. So whenever geometry is concerned specifically, there is a clear link between them. For the part of physics i want to discuss I assume we have a both a metric and a connection and they have to be compatible so that whenever we talk about geometry in the model, these don't provide contradicting accounts. A change of metric hence requires to find a new connection compatible with the new metric. I'm on it.

When you stick to that definition, your 21404 mile route is a geodesic, just not a mimimizing geodesic.