If one has a given connection object field, then changing the metric tensor field changes the covariant derivative of the metric tensor field. One way to look at this (though I make no claim to its equivalence) is that one can change the metric tensor field in which the covariant differential operator is equal to the partial differential operator. Usually, (for spacetime) this is the Minkowskian metric tensor field, but it need not be. This is a change in definition, but it results in corresponding adjustments that keeps everything correct.

 

Edited March 31Mar 31 by KJW

One question that interests me, though I have yet to fully explore, is: Given only a nondegenerate symmetric tensor field [math]a_{\mu \nu}[/math] and its covariant derivative [math]\nabla_{\lambda} a_{\mu \nu}[/math], is it possible in general to determine the metric tensor field [math]g_{\mu \nu}[/math] (with zero covariant derivative)?
 

1 hour ago, KJW said:

One question that interests me, though I have yet to fully explore, is: Given only a nondegenerate symmetric tensor field aμν and its covariant derivative ∇λaμν , is it possible in general to determine the metric tensor field gμν (with zero covariant derivative)?

Naively i would think you should be able, since you know how \( a_{\mu \nu} \) changes in length in any direction.
I would multiply some scalar function f and then try to find a solution such that  \( \nabla_\lambda (f a_{\mu \nu}) = 0 \). Applying the product rule you get the covariant derivative of a as a term, which is known. So at least in the case of 1 dimension, this is a differential equation that can be solved. This gives you an adjustment f a that preserves length along lambda, which reconstructs a part of the metric.

Anyhow, i dead tired and probably got something wrong. stupid of me to check the forums just before i was going to sleep.

10 hours ago, Killtech said:

would indeed require an integration of the now location dependent decay rate along the worldline to get an answer.

Well yes, essentially you are understanding the problem - namely that the outcome of experiments performed in u-time depend on where (and also when) they are performed. So it becomes very difficult to relate predictions from the model to physical outcomes in the real world.

11 hours ago, Killtech said:

So, in u time standard, the corrections from gravity fields should result in producing some kind of physical 2-tensor field (at least) representing gravity

In u-time, no two clocks can be dilated wrt one another, so you always have \(g_{\mu 0}=g_{0\nu}=\pm 1\). This creates another issue, consider the following:

Suppose you have two u-clocks that start off together at the same place near some very massive, rotating object, like a pulsar. They are initially synchronised and at relative rest. Now these clocks travel along a closed trajectory around the pulsar’s equatorial plane, and come to rest again at the same place afterwards. Both clocks travel along the exact same spatial trajectory with the exact same speed profile, but in opposite directions. Because of the way you defined your u-time, at the end of the experiment both clocks read the exact same amount of total accumulated u-time.

Unfortunately, in the real world this isn’t what happens. If you perform this experiment with ordinary t-clocks, their readings will differ when they come together again. 

So in order to relate predictions calculated from your model to the real world, you need to account not just for location and time, but also for the history of the physical system in question.

11 hours ago, Killtech said:

which should be straight forward in the flat spacetime u produces

The spacetime isn’t flat, unless you want to demand that your metrics must be isometric to the Minkowski metric. Is that what your doing?

Furthermore I suggest we stick to established classical physics for now, and not introduce unnecessary complications and speculations.

11 hours ago, Killtech said:

Note that for g-LC connection ∇ we have ∇g=0≠∇g′ . Yet ∇′g′=0 using the g'-LC connection ∇′ . 

Im afraid this makes no sense. Like I have said several times now, it’s not the connection itself that changes, only the connection coefficients.

13 hours ago, Killtech said:

we will get a very different Lagrangian

I’m afraid it’s much more subtle than this. You have to remember that on your u-spacetime not only time and space are redefined, but also any derivatives taken with respect to them, and any quantities integrated from them. Thus, while it probably still is possible to define the concept of a “Lagrangian”, this won’t have the same physical meaning any longer. In addition, the Euler-Langrange equation will be of a different form too; in fact, I think its form would be different at every event on your manifold.

And it gets worse still. Your u-manifold still formally admits a notion of invariance with respect to translations/rotations in u-space as well as translations in u-time. It should also be possible to formally find an corresponding notion of Noether’s theorem, so you might have some form of conserved Noether currents. In particular, u-time translation invariance will lead to some concept of “u-energy-momentum”. Unfortunately this object bears no relation to the energy-momentum tensor in ordinary physics, so you cannot use it to describe distributions of energy-momentum in the real world. Translating that into your formalism will yield something that has a different form at every point in your spacetime.

14 hours ago, Killtech said:

Maybe this makes my approach appear a little bit more reasonable?

I’m afraid not, because these “gravity corrections” cannot be calculated from your formalism. Since the form of all physical laws is explicitly dependent on the gravitational background, you cannot describe any real-world physical situation in terms of your formalism, unless you already know the gravitational corrections. Thus it is not possible to write down a gravitational field equation with this formalism, as any such attempt would be self-referential. You would need to start with prior knowledge of the ordinary gravitational metric, and, based on this, you can then figure out “corrections” and use those to write down equations in your formalism. So you’re effectively multiplying your workload - first you have to use GR to find a metric, then you use it to model things in your formalism, and in the end you have to translate the result back again to actually relate it to real world measurements. Note also that finding the correct form of physical laws for a given set of gravitational corrections is not a trivial task (how would you even approach that problem, since it affects all laws of physics?).

Also think about what this implies - not only would the form of physical laws be different for different observers in different gravitational environments, but that form might also vary for the same observer over time. While it may be formally possible to do such a thing, actually working with such a formalism in a practical sense would be an absolute nightmare. You asked earlier why no one seems to consider this type of formalism - well, this is part of the answer. We really need one set of physical laws that has the same form for all observers, irrespective of their states of relative motion or their location in space and time; anything else just isn’t very useful when it comes to describing the real world, except perhaps in highly idealised scenarios. That is why general covariance is such a fundamental part of contemporary physics.

12 hours ago, KJW said:

One question that interests me, though I have yet to fully explore, is: Given only a nondegenerate symmetric tensor field aμν and its covariant derivative ∇λaμν , is it possible in general to determine the metric tensor field gμν (with zero covariant derivative)?
 

From the looks of it we have enough equations to determine all unknowns, but only if the connection is of type Levi-Civita. If the connection is not guaranteed to be torsion-free, then we need at least one additional constraint to solve this system of equations. So the answer looks to be yes, so long as we know it’s an LC connection.

Whether the metric thus obtained is unique is a different question again - my feeling is that it might only be determined up to an isometry, but I might be wrong; I haven’t actually sat down and attempted a formal solution.

2 hours ago, Markus Hanke said:

From the looks of it we have enough equations to determine all unknowns, but only if the connection is of type Levi-Civita. If the connection is not guaranteed to be torsion-free, then we need at least one additional constraint to solve this system of equations. So the answer looks to be yes, so long as we know it’s an LC connection.

Whether the metric thus obtained is unique is a different question again - my feeling is that it might only be determined up to an isometry, but I might be wrong; I haven’t actually sat down and attempted a formal solution.

While the question of uniqueness is a reasonable question, my only interests are whether the solution metric tensor field exists, and if it does not exist, what the obstruction is to its existence. The question is basically asking whether an arbitrary connection object field admits a metric tensor field (with zero covariant derivative). As for whether or not the torsion tensor field is zero, I don't think this actually matters to the question being posed. I believe that [math]\nabla_{\lambda} a_{\mu \nu}[/math] is independent of the torsion tensor field in the sense that the expression for the connection object field in terms of [math]\nabla_{\lambda} a_{\mu \nu}[/math], [math]\partial_{\lambda} a_{\mu \nu}[/math], and [math]a_{\mu \nu}[/math] will contain the torsion tensor field as an arbitrary field (similar to the expression for the connection object field in terms of [math]\partial_{\lambda} g_{\mu \nu}[/math], [math]g_{\mu \nu}[/math], and [math]\nabla_{\lambda} g_{\mu \nu} = 0[/math], where the torsion tensor field is undetermined or asserted to be zero).
 

15 hours ago, Markus Hanke said:

So in order to relate predictions calculated from your model to the real world, you need to account not just for location and time, but also for the history of the physical system in question.

I am very well aware of that. The backtracing along world lines is however only relevant when translating results from u-standard model to t-standard experimental data. That is indeed expensive. Yet in the case we do for example numerical simulations of galaxies moving over long time spans, we can stay within one standard without the need for switching. Events like collisions will not depend on whatever time standard and its physical laws we use to calculate them - if they are consistent. 

15 hours ago, Markus Hanke said:

The spacetime isn’t flat, unless you want to demand that your metrics must be isometric to the Minkowski metric. Is that what your doing?

This in particular would depend a bit what specific approach would make most sense for quantum field quantization. An isometry to either Minkowski or Euclidean metric would certainly make things simpler. 

12 hours ago, Markus Hanke said:

I’m afraid it’s much more subtle than this. You have to remember that on your u-spacetime not only time and space are redefined, but also any derivatives taken with respect to them, and any quantities integrated from them. Thus, while it probably still is possible to define the concept of a “Lagrangian”, this won’t have the same physical meaning any longer. In addition, the Euler-Langrange equation will be of a different form too; in fact, I think its form would be different at every event on your manifold.

And it gets worse still. Your u-manifold still formally admits a notion of invariance with respect to translations/rotations in u-space as well as translations in u-time. It should also be possible to formally find an corresponding notion of Noether’s theorem, so you might have some form of conserved Noether currents. In particular, u-time translation invariance will lead to some concept of “u-energy-momentum”. Unfortunately this object bears no relation to the energy-momentum tensor in ordinary physics, so you cannot use it to describe distributions of energy-momentum in the real world. Translating that into your formalism will yield something that has a different form at every point in your spacetime.

An approach to change the geometry merely moves real physical information between geometric object and fields of similar nature. This is why it is guaranteed to produce an analogue of the Euler-Lagrange equations. But indeed all geometry dependent quantities are redefined by analogues and this of course includes energy and its conservation.

I am happy that we finally start to understand each other. I find it very interesting as this shows how much relativity there is any model of reality, simply based on the standards and conventions required to map it into a mathematical description. This mapping is just highly non-unique, non-trivial and very influential in terms of the model it produces. I am aware how tedious this may look in the first place, yet on the other hand this line of thought also adds so much more degrees of freedom to model the very same reality. It also adds quite a different perspective on physics, as it is a lot more relative then one would think.

12 hours ago, Markus Hanke said:

I’m afraid not, because these “gravity corrections” cannot be calculated from your formalism

I haven't yet concretized specific corrections. The Schwarzschild example i used was merely to demonstrate that i really want to change the definition of time standard and not just coordinates.

For the concrete corrections, i was considering something along the lines of using a coordinate time concept like Barycentric Coordinate Time TCB and uplifted to serve as a time standard, or alternatively a clock concept basing the corrections on radio clocks based on a common specially placed signal source - both ideas are somewhat related. The latter would allow to determine the corrections experimentally, the former gives a recipe how to do it from the model. Either one should allow to find a relation between the regular GR geometry and the corrected one for any scenario, hence this would allow to deduct field equations / laws for all new physical fields from the Einstein field equations.  

I might note that the way TCB seems to be used in practice for calculation in the solar system is actually not that far from what i intend to do. And there are in fact practical reasons for doing that. It just doesn't go all the way.

13 hours ago, Markus Hanke said:

Also think about what this implies - not only would the form of physical laws be different for different observers in different gravitational environments, but that form might also vary for the same observer over time. While it may be formally possible to do such a thing, actually working with such a formalism in a practical sense would be an absolute nightmare. You asked earlier why no one seems to consider this type of formalism - well, this is part of the answer. We really need one set of physical laws that has the same form for all observers, irrespective of their states of relative motion or their location in space and time; anything else just isn’t very useful when it comes to describing the real world, except perhaps in highly idealised scenarios. That is why general covariance is such a fundamental part of contemporary physics.

Weather laws of physics are differ depending on location is a thing of interpretation. state of physical fields and geometry differs by location but the overall laws won't if you keep your definitions of what is a force clean. Even if you would purely arbitrarily add some curvature to a region that is normally flat, in the resulting model it would look like there is some weird dark matter sitting there and messing up physical fields and geometry - yet apart from the concrete distribution of the dark matter in your model, the new laws of physics can retain the same form at all locations - everything just needs some special interactions with the dark matter when passing through.

Finally, there is also another case of study for this idea and it goes the other way around - that is where physical laws appear complicated and location depended (in the sense of extra fields appearing), yet could be reduced in apparent complexity if a different time standard were to be chosen. Of course there are good reasons not to, but next time it might be of interest to discuss these situations to understand under which circumstances this is advantageous. 

 

 

6 hours ago, Killtech said:

Yet in the case we do for example numerical simulations of galaxies moving over long time spans, we can stay within one standard without the need for switching.

If you don’t switch, you can’t compare it to real world observations.

6 hours ago, Killtech said:

yet on the other hand this line of thought also adds so much more degrees of freedom to model the very same reality.

The number of degrees of freedom need to be the same, or else any system of equations in the formalism is either overdetermined, or unsolvable.

6 hours ago, Killtech said:

Either one should allow to find a relation between the regular GR geometry and the corrected one for any scenario

Yes, like I said, you need ordinary GR first in order to actually determine the gravitational environment. It can’t come out of your formalism. Only after you know all corrections can you formulate things. 

6 hours ago, Killtech said:

Weather laws of physics are differ depending on location is a thing of interpretation.

The comment was about the form of the laws in your formalism, not the physics themselves. If you change the meaning of clocks and rulers, the form of all laws changes too.

6 hours ago, Killtech said:

I find it very interesting as this shows how much relativity there is any model of reality

I’m sorry to say I don’t share your enthusiasm. To me this is at most a mildly curious intellectual exercise, but like I said, such a formalism would be a complete nightmare to actually work with for any kind of practical application.

I think the best will be for you too keep working on it, and see for yourself what comes out of it. If you can come up with something of value, then great, I’ll be the first to congratulate 👍 What I predict will happen though is that you’ll very quickly find yourself in a world of mathematical pain, while at the same time loosing all physical intuition, since none of the quantities you are working with directly correspond to ordinary measurements in the real world. But as an intellectual exercise at least, this project can be very instructive.

15 hours ago, KJW said:

The question is basically asking whether an arbitrary connection object field admits a metric tensor field (with zero covariant derivative).

(Highlight is mine)

Ok, I get you now. That’s an interesting question, I’ll have to think about that for a bit. My immediate guess would be no, not every arbitrary connection necessarily admits a metric tensor field with vanishing derivative - I think if you mix time-like and space-like parts in the definition of the covariant derivative in the right way, the inner product will no longer vanish under parallel transport. This is to say that connections should exist that never preserve any metric. If I have time, I’ll try and construct an explicit example of such a connection.

19 hours ago, Markus Hanke said:

The number of degrees of freedom need to be the same, or else any system of equations in the formalism is either overdetermined, or unsolvable.

Of course. What i meant is that the definition of the standard for clocks and rulers is itself a significant degree of freedom,  as it does not affect any real physics itself. yet massively reshapes the resulting model.

19 hours ago, Markus Hanke said:

The comment was about the form of the laws in your formalism, not the physics themselves. If you change the meaning of clocks and rulers, the form of all laws changes too.

I understood you. I am just not sure i would strictly call e.g. the form of Navier-Stokes equations - in their general form - to depend on location or the observers frame. There is also a general form for Maxwell equations under Galilean trafos. What is nasty though it the transformation between frames. But i know what you mean and you are right that in practice when describing e.g. physics in a medium there is always a frame for which the medium is (at least locally) at rest, making some terms to vanish and the equations of interest simplify. 

19 hours ago, Markus Hanke said:

Yes, like I said, you need ordinary GR first in order to actually determine the gravitational environment. It can’t come out of your formalism. Only after you know all corrections can you formulate things. 

Not sure this is the case. For the corrections in questions it may not be possible to experimentally determine them by local means only, but involving global information, this should be possible to do without any knowledge of GR itself. For example the extremal clock signal a radio clock receives compared against a local running clock of same type allows to extract data about the gravitation environment the signal travelled and used that for corrections. Therefore we should be able to construct such u-standard clocks in reality even without the knowledge of GR.

But since we do not want to experimentally and empirically discover the laws physics anew, we can deduct the new laws from GR, if we can express the correction in terms of GR. Once we know the equations of motions of all fields and the geometry in the new standard, we technically won't need any knowledge of GR any longer.

19 hours ago, Markus Hanke said:

I’m sorry to say I don’t share your enthusiasm. To me this is at most a mildly curious intellectual exercise, but like I said, such a formalism would be a complete nightmare to actually work with for any kind of practical application.

The enthusiasm is indeed a lot on a theoretical level. because if the standard for time and length allows to rewrite the laws of physics with quite some degree of freedom, we could ask if we can postulate certain laws to have a certain form and then find a standard that makes these laws comply with reality.

Applying this idea to Maxwell equation renders it infallible by construction and the nature of c becomes a different interpretation in this concept. Consequently its perfect constancy is a logical consequence independent of physical reality.

But this idea seems to works for a broad set of wave equation without dispersion by frequency. So for example, if we take an ideal medium and look at the acoustic wave equations, we can find a time standard that allows to treat it with the same framework as SR - rendering the speed of sound a perfect constant which need to be set and perfect Lorentz invariance but around the set speed of sound. Even if we include regions with a non constant refractive index, in the new standard such local anomalies vanish from the wave equation in covariant form and instead wander into the geometry / are handled by the connection. 

Edited April 2Apr 2 by Killtech

7 hours ago, Killtech said:

Of course. What i meant is that the definition of the standard for clocks and rulers is itself a significant degree of freedom,  as it does not affect any real physics itself. yet massively reshapes the resulting model.

 

Can always choose an alternative unit system.

As a physical constant, it simply makes sense to have redefined metric units in terms of it.

 

16 hours ago, Killtech said:

The enthusiasm is indeed a lot on a theoretical level. because if the standard for time and length allows to rewrite the laws of physics with quite some degree of freedom, we could ask if we can postulate certain laws to have a certain form and then find a standard that makes these laws comply with reality.

Applying this idea to Maxwell equation renders it infallible by construction and the nature of c becomes a different interpretation in this concept. Consequently its perfect constancy is a logical consequence independent of physical reality.

But this idea seems to works for a broad set of wave equation without dispersion by frequency. So for example, if we take an ideal medium and look at the acoustic wave equations, we can find a time standard that allows to treat it with the same framework as SR - rendering the speed of sound a perfect constant which need to be set and perfect Lorentz invariance but around the set speed of sound. Even if we include regions with a non constant refractive index, in the new standard such local anomalies vanish from the wave equation in covariant form and instead wander into the geometry / are handled by the connection. 

The thing is, the equations in physics require more than being mathematically self-consistent. They have to be consistent with experiment/observation. You can’t just define the speed of sound to be invariant and expect to construct valid laws of physics, because the speed of sound is not invariant.

3 hours ago, swansont said:

You can’t just define the speed of sound to be invariant and expect to construct valid laws of physics, because the speed of sound is not invariant.

Functions of the form \(e^{ikx}\) are invariant under Lorentz-like coordinate trafos. This is a trivial mathematical observation that applies to any simple wave and their equations. So for an ideal uniform acoustic medium you can apply these and notice that the form of the equation remains invariant. This hints what definition of a time standard we have to use to achieve this in covariant form as well. In general we can use analogue construction of a geodesic clock from GR but utilizing acoustic signals instead of light. For the standard of length its pretty obvious: we just define length by the distance an acoustic wave travels in a fixed time - this is why the speed of sound must become a constant in any circumstance. For either standard we can construct devices that measure them in reality. 

Now we have discussed extensively what happens to the laws of physics when we do change the standards of time and length. Apply it to this case.

Just now, Killtech said:

This hints what definition of a time standard we have to use to achieve this in covariant form as well. In general we can use analogue construction of a geodesic clock from GR but utilizing acoustic signals instead of light. For the standard of length its pretty obvious: we just define length by the distance an acoustic wave travels in a fixed time

This thread is growing ever more Terry Pratchett and ever less sf/home/sciences/physics/relativity.

I sem to remember discussing the speed of light in vacuuo, backalong.

What, pray, is the speed of sound in vacuuo ?

Edited April 3Apr 3 by studiot

4 hours ago, Killtech said:

Functions of the form eikx are invariant under Lorentz-like coordinate trafos. This is a trivial mathematical observation that applies to any simple wave and their equations. So for an ideal uniform acoustic medium you can apply these and notice that the form of the equation remains invariant. This hints what definition of a time standard we have to use to achieve this in covariant form as well. In general we can use analogue construction of a geodesic clock from GR but utilizing acoustic signals instead of light. For the standard of length its pretty obvious: we just define length by the distance an acoustic wave travels in a fixed time - this is why the speed of sound must become a constant in any circumstance. For either standard we can construct devices that measure them in reality. 

Now we have discussed extensively what happens to the laws of physics when we do change the standards of time and length. Apply it to this case.

Now how about addressing my objection, which was to the speed of sound, rather than the form of the wave equation. If you have a source and the sound medium moving at some speed, (like on a plane) the sound wave moves at the speed of sound + speed of the plane. i.e. it is frame-dependent.

 

19 hours ago, swansont said:

Now how about addressing my objection, which was to the speed of sound, rather than the form of the wave equation. If you have a source and the sound medium moving at some speed, (like on a plane) the sound wave moves at the speed of sound + speed of the plane. i.e. it is frame-dependent.

Not sure if i get you right, because a sound wave always moves with the speed of sound + speed of the local medium. To model the situation of a plane where the medium has two distinct regions, i.e. inside and outside, is significantly more complicated. It is indeed very interesting for the analogy it has but quite complicated as a start to exploring this framework.

If you are just talking about observing the wave from a frame with a relative movement v to the rest frame of the uniform medium, observe that we use a special time standard, specifically in the moving frame we use a clock which time is t′=(1−v2c−2)−0.5(t+vc−2x) relative to a clock in the rest frame of the medium. We measure distance by how far an audio signal gets in 1 unit of t', hence the observers frame can use coordinate x′=(1−v2c−2)−0.5(x+v−1t) to easily obtain the distances in the needed standard of length.

Redefining time and length, requires that speed is also redefined correspondingly. This tricky redefinition causes the speed of sound c=Δx′Δt′−1 to become constant in every frame. But it is just a that: a redefinition. The latter is nothing new: it was basically the discoveries made when developing Lorentz Ether Theory which uses almost the exact same framework - except for a slightly more complicated wave equation. Furthermore Poincaré corrected version of LET is equivalent to SR producing identical predictions.

For reference: https://philsci-archive.pitt.edu/5339/1/leszabo-lorein-preprint.pdf

EDIT: ... so formulas, particularly exponents break on editing here?!? any way to prevent that?

Edited April 4Apr 4 by Killtech

On 4/2/2025 at 1:43 PM, Markus Hanke said:

Ok, I get you now. That’s an interesting question, I’ll have to think about that for a bit. My immediate guess would be no, not every arbitrary connection necessarily admits a metric tensor field with vanishing derivative - I think if you mix time-like and space-like parts in the definition of the covariant derivative in the right way, the inner product will no longer vanish under parallel transport. This is to say that connections should exist that never preserve any metric. If I have time, I’ll try and construct an explicit example of such a connection.

It could actually be just an algebraic problem: Given [math]{R_{pqr}}^{u}[/math] (obtained from [math]{\Gamma_{qr}}^{u}[/math]), find [math]g_{us}[/math] such that:

[math]{R_{pqr}}^{u} g_{us} + {R_{pqs}}^{u} g_{ur} = 0[/math]

[math]{R_{pqu}}^{u} \ne 0[/math] is an obstruction to this, but I'm not sure if it's the full obstruction. (Note: This is NOT the Ricci tensor field)

However, if [math]{R_{pqr}}^{u} = 0[/math], then all [math]g_{us}[/math] satisfies the above equation, but not all [math]g_{us}[/math] satisfies [math]{R_{pqr}}^{u} = 0[/math], so a solution to the above equation is not necessarily a metric corresponding to the given connection object field. Nevertheless, [math]{R_{pqu}}^{u} \ne 0[/math], which is derived entirely from the given connection object field, implies that a metric (with zero covariant derivative) does not exist.

The significance of this is that the connection object field emerges to transform a non-tensorial partial differential operator to a tensorial covariant differential operator, but this does not suffice to produce the metric tensor field (with zero covariant derivative, a seemingly important property of the metric tensor field).
 

3 hours ago, Killtech said:

Not sure if i get you right, because a sound wave always moves with the speed of sound + speed of the local medium. To model the situation of a plane where the medium has two distinct regions, i.e. inside and outside, is significantly more complicated. It is indeed very interesting for the analogy it has but quite complicated as a start to exploring this framework.

You said “So for example, if we take an ideal medium and look at the acoustic wave equations, we can find a time standard that allows to treat it with the same framework as SR - rendering the speed of sound a perfect constant which need to be set and perfect Lorentz invariance but around the set speed of sound”

But you admit that the speed of sound isn’t invariant.

3 hours ago, Killtech said:

If you are just talking about observing the wave from a frame with a relative movement v to the rest frame of the uniform medium, observe that we use a special time standard, specifically in the moving frame we use a clock which time is t′=(1−v2c−2)−0.5(t+vc−2x) relative to a clock in the rest frame of the medium. We measure distance by how far an audio signal gets in 1 unit of t', hence the observers frame can use coordinate x′=(1−v2c−2)−0.5(x+v−1t) to easily obtain the distances in the needed standard of length.

Redefining time and length, requires that speed is also redefined correspondingly. This tricky redefinition causes the speed of sound c=Δx′Δt′−1 to become constant in every frame. But it is just a that: a redefinition. The latter is nothing new: it was basically the discoveries made when developing Lorentz Ether Theory which uses almost the exact same framework - except for a slightly more complicated wave equation. Furthermore Poincaré corrected version of LET is equivalent to SR producing identical predictions.

For reference: https://philsci-archive.pitt.edu/5339/1/leszabo-lorein-preprint.pdf

How do you build such a clock that “knows” how fast it’s moving with respect to some arbitrary reference frame, and its location with respect to some arbitrary origin? (though your equations can’t possibly work; unit analysis shows this)

On 4/1/2025 at 9:57 AM, Killtech said:

I would multiply some scalar function [math]f[/math] and then try to find a solution such that [math]\nabla_{\lambda} (f a_{\mu \nu}) = 0[/math].

A transformation of the type:

[math]g'_{pq} = \phi\ g_{pq}[/math]

where [math]\phi[/math] is a scalar function of the coordinates, is called a "conformal transformation". Conformal transformations vary the scale over spacetime location but do not change angles. In spacetime, angles include the speed of light in a vacuum, which is also invariant under conformal transformations. An important property of conformal transformations is the invariance of the Weyl conformal tensor field of the form [math]{C_{pqr}}^{s}[/math] (other forms of the Weyl conformal tensor field transform according to the metric tensor field used to raise or lower indices). The Weyl conformal tensor field is the part of the Riemann curvature tensor field that describes the tidal effect associated with pure gravitation (external to any distribution of energy-momentum). It should be noted that the connection object field transforms such that the covariant derivative of the metric tensor field remains zero (a non-zero covariant derivative of the metric tensor field transforms covariantly):

[math]\nabla'_r g'_{pq} = \phi\ \nabla_r g_{pq} = 0[/math]

If a conformal transformation is applied to a metric tensor field of flat spacetime, the resulting spacetime is called "conformally flat". A conformally flat spacetime has zero Weyl conformal tensor field, but non-zero Ricci tensor field in general. An example of a conformally flat spacetime is a flat-space Friedmann-Lemaître-Robertson-Walker (FLRW) metric used in cosmology. By coordinate-transforming time to expand the same as space, the conformal flatness of the FLRW spacetime becomes explicit. In this case, lightlike trajectories become simply described as straight lines, simplifying the calculation of cosmological redshift.
 

 

Edited April 4Apr 4 by KJW

24 minutes ago, swansont said:

But you admit that the speed of sound isn’t invariant.

You should read the article i liked in my post before. Invariance has no absolute meaning but only within a given model - that is invariances depend on the definition of time and length standard, as was discussed in this thread already:

On 4/1/2025 at 9:29 AM, Markus Hanke said:

And it gets worse still. Your u-manifold still formally admits a notion of invariance with respect to translations/rotations in u-space as well as translations in u-time. It should also be possible to formally find an corresponding notion of Noether’s theorem, so you might have some form of conserved Noether currents. In particular, u-time translation invariance will lead to some concept of “u-energy-momentum”.

.

28 minutes ago, swansont said:

How do you build such a clock that “knows” how fast it’s moving with respect to some arbitrary reference frame, and its location with respect to some arbitrary origin? (though your equations can’t possibly work; unit analysis shows this)

For LET, this question was a difficult one, as the ether wind cannot be measured (at least not if the medium is perfectly uniform). For a classic medium however it is trivial to measure the flow velocity wrt. the observer frame.

The equation i wrote is the simple Lorentz trafo in one dimension. i expanded gamma and beta to not mistake them for their analogue - not sure if you call it a Lorentz trafo is c is exchanged for the speed of sound. I also used used ^{-1} notation instead of \frac to make it not stand out within the text. but when i edited the post, the equations broke the ^ and all exponents fell down. i could not repair them.

1 hour ago, Killtech said:

You should read the article i liked in my post before. Invariance has no absolute meaning but only within a given model - that is invariances depend on the definition of time and length standard, as was discussed in this thread already:

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In the context of the OP, invariant means the same in all inertial reference frames. It has nothing to do with the definition of the time and length standard; these have changed over the last ~120 years, and the theory of relativity did not change as a result. 

 

 

7 hours ago, swansont said:

In the context of the OP, invariant means the same in all inertial reference frames. It has nothing to do with the definition of the time and length standard; these have changed over the last ~120 years, and the theory of relativity did not change as a result. 

Have you read the article? Do you unterstand by what means LET and SR are equivalent? You understand that in LET the speed of light in a frame = speed of light + ether velocity, hence depends on the frame due to the Galilean nature of the theory? Despite this, both interpretation maintain the equivalence making identical predictions. This is same as in my case. This change of interpretation is fundamental for understanding the issue and ideas in this thread - and the reinterpretation of time and length is the key for this.

I seem to fail to explain it well enough with my words. Yet the equivalence between LET and SR are well established, so maybe you should check out better sources then me for an explanation why this works. I admit though, it is a bit mind-boggling to get the head around it at first. Or please read the article, it explains all your questions in detail.

2 hours ago, Killtech said:

Have you read the article?

Have you read the rules? (2.7 in particular)

2 hours ago, Killtech said:

Do you unterstand by what means LET and SR are equivalent? You understand that in LET the speed of light in a frame = speed of light + ether velocity, hence depends on the frame due to the Galilean nature of the theory? Despite this, both interpretation maintain the equivalence making identical predictions. This is same as in my case. This change of interpretation is fundamental for understanding the issue and ideas in this thread - and the reinterpretation of time and length is the key for this.

I seem to fail to explain it well enough with my words. Yet the equivalence between LET and SR are well established, so maybe you should check out better sources then me for an explanation why this works. I admit though, it is a bit mind-boggling to get the head around it at first. Or please read the article, it explains all your questions in detail.

LET is discredited because you can’t experimentally confirm that one aspect of it that distinguishes it from SR. It also has nothing to do with any “convention tied to our choice of units” which was the premise of your first post.

Since the results are equivalent to those of SR, nothing changes in regard to unit definitions. You still get the same results. I don’t see how adopting LET changes the length of a platinum-iridium bar or the rotation rate of the earth, and you haven’t shown that it does. Since the modern definitions are based on those, how does anything change with a change in the laws of physics that give results the must be consistent with SR?

 

Have you read the rules? (2.7 in particular)

I am pretty sure the University of Pittsburg does not count as a commercial site, nor could a link to a scientific paper highly relevant to the discussion be considered advertising.

LET is discredited because you can’t experimentally confirm that one aspect of it that distinguishes it from SR.

By that standard all interpretations of quantum mechanics discredit one another? No, it is discredited because it has a preferred frame and in most situations, there is no experimental way to find which it is.

Apparently you lack a bit of knowledge about this part of physics and its history, so it would be quite a bit easier for the discussion if you would read up on it. I understand that this isn't highly relevant for popular modem physics, though i do think it would broaden your perspective on SR understanding its history better.

LET differs from SR that both time and length remain invariant, whereas they differ in SR by frame. And when it comes to the speed of light, it is frame dependent in the prior but not so in the latter. Those statements are however in no contradiction because the definitions of time and space differ between those two interpretation, i.e. they are not the same quantities and hence neither are speeds.

In LET Maxwell transforms by Galilei, hence they equations looks different for every frame. The frame dependence can reduced to a physical field called the ether and its movement speed. One might be naively lead to believe that this difference by frame should easily distinguished experimentally. However, we have to consider that our clocks and rulers are build out of atoms - which are electromagnetic in origin. So we have to account to what happens to atoms moving relative to the aether as well. Michelson-Morey explains how we expect the electromagnetic waves to be affected by this. We do know for example of the Stark effect when an atom is exposed to an external electric field, so similarly we expect effects due to the aether wind. And indeed, using some special coordinates we can simplify the problem to a known case, easily obtaining the answer that the atom has to contract in this instances and also its energy levels shift - the factors are a combination what we know as time dilatation and length contraction from SR, so if one was to use these atoms to as a base for a clock or line them up to produce a ruler, they will be affected in the very same way as SR prescribes. So LET makes the very same predictions as SR but explains them by the clocks and ruler we use being distorted by a physical field and thus not actually showing the proper time.

And indeed we cannot say how the "true" time ticks, because in reality all measurement is relative: a comparison between the measured object and a reference. So all we can measure is whether one object changes relative to another. If both change in the same manner, we are not able to see that, hence we can never say if an object changes in an absolute way. But we don't need to know that to make all the prediction we are interested in.

I am pretty sure the University of Pittsburg does not count as a commercial site, nor could a link to a scientific paper highly relevant to the discussion be considered advertising.

You have to read past the first couple of sentences.

“Links, pictures and videos in posts should be relevant to the discussion, and members should be able to participate in the discussion without clicking any links or watching any videos”

By that standard all interpretations of quantum mechanics discredit one another? No, it is discredited because it has a preferred frame and in most situations, there is no experimental way to find which it is.

QM interpretations are not theories. They are a way to think about QM to make it make sense to a person, not QM itself.

And indeed we cannot say how the "true" time ticks, because in reality all measurement is relative: a comparison between the measured object and a reference. So all we can measure is whether one object changes relative to another. If both change in the same manner, we are not able to see that, hence we can never say if an object changes in an absolute way. But we don't need to know that to make all the prediction we are interested in.

In what way does this relate to, and depend on, the definition of the second?