There appear to be just are two ways to reconcile lightspeed constancy with relativity under classical theory:

• METHOD 1 is the "physicalist" approach, where we treat matter-physics as being all about physical bodies. We say that the speed of light is constant for every physical observer-mass ... but only in their immediate vicinities. The velocities of light in the regions between differently-moving bodies then vary with the proximity to those bodies (gravitomagnetic fields). This gives a dynamic lighbeam geometry reminiscent of the dragged-aether theory of Hertz (1890).
• METHOD 2 is the "graphpaper" approach, where we decide that matter-physics is entirely defined by the properties of space alone, and that matter is a passive occupant of spacetime. The presence and behaviour of matter has no effect on light, and there is a preexisting, absolute lightbeam-geometry that is fixed and flat rather than dynamic. We delete all proximity field effects from the description by abstracting away physical matter and instead dealing with the relationships between flat extrapolated frames (which hold no proximity data). Since we've decided that the laws of physics are already entirely pre-built into the structure of empty space, we invoke Maxwell's equations for empty space. With this method, lightspeed constancy is global, and disagreements between frames as to measured distances, times and velocities are resolved by Lorentz rescalings. This gives a fixed, static lightbeam geometry echoing Lorentz aether theory.

METHOD 1 gives the sort of dynamic "acoustic metric" proposed by Visser in 1998, which meshes with QM, and it works for inertial physics, relativistic gravitation and general relativity.

But it involves going straight to curved-spacetime geometry, which freaks people out.

METHOD 2 borrows global c-constancy from Lorentz aether theory, but its uniform-rescaling approach only works for inertial physics, and is incompatible with relativistic gravitation, general relativity, and the principle of equivalence of inertia and gravitation. It also gives absolute event horizons whose behaviours are incompatible with QM. Method 2 is deeply incompatible with subsequent, more advanced physics paradigms.

But it's convenient to teach, because it turns more advanced physics into a "two-step" system, first flat spacetime, then curved.

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TRANSFORMATIONS:

Also, because it assumes a single fixed, predefined underlying spacetime geometry, METHOD 2 it lets us discover this geometry by saying that all observers experience aspects of the same single underlying geometry, which merely undergoes transformations as we change how observers and masses move. Deriving the necessary nature of those transformations inevitably gives us Special Relativity and Minkowski spacetime.

METHOD 1 does not let us use the "transformation" approach for deriving how matter interacts with light in its vicinity, because when we repeat an experiment with our observer(s) and objects moving in different ways, the spacetime geometry is different every time. The geometry is responsive . Once we already know the rules of physics in a region, and know the region's shape, we can use transformations to describe how that region appears to different remote onlookers ... but transformations won't give us what those laws are, or how the shape depends on the region's contents.

Transformations describe different views of a single assumed fixed geometry, but dynamic spacetime makes the geometry interactive. Dynamic spacetime and gravitomagnetism inevitably give us something other than Special Relativity and Minkowski spacetime.

Preprint Two Routes to General Relativity

So:

• If you believe that moving matter has an effect on the propagation and motion of nearby light and matter (Fizeau, slingshot effect, general principle of relativity), then you need gravitomagnetic effects to be real, and you probably need METHOD 1 and you need inertial and gravitational physics to be correctly described by an acoustic metric.
• If you believe that moving matter has NO effect on the propagation and motion of nearby light and matter, then you need gravitomagnetic effects to NOT be real, you're looking at METHOD 2, and you need inertial and gravitational physics to be correctly described by the Minkowski and Schwarzschild metrics, instead.

Well, however you do it, it won't give Minkowski spacetime and the SR/LET Doppler relationships. So the "bodies, not frames" approach, with purely local lightpeeds doesn't let us use special relativity as an intermediate stepping-stone. We have to go straight to a non-SR dynamically-curved spacetime, which, if we project it onto a flat background, gives gravitomagnetic fields, so that SR no longer applies.

Robert A. Phillips

I'm saying that the non-standard approach, which some people would think of a dragged-aether approach, but which I prefer to think of in terms of gravitomagnetism, has to use non-SR equations. Because the SR/Minkowswki Doppler equations are specific to flat spacetime and the absence of gravitational and gravitomagnetic fields, and with a dynamic metric, the shape of the lightmetric physically changes when we change the relative velocities of matter passing through the region.

Is that roughly what you're saying, too?

That doesn't fix the conflict with SR.

Special relativity can only apply if a region's "lightbeam geometry" is flat, and if lightspeed is globally constant over the region being modelled.

• If we take a flat coordinate system and superimpose gravitomagnetic fields, then the resulting lightbeam geometry within that coordinate system is no longer flat, and SR no longer applies.
• If we take a flat coordinate system and superimpose smooth, classical lightspeed variations of any other sort, then the speed of light becomes variable as measured against the flat reference-grid, and SR does not apply.
• Any variation in propagation rates as measured against a projected geometrically-flat reference grid can be interpreted as spacetime curvature, and/or the effect of gravitational/gravitomagnetic fields, at which point SR does not apply.
• If the motion of matter has any effect at all on the propagation of light, then SR does not apply.

The intrinsic geometrical properties of lightbeams in a flat, absolute Minkowski metric, which is entirely unaffected by the presence of moving matter, cannot be the same as the properties of lightbeam geometry in a region containing moving matter, where the behaviour of that matter alters the way that light propagates.

No amount of coordinate remapping can dissolve the fact that these are two different systems with different inherent geometrical properties, and physically different characteristics and behaviours.

• Einstein (1920): " ... the entire conceptual system of the theory of special relativity can claim rigorous validity only for those space-time domains where gravitational fields (under appropriately chosen coordinate systems) are absent. The theory of special relativity, therefore, applies only to a limiting case that is nowhere precisely realized in the real world. "

... and the limiting case is not a physical limiting case. It's the case of an infinitesimally small region that contains no interacting bodies. SR is not extensible, and its relationships are not valid in more complex physical (gravitational) theories.

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What Einstein fails to do is to go back and rederive the basic equations in a form that should be precisely realised in the real world. He never develops a curved-spacetime theory that still works when all matter and all observer-masses have associated gravitational fields, he never develops a theory of relativistic gravitation that can apply the relativity principle to the motion of stars and planets, he never develops a workable theory of gravitomagnetism, he never develops a general theory that actually conforms properly to the general principle of relativity.

His general theory never actually works as geometry, and tht's why he keeps trying to tinker with if for the next thirty years until his death. He never solves the problem, because it's not soluble. He's not prepared to give up SR, and since the SR relationships make gravitomagnetism impossible, SR cannot coxist in a larger system with the GPoR.

Special relativity only works for inertial physics in perfectly flat spacetime, which either means that it doesn't work when there's any matter present, or it doesn't work if matter has gravitational fields. It can't exist in the same universe as the principle of equivalence of inertia and gravitation. It's essentially an archaic Nineteenth-Century set of flat equations derived from flat aether theory, that don't mesh with modern physics.

To answer the actual question, yes, there is one. And it is more than just an idea, but actually already a fully fledged theory.

The basic idea is as follows: One can postulate, for example, that EM fields do not propagate at the speed of light at all, but have instead a whole continuum of speeds. If we further assume that each point charge can only perceive that part of the field which is not faster than c in its own center-of-momentum frame, then we have explained both of Einstein's postulates even without the Lorentz transformation.

Why? Imagine a point charge that generates a radial symmetric field that is moving with different velocities from the generating point charge. However, another point charge can only perceive that part of the field which has subluminal velocity in its own center-of-momentum frame. This means that every receiver charge, regardless of the speed of the source charge, must be of the opinion that information from the source charge propagates at a maximum speed of c in its own rest frame. This explanation fits in a natural and logical way with Einstein's postulates. Spacetime becomes superfluous.

Interestingly, this interpretation also fits very well with Maxwell's equations. This is because Maxwell's equations can be solved analytically for two point charges (using the Lienard-Wiechert potentials). This produces a mathematically equivalent representation without differential equations, which is much easier to interpret. In fact, there are only two equations left: One that is similar to Coulomb's law (Weber-Maxwell force) and one that describes the time delay of the force propagation.

This general solution of Maxwell's equations shows that Maxwell's equations do indeed fulfill Einstein's postulates. At the same time, one can also see that Newton's third law is satisfied. Remarkably, this is also the case when point charges are accelerated and they emit electromagnetic waves. Furthermore, this hypothesis explains the origin of magnetism and bypasses the definition of the magnetic field.

In the end, classical electrodynamics is just ordinary Newtonian mechanics. The fact that forces do not propagate infinitely fast does not affect this. The conservation of momentum is always fulfilled.

You can find more on this subject here (at the end of the article is the proof that the Weber-Maxwell force is the general solution of Maxwell's equations):

https://www.researchgate.net/publication/377495637_Weber-Maxwell_electrodynamics_classical_electromagnetism_in_its_most_compact_and_pure_form

There is also a new simulation software that uses the general solution of Maxwell's equations:

https://github.com/StKuehn/OpenWME

There are many examples (with videos) on all kinds of themes, such as Lorentz force, induction, magnetic force, interference, diffraction, bremsstrahlung, scattering and also on relativistic phenomena, such as Einstein's light clock.

Robert A. Phillips

They can coexist, just not in an SR-centric framework.

Gravitational time dilation:

Newtonian theory generates gravitational shifts, and Einstein's 1911 argument showed that the inescapable consequence of Newtonian gravitational shifts was gravitational time dilation.

So gtd is general, and not specific to an SR universe.

Velocity-dependent time dilation:

Regarding the time dilation interpretation and motion shifts:

• If the s.o.l. is considered fixed for the observer, the expected recession redshift is E'/E = c/(c+v)
• If we use SR, the expected recession redshift is E'/E = sqrt{ (c-v)/(c+v) } = c/(c+v) × LorentzRed
• If we use C19h Newtonian theory, the expected recession redshift is E'/E = (c-v)/c × LorentzRed × LorentzRed

Special relativity's physical predictions for a given nominal velocity are the average (geometrical mean) of the predictions associated with [A] fixed c for the observer, and [B] those associated with fixed-c for the emitter.

[SR] = root[A × B]

So as a broad statement, we can say that any physical effect that exists under SR, and doesn't exist under [A] must exist more strongly under [B].

This makes time dilation quite a general effect. Any physical procedure or protocol that we can use to demonstrate the existence of time dilation under SR, will also report a positive result for time dilation if the SR equations of motion and Doppler equations are swapped out in favour of the C19th Newtonian equations.

Newtonian theory generates both gravitational and velocity-based time-dilation effects.

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Metric-compatibility:

What doesn't work in traditional Newtonian theory is wave-compatibility. If light leaves an emitter travelling at at cEMITTER, compatibility with wave and metric principles requires it to arrive at a detector already moving with a speed of cDETECTOR. To impose wave-compatibility onto Newtonian mechanics requires full gravitomagnetism: we have to invent a gravitomagnetic field that exists between any pair of bodies with relative velocity v, so that the velocity-differential associated with the field is also v.

Gravitomagnetism:

At this point, C19th Newtonian theory morphs into a full-blown gravitomagnetic theory that is no longer Newton's. In the resulting system, kinetic and gravitational TD co-exist rather better than they do under Einstein's system, because a motion shift can now also be thought of as a gravitational (gravitomagnetic) shift. Under existing theory, we can already calculate gravitational-shifts-as-motion-shifts, but the correspondence is one-way, because motion shifts are supposed to be describable in flat spacetime. Under the revised system, the shift equivalence goes both ways, and we also have motion-shifts-as-gravitational-shifts. The Doppler effect then exists in the time domain as the result of relative motion, but also exists outside the time domain as a gravitomagnetic curvature effect.

Cosmological shifts:

With gravitomagnetic theory, the "shift equivalence" idea also then extends to include cosmological shifts, as the non-SR equations governing motion and gravitational shifts turn out to be the same non-SR equations used for cosmological expansion redshifts.

This is the right result, as an expansion redshift can also be considered as a gravitational redshift on a signal that originates in a smaller, denser universe, and has therefore climbed "uphill" across a field density gradient to reach us.

The expansion-shift-as-gravitational-shift trick doesn't work under Einstein's system, as it assigns different shift equations to cosmological and gravitational shifts. Einstein's system was designed for a fixed-size universe and can't be properly adapted to cope with a variable-size universes.

Redundancy of SR:

The gravitomagnetic approach makes special relativity redundant (because gravitomagnetic lightspeed regulation gives local c for every observer-mass), and merges together three different categories of shift effects — motion, gravitation and expansion — under a single common set of equations.

Quantum mechanics:

As a bonus, the non-SR equation-set turns out to be the same equation-set required to bring GR into line with QM, by turning Wheeler's absolute event horizons into leaky relative horizons, allowing the classical counterpart of Hawking radiation.

Reconciling Newtonian theory with local lightspeed constancy doesn't just eliminate special relativity and solve the internal problems with general relativity, it also seems to make general relativity QM-compliant.

So ... quantum gravity! Yaay!

Robert A. Phillips

Fluid dynamics sounds good! But it's a way outside my area of expertise, so I'll have to leave that to other people. There's also a strong overlap with acoustic metrics:

Article Analogue Gravity

I do accept that imposing strong gravitomagnetism just to fix Newtonian theory is a bit extreme, and a slightly "ad hoc" solution. But there are also quite a few other reasons why we seem to need gravitomagnetism:

Preprint Principles of Gravitomagnetism I - Unavoidability