We may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether. According to the general theory of relativity space without aether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this aether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it. 
Brady, in the paper “The irrotational motion of a compressible inviscid fluid” hypothesizes something different – that the universe is made of a non – relativistic compressible fluid, and that this fluid generates General Relativity.
Einstein’s inertial medium behaves as a nonrelativistic barotropically compressible inviscid fluid.
Although my model of the electron and quantum effects is very similar to Brady’s, I diverge with him on the essence of the aether. I hypothesize that Brady and Einstein’s ether are the same thing, so that instead of Brady’s concept of generating GR from aether, we instead start with Classical General Relativity (with ‘no matter’, so the stress tensor T = 0), and then create Sonons as solutions of GR. The aether is that of Einstein’s GR.
Einstein’s Aether in Fluid Dynamics terms
Einstein’s aether is inviscid – which means it has no viscosity (rocks travelling through empty space experience no drag…). Is it compressible? Certainly – this is what constructs such as black holes are. Is it irrotational? – that is a not a property that we need to determine, since without viscosity, an irrotational flow will stay that way.
No. GR is non-linear, which makes the inviscid property only an approximation – it’s a good approximation, though! Waves generated on an ocean or an oil puddle in a lab travel a limited distance, while the waves of GR can easily travel the universe. But they don’t travel ‘forever’.
Consider now the construction of a Brady like sonon out of pure GR. We follow Brady’s paper until section 1.1, where he states:
When an ordinary vortex is curved into a smoke ring, this force is balanced by Magnus forces (like the lift of an aircraft wing) as the structure moves forward through the fluid . However a sonon cannot experience Magnus forces because it is irrotational, and consequently its radius will shrink, causing the amplitude A in (5) to grow due to the conservation of fluid energy. Nonlinear effects will halt the shrinking before A reaches about 1 since the density cannot become negative.
Intriguing. Look now at a completely classical general relativistic object – a spinning Kerr solution. We have a tightly spinning GR object that can shrink no further. Since we are trying to model an electron here, we use the standard black hole values (for an electron model this is a ‘naked’ a > m Kerr solution )
Brady’s sonons interact with the surrounding aether – how would that work in GR? We are after all taught that all GR objects like black holes have no hair. But of course they can have hair, its just that it will not last long. That’s the point here. Sonons can and will stop interacting if the background incoming waves die down below a certain point. But above a certain point black holes become perturbed, and things like ‘superradiance’ as Teukolsky and others discovered come into play.
So pure GR has at least the ability to interact in interesting ways, but are the numbers there? What frequencies do we need for Brady like Sonons constructed from GR (I’ll call them geons from now on) to get to the point where there are electromagnetic strength interactions are taking place?
Bradys interactions occur with mass transfer – the compressible fluid carries away mass to and from each Sonon in a repeating manner. Not a problem for any GR ‘blob – geon’. If they interact, then energy must be flowing in and out – that’s the definition of interaction.
An Electron Model
A previous post here - An Electron Model from Gravitational Pilot Waves outlines the process.
We take a small region of space (e.g. containing a Kerr solution) and assume that this region of space is exchanging gravitational energy with its surroundings. Call it an geon-electron.
Assuming that the exchange takes place in a periodic fashion, the mass of this geon-electron (energy contained inside of the small region of space) is given as
me(t) = me*((1 – f) + f*sin(vt))
where v is some frequency, and f is the proportion of mass that is varying, so f is from 0 –> 1.
This varying mass will give rise to changes in the gravitational potential outside the region. But gravitational effects do not depend on the potential, rather they depend on the rate of change of the potential over spacetime intervals. So it’s not the potential from this tiny mass that is relevant, it is the time derivative of the potential that matters.
Potential = -G*me(t)/r
Look at the time derivative of the potential
dP/dt = -G*me*f*v*cos(vt)/r
This gradient is what one can think of as the force of gravity. This force rises linearly with the frequency of the mass oscillation.
The EM force is some 10^40 times that of gravity, so we just need to use this factor to figure out an order of magnitude estimate of the frequency of this geon mass exchange rate.
This is detailed in the ‘Coulomb Attraction’ section of an earlier post.
Using de Broglie’s frequency – he considered the Compton value of 1.2356×1020 Hz as the rest frequency of the internal clock of the electron, one arrives at an electron model with these properties:
- Entirely constructed from classical General Relativity
- Frequency of mass exchange is the Compton frequency
- Electromagnetic effects are a result of GR phenomenology
- Quantum effects such as orbitals and energy levels are a natural result of these geons interacting with their own waves, so QM emerges as a phenomenon too.
The State of Physics today -------------------------- Obviously a sea change in fundamental physics would be needed to allow for anything like these ideas to be considered. In fact its not that the ideas here might be correct - but rather that Brady and others who toil on actual progress in physics are sidelined by the current 'complexity is king' clique that is the physics community today. The physics community is more than it ever has been in the past, a tightly knit clique. This may be the fault of the internet and the lock in group think that instant communication can provide. This clique gives rise to ideas like 'quantum mechanics is right' and other absurdities, such as the millions of hours spent on String Theory, when it's 'not even wrong'.
Tests and Simulations
Given the entrenched frown on the subject of alternative bases for the underpinnings of our physical world, we need to look for experimental evidence to support these kinds of theories.
The work of Yves Couder and his lab in one kind of essential experiment. They have shown conclusively that quantum like behaviour can emerge from classical systems.
Another path – one that in my opinion has been somewhat neglected in this field is that of numerical techniques.
Here I outline some steps that might be taken to construct a GR based model of an electron. Excuse the more colloquial manner, I am making notes for a future project here!
There are only about 22 Compton wavelengths within the Bohr radius. So if one goes to a 100 Compton wavelength simulation zone, with 1000 grid points on a side, thats 1e9 grid points, and each point needs only four 8 byte doubles, so 32 bytes, so 32 GB.
The equations to solve on this simple grid are those of fluid dynamics: Compressible Isothermal Inviscid Euler equations. : As from I do like CFD.
With a 32GB data set, 1e9 data points, and about 1000 computer FLOPs per visit, we have 1e12 FLOPs per time step, and an algorithm that gets 10GFlops, I get about a minute per time step. Each time step needs to cover about 1/100th of the Compton time, or about 1e-22 secs, and we need to let light cross the atom (3e-19 secs) hundred times to get things to converge, or about 3e-17secs, so 300,000 time steps. (Better speed up the algorithm! Should be easy to get 20GFlops over 8 processors, and perhaps cut Flops/grid point down, which could mean a day or so on a 8 core Intel).
- Cannot use already built CFD codes, as they are not built for the high dynamic range ‘Couder like memory’ like I am looking for.
- However Euler equations and physical setup is trivial for a CFD problem.
- 1000 resolution seems high enough.
- Two ‘electrons’ (one out of phase so will be a positron, or perhaps heavier and a proton) will be modelled as a point like particle that provides expansion/contraction of the aether, while at the same time each of these masses responds (‘surfs’) on the generated field.
- The goal then is to create a model of the hydrogen atom (or positronium).
- The electron should orbit the proton with Bohr like energy levels, determined numerically without recourse to QM.
- Photon/Atom interactions could also be modelled – photons should emit from the assembly when its started with ‘too much energy’.
Note on the Fine Structure Constant (useful in a numerical model)
The quantity was introduced into physics by A. Sommerfeld in 1916 and in the past has often been referred to as the Sommerfeld fine-structure constant. In order to explain the observed splitting or fine structure of the energy levels of the hydrogen atom, Sommerfeld extended the Bohr theory to include elliptical orbits and the relativistic dependence of mass on velocity. The quantity , which is equal to the ratio v1/c where v1 is the velocity of the electron in the first circular Bohr orbit and cis the speed of light in vacuum, appeared naturally in Sommerfeld’s analysis and determined the size of the splitting or fine-structure of the hydrogenic spectral lines. [*]
See also the Wikipedia physical interpretation section.