Contents:

1. Ether and Einstein

2. An Ether Compatible with Relativity

3. Klein Gordon and Dirac Equations

4. Interference Properties

5. Interference Effects and Quantum Mechanics

6. Experimental Support and MOND

7. Afterword

A. Appendix: Special Relativity Computations in the Proper Time
topology

The concept of a "luminiferous ether", (that there must be a firmament that carries the vibrations of light, and there must therefore also be a preferred frame of reference) has been largely ignored in physics since Einstein so beautifully explained the Michaelson-Morley light speed measurements in his 1905 paper on Special Relativity. So in a paper that postulates the existence of an ether, I should begin with a critique of the reasoning that first suggested that ether not exist.

Einstein's reasoning was based on two postulates:

(1a) *The laws of nature are the same in all inertial frames of reference.*

(1b) *The speed of light is the same in all inertial frames of reference.*

Based on these principles, Einstein shows that Galilean relativity is
inadequate. This we all agree with. But even the assumption of
postulate (1b), as a perfect fact of nature, does not in itself disprove
the absence of a luminiferous ether. Instead, it shows is
that if there is a preferred inertial frame of reference attached to the ether,
an experiment in that frame will measure the speed of light as *c*.
Throughout this paper I will assume "inertial" when talking of frames,
unless otherwise noted, and by "preferred", I mean the rest frame which
is connected to the ether and so is unique subject to an arbitrary
rotation (but not to a boost).

Einstein's special relativity paper shows that time is relative. More precisely, what he shows is that the measurement of the passage of time is dependent on the movement of the clock that measures it. This is less than showing that there is no absolute time. What it shows is that there is a relation between movement and the perceived passage of time. If there is a preferred frame of reference, then the passage of time in that frame would be a logical choice for absolute time, at least in the "Special" theory. All other frames would experience the same absolute time, but a slower perception of time.

Modern text books teaching the theory of Special Relativity give a series of arguments showing why ether was not acceptable, but these arguments do not apply to all varieties of ether, as will be shown later. They typically cover (a) Galilean ether, (b) a version where matter drags frames around, and (c) a version where light travels at different speeds depending on the velocity of the source. But not all models for ether are subject to the same deficiencies as these three. In the next section of this paper, I will present a version of ether that is compatible with both special and general relativity.

The logical argument for the existence of ether was the belief that waves require a medium in order to be transmitted. That is, there must be something that is affected, in order to have a wave. This is a common observation of our physical world, and it was natural to extend it to the subject of light. But the simple versions of ether that the 19th century physicists tested were inadequate.

A world of Galilean ether would be a tough world to live in. This is a fact that was probably not appreciated in the 19th century. But if the speed of light depended on the absolute velocity of the frame it was measured in, our modern technological apparatus would be subject to small effects that would have complicated periodicity due to the motion of the earth. What's worse, life itself, with its strong dependency on finely tuned molecular interactions, would be very difficult to produce in a world with Galilean ether. And an ether where the velocity of waves depends on the velocity of their source is a difficult and complicated theory, as is one where the ether is dragged around by matter.

19th century physicists should have known that the Galilean ether was impossible. Some crystal forms, for example, should have shown growth dependencies on the time of day, or the time of year, that they were crystalized. As the magnitude of the ether wind changed with the millenia, there should have been differences in the crystal habitats of the minerals that were laid down. They had no such observational evidence of ether, but still they expected that the ether was out there.

The absence of evidence for ether, combined with the extensive search for it, suggests that the logic behind the existence of the ether was stronger than the modern physics mind realizes. It was only with great difficulty that physics was weaned off of ether, and the separation was effected only by the repeated failures of its predictions. Alone among modern physics theories, the ether was supported without any experimental evidence at all, but instead on the basis of its compelling logic alone. Consequently, a version of ether that supports relativity needs to be seriously considered. Such an ether would presumably have received strong support 100 years ago, and if it is compatible with experimental observations between then and now, it should be as seriously considered now.

It is at least logically consistent to arbitrarily add ether, with an implied preferred frame of reference, to Special Relativity (or General Relativity, as will be shown in a section 2), as Einstein did not prove that no such frame existed. What he showed instead, was that given his postulates, there was no apparent need for such a reference frame. Adding an additional postulate, to the effect that a preferred frame of reference exists, gives a slight change to the interpretation of some of Einstein's results, but no changes to the mathematics or its predictions. One change is that the preferred frame of reference provides a consistent time ordering of spacelike separated events. Another is that the Lorentz rotations are interpreted not as mixing space and time, but instead as indicating the subjective rate of passage of time (i.e. proper time) in a moving reference frame. This means that boosts will no longer mix space and time, but instead will simply alter the relationship between absolute time, and the measurements of clocks.

So this paper is written with an additional postulate to Einstein's, one that implies the existence of a preferred frame of reference, but rather than make that postulate alone, I will instead postulate the reasoning behind the original belief that ether should exist:

(2) *There is only one true description of nature.*

This is not to be interpreted in the sense of coordinate transformations, or in terms of translational and rotational invariances, but instead in the sense of geometrical content. For example, a description that uses polar coordinates is as true as one that uses Cartesian coordinates. But an explanation of electricity and magnetism that uses a potential that results in the same predictions when the potential has a constant added to it, cannot be true description of nature. The postulate implies that if nature has a potential, then that potential has a value. Our inability to define that potential is an indication that our understanding of the potential is incomplete, not that we live in more than one physical reality.

As an example of the use of the above postulate, consider the success of gauge field theories. The vector potential for E&M in quantum mechanics is not defined in terms of particular values, but instead a wide range of possible potentials are equivalent. The various potentials are connected by "gauge transformations", which redefine the vector potential in geometrically non trivial ways. According to the above postulate, only one of these gauges can be "true". The others must therefore either be accidents of our choice of coordinates, or possibly be created by a force that imitates the symmetry.

Note that our inability to guess the true gauge does not prevent us from making deductions based on the assumption of its existence. Indeed, we choose any gauge that appeals to us, quantize the others that we didn't choose (and are therefore assumed to be forces mediated by the exchange of particles), and when we make calculations for things we can observe, the results will turn out to not depend on our choice of gauge. Under the above postulate, the success of this program does not prove that there is no true choice of gauge, but instead shows that nature is so good at imitating the gauge symmetry that we cannot distinguish between situations of nature connected by that symmetry, and so cannot (at least at the present time) know when we have chosen the true gauge, and therefore don't need to know what it is. But it was the existence of a true gauge that allowed us to deduce the forces that we quantized.

One might ask, "if nature is so good at hiding her details from us, why should we care?" The first answer to note that no amount of experimental evidence can prove that a symmetry truly is exact. We are very finite creatures, so we should be aware of what to look for that would indicate that a symmetry of nature that we assumed was perfect, was, in fact, only approximate. The second answer is to note that at the present time, the assumption that the observed (and therefore at least approximately accurate) symmetries of nature are perfectly accurate has led us to an understanding of quantum mechanics and relativity that is not only difficult to reconcile with each other, but is impossible to reconcile with our own experience as physical creatures. This has resulted in bizarre notions as to what the nature of the universe is.

Arguably the worst example of nature showing two apparently equally true faces, at least apparently, is in the concept of "duality" in quantum mechanics. It's said that quantum objects (which I will call particles from here on) are both particles and waves. But the particle form occurs only after the wave form is "collapsed", so these two forms cannot coexist at the same time. Instead, the wave form is what describes an experiment before it is run, while the particle form describes the result of the experiment after it is complete.

This division of labor between the particle and wave descriptions of matter and energy suggests that time must pass, from the future to the past, for a wave to be collapsed to a particle. But the equations of physics do not describe this collapse in detail. Instead, quantum mechanics associates the wave function amplitude with a probability of a given particle position (or other measurement). Without a true global time, quantum mechanics shows violations of causality. Bringing back the ether solves this problem, though it does leave locality broken, if not in the passage of information, then in some sort of influence. More will be discussed on this later, along with Bohmian mechanics, an alternative interpretation of quantum mechanics.

The fact that it is the passage of time that collapses wave functions, along with the obvious observation that the passage of space has no effect on wave functions, suggests that the mixing of space and time by relativity is not nature's truth, but is instead a result of an ether with a symmetry that imitates the mixing of space and time by altering clock rates.

In any given rest frame, all light travels at the same speed. This is compatible with our usual understanding of waves, in that when we want to model waves, and to make them as simple as possible, this is exactly what we assume; that the speed of a wave does not depend on its wavelength. The 19th century physicists, following Maxwell, noticed this simplicity in light's behavior, and assumed that light was simply another one of the many physical waves following such a law, and so began the search for Galilean ether.

But it is not in the motion of light that the world is complicated. Instead the complexity is in the motion of the clocks that measure distances and the speed of light, and in the relationships between relative rest frame velocity and measurements of distances and times. Thus it is not light itself that is complicated, but instead the motion of the massive bodies that are used to measure light. In any one given rest frame, and in particular in the preferred rest frame of the ether, light is very simple.

So if we assume a preferred rest frame, the motion of light in it
is natural and exactly as expected. Instead it is the motion of
massive objects that must be bizarre and unexpected. Where the searchers
for the ether ensured their failure was in searching only for the
"luminiferous" ether, rather than a more general ether whose waves
define the motion of *both* energy *and* matter.

Minkowski space itself makes a poor choice for the ether because it intrinsically mixes space and time. That is, time is not treated as a special coordinate distinct from the spatial dimensions, but instead, at least locally, is indistinguishable from them. But the metric that defines Minkowski space is very important to the observed motion of matter, so it's a good place to start in a search for an ether compatible to relativity:

(3) *ds*^{2} = *dt*^{2} – (*dx*^{2} + *dy*^{2} + *dz*^{2}).

If time is going to be held distinct from space, we need to bring
*dt* to one side of the equation:

(4) *dt*^{2} = *ds*^{2} + *dx*^{2} + *dy*^{2} + *dz*^{2}

The above metric defines how much absolute time has passed given that the
object has changed its position by *dx*, *dy*, and *dz*, and
has experienced a passage of its own proper time by *ds*. An ether
that exhibits this metric will naturally exhibit the relationships
between proper time and absolute time that special relativity predicts
(and therefore also Lorentz contractions, see Appendix A), but there
is the small problem of how to interpret *ds* as a dimension
of the ether.

Einstein's equations mix space and time, the above metric mixes
space and proper time. This seems to make no sense, as different
objects have different proper time. But make the *s*
dimension cyclic, with radius *R*, and if *R* is small
enough, we will get an ether that appears to large ungainly creatures
such as ourselves as the standard **R**^{3} that we all
perceive.

In several ways, this is a simplification of the metric defined by Einstein.
Instead of a Minkowski space, this is locally a simple
Euclidian **R**^{4} space, and is globally a standard
differentiable manifold, **R**^{3}x**S**^{1}.
And since *ds* is defined for the movement of all objects, no
extraneous degrees of freedom have been introduced.

Another simplification is that in this topology, all objects, massive or massless, travel at the speed of light. The old ether, the one that was rejected by experiments 100 years ago, would have had difficulty supporting the movement of matter waves, given that matter travels at various speeds. But this ether greatly simplifies that problem by instead assuming an ether where all objects (i.e. matter waves) travel at the same speed. The additional dimension is an extra degree of freedom, but the restriction of speeds to that of light removes one degree of freedom, and leaves the equations of motion unchanged from classical special relativity.

To convert an event in this topology over to an event as defined in special
relativity, simply ignore the *s* component. The remaining
four components (one being the absolute time), provide the four vector
for the event (in the reference frame of the supposed ether). But
converting an event the other way is subject to a slight problem, what
to choose as the *s* component.

In working practical problems in classical special relativity, it will be
found sufficient to define the arbitrary *s* component as zero. This
may not be an obviously successful tactic, and since the reader may also
have difficulty appreciating
the fact that this topology gives the same results
as special relativity, I've included sample calculations for time dilation
and Lorentz contraction in Appendix A, showing them to be identical
between the two topologies.

The arbitrariness of the *s* component should not be surprising
given that the uncertainty principle shows that defining the position
of an object to perfect accuracy will result in its momentum being
completely uncertain.

That matter, or at least electrons, always travel at the speed of light is not a concept first introduced by this paper. The idea was a feature of the zitterbewegung theory of Schroedinger in 1930. A good modern paper along this line is "The Zitterbewegung Interpretation of Quantum Mechanics" by David Hestenes, Found. Physics, Vol. 20, No. 10, (1990) 1213-1232. The zitterbewegung idea is that stationary electrons move at the speed of light in circles just large enough to provide the spin angular momentum of the electron. The difference between the Zitterbewegung concept and this topology is that the Zitterbewegung helices are assumed to be in the usual 3 space dimensions.

The generalization of this topology to General Relativity follows the example of Special Relativity. We need to get the metric into a form where all the curvature is in the space coordinates:

(5) *ds*^{2} = *dt*^{2} -
*g*_{ij}*dx*^{i}*dx*^{j}.

The above metric gives coordinates that are called a "synchronous coordinate system" or a "Gaussian normal coordinate system". How to set up such a coordinate system for an arbitrary spacetime manifold is shown in exercise 27.2 in the classic text "Gravitation" by Misner, Thorne and Wheeler (1971), page 717. The existence of such a coordinate system is similar to the choice of gauge in gauge field theories.

When one performs the above program on the standard Schwarzchild metric the resulting geodesics all eventually terminate by falling into the black hole's event horizon. The natural ether interpretation is that gravitation is caused by matter's general tendency to destroy ether. A black hole is then a region where ether is being destroyed faster than the speed of light would allow light waves to escape the destruction. We sometimes see a similar effect in rivers; when the current is faster than the wave velocity, waves cannot propagate upstream. Note that the frame dragging effects of rotating black holes show that matter's interaction with ether is not a simple scalar destruction.

Since it is clear that matter has a distorting effect on the ether, one wonders whether it would also modify the radius of the hidden dimension. If presence of matter increased the dimension this would give an attractive force that would be felt by massive objects but presumably ignored by massless ones. A decrease in the dimension would cause a repulsive effect. But neither this, nor the details of the frame dragging from the previous paragraph will be further discussed in this paper, which instead assumes a flat ether, except in section 6 where the MOND effect is calculated.

Motion in the hidden dimension *s* implies the experience of
proper time. Therefore this topology will be called the "proper
time topology".

In some ways, the Klein Gordon equation seems to be the only wave equation in particle physics, in that the components of the Dirac and the Maxwell equations also satisfy the Klein Gordon equation. Where the Maxwell and Dirac equations differ from the Klein Gordon equation is in the number of field components, and in how those components interact. Maxwell's equations have already been derived as a consequence of a hidden dimension. This is known as the Kaluza-Klein derivation, and need not be repeated here.

The assumption of this topology is that all particles travel at light
speed, and have zero mass, therefore the natural wave
equation to postulate is the massless Klein Gordon equation,
with, of course, the extra space dimension *s*:

(6) ∂^{2}/∂*t*^{2} Ψ =
*c*^{2}[ ∂^{2}/∂*x*^{2} +
∂^{2}/∂*y*^{2} +
∂^{2}/∂*z*^{2} +
∂^{2}/∂*s*^{2} +
*V*(*x*,*y*,*z*,*s*) ]
Ψ(*x*,*y*,*z*,*s*,*t*).

Taking a Fourier decomposition of Ψ in the *s* dimension,
and keeping only the lowest order terms *cos*(*s*/*R*)
and *sin*(*s*/*R*), turns the
∂^{2}/∂*s*^{2} term into
a mass term, if *m* = *1/R*, giving the standard
Klein Gordon equation in the usual 3 dimensions:

(7) ∂^{2}/∂*t*^{2} Ψ =
*c*^{2}[ ∂^{2}/∂*x*^{2} +
∂^{2}/∂*y*^{2} +
∂^{2}/∂*z*^{2} +
*V*(*x*,*y*,*z*,*s*) -
*m*^{2} ]
Ψ(*x*,*y*,*z*,*s*,*t*).

The usual matrix form of the Dirac equation is sufficiently complicated that playing with it was my evening entertainment for several years. The articles by David Hestenes are easily the most illuminating on the subject, but unfortunately these papers require an understanding of "Geometric Algebra". An "algebra" is collection of "numbers" or "points" that can be manipulated with certain rules and operations. A good introduction to Geometric Algebra, and how to use it to derive Schroedinger's and Pauli's equations from the Dirac equation is Spacetime Physics with Geometric Algebra, David Hestenes, Am. J. Phys, 71 (6), June 2003, especially section VII. The algebraic rules have the flavor of exterior calculus, with the notation a little less revealing, and therefore more confusing, but most importantly it is much more inclusive.

I'm sure that most of the readers of this article have failed to click on and read the above link, and that's too bad, but perhaps I will give a few more reasons to go back and click on it, and to at least read it through once. The "spacetime splits" that Hestenes talks about are equivalent to my assumptions of specific ethers, though these splits are limited to flat Minkowski space, a limitation that I expect the author to soon repair. I should note that Hestenes is not tainted with any connection to me or the theory of this paper, though I have found his work particularly illuminating.

In my own words, what Hestenes shows, in the above paper
and in a few others, is that Schroedinger's and Pauli's
wave equations are derived directly from the Dirac equation,
and that Schroedinger's equation is that of a particle
in an eigenstate of spin, unperturbed by any spin interactions.
The γ^{μ} matrices in Dirac's equation are
reinterpreted as defining a set of orthonormal coordinate
axes in Minkowski space.

The reinterpretation shows that a spinor should be thought of as one half of a proper Lorentz rotation. That is, it is usual to imagine a Lorentz rotation as a 4x4 matrix, but this wastes a lot of coordinate information because of redundant information. His algebra breaks a Dirac wave function into the product of the square root of a probability density, a phase factor, and a "rotor". A rotor is an object that defines a Lorentz rotation. The phase factor is the traditional cause of interference in quantum mechanics, and is not explained in Hestenes papers. In section 5, I will return to this subject.

In equation (192) of the above reference, Hestenes writes
the Dirac equation in "coordinate free" form. The equation
explicitly assumes a time like vector γ_{0}, and
a bivector **i** = γ_{2}γ_{1},
where γ_{1} and γ_{2} are
perpendicular to γ_{0}, but it does not
use any coordinates as defined by these vectors. I guess
this is "coordinate free", but it does require the
assumption of selected coordinate axes, and therefore
is quite compatible with an ether requirement. He shows
that the equation is Lorentz invariant, in that the
results do not depend on the choice of coordinate axes.
The situation is reminiscent of gauge field theories, in that
geometrically distinct conditions result in the same
physical observations, but despite this symmetry, a choice
of gauge must be made. Note that the equations of general
relativity, on the other hand, in addition to being
coordinate free, are also axis free. This suggests that
other than splitting off the time coordinate, we should
not try to gauge general relativity as it could be a true
description of nature. This assumption will be made for
the remainder of this paper.

Note that to use a rotor to perform a Lorentz rotation on an object requires that the rotor be used twice. This explains why the probability density shows up as the square of the quantum wave function better than any other explanation I've seen. Now go, read the above link at least once, and don't come back until you're done. You'll thank Hestenes (and maybe me too) for it.

You may have noted that section VIII of the above Hestenes paper includes a short introduction to Bohmian mechanics. This paper takes a slightly different interpretation than Bohmian mechanics, in that the wave and particle parts of the complete description of a quantum object are here assumed to be true only at different "real" times. That is, if the experiment is to be run at 3:00PM, then at 2:45PM a "true" description of the particle's prospective tracks is given by the wave function, while at 3:15PM, a "true" description of that same 3:00PM event, will now be given by the actual particle track. Thus the two conflicting descriptions of the same event are both true, but only at different moments in time. It's not that nature is dual, instead that with the passage of time, the true nature of the event changes from a prospective wave to a retrospective particle.

Because of this interpretation of time, this paper disagrees with Bohmian mechanics over the question of "wave function collapse". Instead, this paper agrees with the Copenhagen interpretation, where a wave function collapses to the result of a measurement. But instead of looking for a mysterious "measurer", I look for the simple passage of time. And the time that must progress is proper time. This was the original motivation for exploring the proper time topology, an effort that is documented on the web in this link: Searching for a fundamental reality in physics

This paper's concept of the passage of proper time and wave function collapse indicates that our universe truly cannot be described in the way that classical mechanics implicitly assumes. Classical mechanics uses the fact that there is a situation, and that situation changes with time. But even with the assumption of an ether, and therefore a global time coordinate, the observed facts of quantum mechanics prevents the universe from reasonably being described by a situation that evolves from moment to moment. As an example, consider the astronomical case of galactic lensing. A distant galaxy acts as a magnifying lens, showing an even more distant galaxy. In this situation, one can set up an experiment where a photon is arranged to interfere with itself despite the passage of billions of years of space, and billions of years of time. That the photon has not been collapsed, and so is capable of interfering with itself over such a magnificently huge two-slit experiment, is an indication that the wave function collapse is tied to the passage of proper time, not that of the global time. If, for example, another photon had been simultaneously emitted in an entangled state, it would not be possible for a polarity measurement of the first photon to complete before billions of years of global time had gone by.

Experiments showing that uncollapsed quantum waves can be arranged at significant distances in space and time are, under the assumptions of this paper, assumed to result from the particles having been accelerated to such high speeds that they experience little of the passage of proper time. Mother nature's demonstrations of the same effect, but with light, are more impressive.

With the picture of wave functions as the true description of future experiments, and particle tracks as the true description of past experiments, one must consider the action of the passage of proper time that changes one to the other. Physics does not deal with this action much. There is quite a lot of literature on the "Quantum Zeno Effect". An example of this effect would be a radioactive particle that emits a gamma ray with some half life. Half lives imply that the probability of the particle decaying must follow an exponential curve. But quantum mechanics shows that the particle, just after being "observed", has a smaller probability of decaying. The mathematics follows from the fact that wave functions are square roots of probability densities, and therefore always show up squared. The probability of a very early decay ends up squared, and therefore smaller than an exponential decay would indicate. By the way, I figure that the mathematics behind the MOND effect is similar, as will be discussed in section 6.

This action of the passage of time, along with the assumption that true descriptions of nature exist, suggests that there must be a topological relation between a description of a quantum object before and after the passage of time. This assumption can be made even in the absence of an equation relating the two descriptions. Since there are many different ways of describing a particle or a wave, the requirement that they be topologically related reduces the number of descriptions that we need to consider. Also note that with this interpretation of quantum mechanics, the action of wave function collapse is, in some ways, analogous to the reverse of the action of quantization. This allows these two seemingly unconnected processes to be justified with respect to each other. But quantization, and also "second" quantization, deserve another paper because this one is already more than long enough.

It's pretty clear that we can describe a particle position with a scalar field, providing we allow a delta function in the field. But Hestenes' description of the Dirac equation indicates that a Lorentz rotation vector should also be included. For this reason, it's more natural to describe the particle track by a Lorentz vector field, where most of the vectors are zero. The above link "Searching for a Fundamental Reality", includes an intuitive justification for the choice of a velocity vector field based on a classical analysis of Schroedinger's equation, but there is no question that Hestenes' more general and elegant analysis is to be preferred. (Now are you finally going to go back and read that paper?)

But there is one part of Hestenes' analysis that is lacking, and that is a derivation of the quantum interference effects. Having only learned the rudiments of the geometric algebra a few days ago, I will refrain from deriving the results in the Dirac equation, and instead return to the derivation of interference effects in Schroedinger's equation in section 5. But first, the next section covers some interesting and useful self interference effects in the proper time topology.

In the interest of fulfilling the principle of simplicity, as well as in recognition of the fact that all types of matter and energy require only a single Plank's constant, I will make the assumption that there is only one fundamental subparticle that makes up all of matter and energy. This is not so radical as it looks. All of both matter and radiation uses the same Plank's constant. And the zitterbewegung model of the electron requires that they travel in tightly helices. Maybe those helices are tightly bound states. That is the assumption of the rest of this paper, but this is more for the convenience of not having to keep track of the variety.

If the electron has substructure, our intuition would suggest that the constituent particles should be lighter than the electron, and therefore observable in the lab, an observation that has not been seen. But our intuition is honed on bound states where the binding energy is very small. Even in those states the combined masses of the constituents of a bound state is greater than the mass of the bound state itself, as Einstein's relation between mass and energy shows. This loss of mass is undetectable at chemical energy levels, but can be detected at nuclear energy levels. At the level of nuclear subparticles, such as the quark, the missing mass eliminated by the bound state is so large that quarks are permanently bound.

Renormalization causes the rest mass of particles that have very strong charges to be very high. Since the posited subparticle is to be massless, this means that it must be characterized with a frequency which is very high. In this scheme, there is only one very strong fundamental force in nature. That force binds subparticles with an extremely high fundamental force. The Strong, electroweak, and perhaps the gravitational force are then the remanants of that fundamental force, after most of its field is cancelled in bound states. After renormalization, this makes the more deeply bound states appear to be lighter.

Even if we do not know how the fundamental subparticles form bound
states that define standard particles, we can make some deductions
based on interference effects in the proper time topology. These
effects are due to the requirement that the wave states for the
subparticles must be single valued around the *s* dimension.

So let *W*(*x, y, z, s, t*) be a plane wave
travelling at speed *c*, and representing this postulated
fundamental subparticle.
To model a massless particle travelling in an arbitrary
4-dimensional direction, define a wave vector * k*
and frequency ω with |

(8) * k* = (

(9) *Ψ*(*x, y, z, s, t*) =
*exp*(*i* (* k* •

Note that in this theory, ω is a fundamental constant
of nature. That is, if the particle is to experience 4 dimensions
of travel, its movement in the *s* dimension must be similar to
its movement in the usual 3 dimensions. Thus its frequency ω cannot
depend on its speed in the usual 3 dimensions. And since this is
the fundamental particle, its frequency is fundamental.

To be single valued the wave must match itself in the *s*
dimension. This is similar to the requirements
for bound solutions to a central force problem, but in this
case we have the unfamiliar consequence that the
spectrum of free space solutions is not continuous. Instead,
the spectrum is discrete, with the wave vector's *s* wave
number defined by an integer number:

(10) *Ψ*(*x, y, z, s, t*) =
*Ψ*(*x, y, z, s *+ 2 π* R, t*)

=> *k*_{s} = *n* / *R*.

This implies that the time dilation ratio is quantized:

(11) *R*_{td} = *k*_{s} / *k*
= *n c* / *R*ω.

And that the speed in real space is also quantized:

(12) *v*_{n} = *c*
√ (1 - (*nc* / *R*
ω)^{2}).

The number of different possible speeds is no longer infinite as in special relativity, but instead is finite:

(13) *N*_{s} = [ω *L* / (2 *c* π)].

In order for this theory to be realistic, *N*_{s} must
be very large, so in the remainder of this article, it is so assumed.
Since there are only a finite number of different speeds, there must be a
maximum speed, and this maximum is attainable:

(14) *V*_{max}
= *c* √(1 - (*c*/*R*ω)^{2})
~= *c* - *c*(*c*/*R*ω)^{2}.

There is also a minimum attainable speed, which may be zero, or
may be larger, depending on the exact values of ω, *c*,
and *R*. But the minimum attainable speed will be bounded
above by:

(15) *v*_{min} < *c* √( 2*c*/*R*ω).

The above equation (15) gives the quantization of speeds for speeds near zero. That is, low speeds are quantized approximately according to:

(16) *v*_{slow}
= *n* √( 2*c*^{3}/*R*ω) + *v*_{min}.

Since the topology has only a finite number of possible momenta, there is no need to pick an arbitrary momentum cutoff, or to cancel infinities due to ultraviolet catastrophes. Also, the zero point vacuum energy is finite, another property that a fundamental theory of nature should satisfy. The source of the divergence in standard physics are integrals over all possible momenta, these become finite sums in the proper time topology:

(17) (2π)^{-3} ∫ ∫ ∫
*F*(* p*)

=> (2π)

Since the *s* dimension is so small, and the Lorentz symmetry
so good, none of the above effects is going to be easy to measure.
So other than justifying doubtful regularization procedures, we
haven't done anything useful here. But the quantization of
velocities does suggest a justification for the fact that quantum
mechanics involves probabilities. Perhaps chance enters
into the process when a particle finds itself induced by forces
(or the distortion of space) to travel at velocities that are
incompatible with its momentum or energy. It must then choose another
velocity that is legal, and it does so randomly, or perhaps by
exchanging a very small energy particle of some sort.

Under this assumption, the fact that particles stay in stationary states despite the induced randomness is due to the fact that particle interference (in the wave form) extends across time. By "across time", I mean that according to the Feynmann prescription, the interference is calculated by summing the entire path of the quantum object. If it were instead the case that the inteference only applied at the point where the quantum object existed as a particle, and so was choosing which random path to follow, then the path could be described with a classical description where the system state changes with global time.

The previous section covered self interference effects in waves due to the cyclic nature of the proper time topology. Because the fundamental frequency ω is so high, the self interference effects are extremely difficult to directly detect. But interference effects between two waves, or between a single wave and itself (as in the 2-slit experiment), will result in much lower "beat" frequencies, and these effects are easily within the reach of physics today. This section explores these interference effects and shows that they are equivalent to quantum mechanical interference effects.

Classically, one of the oddest features of quantum theory, is the Pauli exclusion principle. This principle states that two fermions may not occupy the same state. This is a generalization, by fourier transform, of the more basic concept that two identical fermions cannot occupy the same position at the same time. As stated, this is a very general constraint on nature. A less restrictive principle, but with the same end effect, is to require that fermion waves interfere with each other in such a way as to cause the joint probability of their being found at the same point to be zero. In this form, the Pauli exclusion principle can therefore be derived from the interference effects between two Dirac waves with the same spinor state. Since Hestenes, in the above cited paper, has shown that Schroedinger's equation describes this limiting case of the Dirac equation, I will choose Schroedinger's waves to analyze the interference effects of quantum mechanics. In some future paper, after I've studied the Geometric Algebra for a few more weeks, I'll try to update this calculation to use the full Dirac theory.

Schroedinger's wave equation is a fundamentally complex
equation, and according to Messiah [ Quantum Mechanics,
first English edition (note: the link is to a later edition
which I do not have) (John Wiley ~1961), pages 222-224,
Volume I], the natural way to show the correspondence between
this equation and classical mechanics is to write
*Ψ* = *A* *exp*(i *S*/* h* ).
This is similar to what Hestenes does in his decomposition of
the Dirac equation, but, due to familiarity, I will follow
Messiah's notation. This is in the beginning of the section
entitled "Classical Limit of the Schroedinger's Equation".
Making this substitution, seperating into real and
imaginary parts, and multiplying by 2

(18a) ∂*S*/∂*t* + *grad*^{2} (*S*)
/(2 *m*) + *V*
= (~~h~~^{2} / 2*m*) (Δ *A*) / *A*,

(18b) *m* ∂(*A*^{2})/∂*t*
+ *div*(*A*^{2} *grad* *S*) = 0.

Equation (18b) is simply the continuity equation for the probablity
density *A*^{2}. It can be interpreted in a purely
classical manner. Messiah interprets *S* to be a potential
that generates a velocity field ** v** for that probability
density (where

(19) ** v** =

Messiah then shows that (with * h* = 0) the velocity
field follows the law of motion for a classical fluid under the
influence of the given potential. But if he had not taken the
classical limit, the conclusion that

Since the probability density acts like a classical fluid, as opposed to a gas, it need have only a single velocity defined at each point. This is not what a statistical ensemble of (non interacting) particles with various positions and velocities would give. Such a collection would be a gas, rather than a fluid. Instead, the "particles" strongly interact, and they interact in such a way as to allow only a single velocity at each point in space. This is compatible with the requirement that the wave function truly be a wave, rather than an ensemble of possible trajectories. It is also a hint as to the nature of quantization.

*S* only shows up with a gradient, except for the term
indicating how it changes with time. It is therefore possible
to take the gradient of that equation (18a), and then make a
change of variables to replace *grad S* with *m v*:

(20) m ∂* v*/∂

As an aside, the above equation suggests a generalization of Schroedinger's wave equation that may be useful in writing out an explicit wave function collapse equation. After both the wave equation and the particle track are described by velocity fields, there is still a very fundamental difference between them, and that is the Heisenberg uncertainty relation. The uncertainty relation does not hold for the particle track. It's a natural inclination to try and put a wave function into Schroedinger's wave equation that does not satisfy Heisenberg's uncertainty relation, just to see what Schroedinger's equation indicates that its evolution is, but one quickly discovers that this is impossible. Any wave function always must satisfy the uncertainty relation. But equation (20), is no longer bound by this restriction. While equation (20) does evolve solutions of Schroedinger's wave equation the same as Schroedinger's equation does, it also evolves velocity fields that do not satisfy Heisenberg's uncertainty relation. This suggests that (20) may be a starting point for an equation that, relates the collapsed wave functions to the uncollapsed wave functions.

Intereference effects in quantum mechanics show up in the joint wave functions, which are joint probability densities. To analyze the quantum effects, as opposed to classical effects, it's necessary to define joint probability densities for both cases, and to compare the two interferences. It's frequently stated that classical mechanics does not allow for identical particles, but this observation does not apply to a wave theory. Even in classical mechanics, waves is waves, and they cannot be distinguished. But different particles can be distinguished, and quantum mechanics gives the joint probability density of distinguishable particles the same as classical mechanics does.

The quantum mechanical fundamental description for a single
particle consists of its wave function, whose informational
content includes a probability density and a velocity potential.
The corresponding description for two particles is a joint
probability density, and a joint velocity potential. The joint
probability density is a function on **R**^{3} x
**R**^{3}, as it has to provide a density value for
the two particles. The two position coordinates will be called
*x*_{1} and *x*_{2}. To be specific,
let *P*(*x*_{1}, *x*_{2}) *d*^{3}*x*_{1} *d*^{3}*x*_{2} be the probability
that the the first particle is found within
*d*^{3}*x*_{1} of position *x*_{1}, and the second particle
within *d*^{3}*x*_{2} of position
*x*_{2}.

I've already made the more sensitive wince, by refering to
"first" and "second" particles just inches from having previously
assumed that they are identical. Because of this defacto
enumeration of the particles, most of our calculated probability
density values will be 1/2 (for *n* particles,
1/*n*! ) the literal value, (but they'll be correct when
*x*_{1} = *x*_{2}). The reason
physicists use this way of counting is that it makes
the wave functions easier to normalize and calculate with. That
is, if our space only had two points, *a* and *b*,
we will have
*P*(*a*,*a*) + *P*(*a*,*b*) + *P*(*b*,*a*) + *P*(*b*,*b*) = 1.

We'll compare the four cases using notation common to the
quantum case. So let *P*_{1}(*x*) and *P*_{2}(*x*) be the probability densities
(i.e. *P* = *A*^{2}), and
*S*_{1}(*x*) and *S*_{2}(*x*)
be the velocity potentials for the corresponding quantum particles.
Classical particles are described likewise, except that there is
no velocity potential.

If the particles are distinguishable, the combined probability density is just the product of the densities. This is true for both classical mechanics and quantum mechanics:

(21) *P*_{dist}(*x*_{1},
*x*_{2}) =
*P*_{1}(*x*_{1}) *P*_{2}(*x*_{2})

If the particles are identical, but are classical, we have pretty
much the same thing, but since the particles are identical, we
have to take this fact into account. Note that the factor of 2
is from our splitting the density for *x*_{1} ≠
*x*_{2} into to two parts; the factor which may have
caused some wincing:

(22) *P*_{class}(*x*_{1},
*x*_{2}) =
(*P*_{1}(*x*_{1}) *P*_{2}(*x*_{2}) +
*P*_{1}(*x*_{2}) *P*_{2}(*x*_{1})) / 2.

The formulas for identical fermions and bosons are more
complicated as they also depend on *A*. In these
equations, the ± indicates bosons (+) or fermions (-):

(23) *P*_{quant}(*x*_{1},
*x*_{2}) =
(*P*_{1}(*x*_{1}) *P*_{2}(*x*_{2}) +
*P*_{1}(*x*_{2}) *P*_{2}(*x*_{1})) / 2

± [*P*_{1}(*x*_{1})
*P*_{1}(*x*_{2}) *P*_{2}(*x*_{1})
*P*_{2}(*x*_{2}) ]^{1/2}
*cos* [ (
*S*_{1}(*x*_{1}) - *S*_{1}(*x*_{2})
) - (
*S*_{2}(*x*_{1}) - *S*_{2}(*x*_{2})
) ].

Note that *S*_{1} or *S*_{2}
could have a constant added to it with no change in the result.
This is a global symmetry of *S*. But there is also
a local symmetry in that local changes to *S* by multiples
of 2π result in no change in the joint probability density.
These observations also apply to the joint velocity potential,
*S*_{quant}(*x*_{1},
*x*_{2}). Also note that if *P*_{1}
and *P*_{2} share no support, then
the quantum mechanical formula is identical to the classical
formula.

The fact that the joint wave function does not depend, even
locally, on changes in *S* by 2π suggests that
*S* should not be considered to be a real function.
Instead, it's natural range should be **S**^{1},
a hint at a hidden dimension.

Since we are assuming that the fundamental nature of a quantum
field should be a velocity field, we need to convert equation
(23) into a form that depends on * v* instead of

(24) *P*_{quant}(*x*_{1}, *x*_{2}) =
(*P*_{1}(*x*_{1}) *P*_{2}(*x*_{2}) +
*P*_{1}(*x*_{2}) *P*_{2}(*x*_{1})) / 2

± [*P*_{1}(*x*_{1})
*P*_{1}(*x*_{2}) *P*_{2}(*x*_{1})
*P*_{2}(*x*_{2}) ]^{1/2} *cos* [
∫_{1}^{2}*m* (*v*_{1} -
*v*_{2})/* h* •

When one calculates the interference between two real waves, one expects to get a result that is a product of real waves. For example:

(25) *cos*( A ) + *cos*( B ) = 2 *cos*((A + B)/2) *cos*((A - B)/2).

The above equation has the usual interference term that we expect.
There are two sinusoidal waves in the product. The first one,
*cos*((A + B)/2), is an average of the interfering waves, while
the second one, *cos*((A - B)/2), is a difference. In
equation (24), by contrast, only the difference term arises, the
other does not. And the two cosines on the left hand side of
equation (25) do not appear at all in the uninterfered probability
densities. This suggests, along the line of the zitterbewegung theory,
that the missing terms are due to their having such high frequencies
that they are not detectable at our own energy levels. If
Schroedinger's wave equation averages out these high frequencies,
only the low difference frequency appears.

What's really odd about this interference is that it is completely
cyclic in * h*. That is, if you are considering points
that are at a distance Δx, an addition to the momentum

Of course it's an accepted fact that angular momentum is quantized.
But the cosine factor in equation (24) suggests that there is a
milder form of a sort of quantization for momentum itself. That
is, that momentum integrated over a distance interferes with itself
according to multiples of * h*. If it were not for the
proper time topology, this would be a rather bizarre feature to
have to explain.

Accordingly, let *W*_{1} and *W*_{2} be
two waves in the Proper Time topology, and let's compute their
interference patterns. As before, assume that the two waves use
the same ω, but have different wave vectors, so let their corresponding wave vectors be *k*_{1} and
*k*_{2}. Assuming equal magnitudes
for the waves represented as cosines then the interference is:

(26) *W*_{1}(*k*_{1} •
** x** –

~=

=

The (*k*_{1}+*k*_{2})/2 term
is of the same form as the original *W* waves, and approximately
of the same length (and therefore momentum), while the (*k*_{1}-*k*_{2})/2 term is
a lower frequency interference term. Since the wave functions
correspond to particles with velocity *c*, we can convert
*k*_{1} and *k*_{2}
to velocity format using the relation ** v** =

(27) *cos*((*v*_{1} - *v*_{2})
ω/2*c*^{2} • ** x**).

The above term gives the inteference between two plane waves. For more general waves, the above is integrated along a suitable path, and the general interference term is therefore:

(28) *cos*(∫_{1}^{2}
(*v*_{1} - *v*_{2})
ω/2*c*^{2} • *d***η**).

Comparing term (28) to equation (24) shows that interference between real waves in Proper Time topology will be equivalent (at least in the sinusoidal parts) to interference between complex waves in Schroedinger's equation providing:

(29) ω/2*c*^{2} = *m*/~~h~~.

My guess is that when this calculation is redone with the Dirac
equation and geometric algebra, the equivalence will trivially
extend to the amplitude portions of (24) and (28) as well,
but that calculation will await my further exploration of
the geometric algebra.
Consistent with the Proper Time topology, the above equation
corresponds to the frequency of a particle travelling at speed
*c*, but there is now an extra factor of two.
This factor comes from the conversion between quantum and
classical interference. Quantum interference is defined as a
straight subtraction, but classical interference takes the
difference and divides by two.

At the present time, the frequencies implied by the above equation are too high to be observed. But beat frequencies are much lower, so it is in the interference interactions that the Proper Time topology shows itself as quantum mechanical interference.

An argument can be made that the quantization of velocity (16), along with a characteristic particle frequency ω imply that very small accelerations will be modified under the proper time topology. The idea is that velocities are quantized by a constraint that a (non relativistic) fundamental particle will encounter at a rate of approximately ω. If the acceleration is so small that its influence over a time 1/ω will be less than the minimum quantum of velocity, then the effect on the particle's motion is unclear. One would expect that the effect of random perturbations to the particle velocity would cancel out, and the particle would receive an average acceleration corresponding to the classical prediction. But anyone who has dropped a quarter in a pin ball machine knows that a particle, when it encounters a constraint that restricts its motion, will travel so as to minimize its potential energy rather than kinetic. Were this not so, a pin ball game would last considerably longer.

The same principle suggests that a particle in a very weak potential, when faced with a quantized velocity, will choose the velocity that corresponds to a dive into the potential rather than the velocity that corresponds to a jump out of the potential. This could result in a substantial increase in the effective strength of very weak potentials.

Just such an effect has been observed in the galactic rotation measurements. Larger galaxies have their motions analyzed in a newtonian approximation. The calculated accelerations are exceedingly small, the observed accelerations are also low, but much higher than predicted by calculating the potential due to the matter visible in the galaxy.

Most physicists assume that the higher observed accelerations are
due to a form of missing "dark" matter, as only matter which
radiates, and therefore is visible, is easily toted up by earth-bound
astronomers when computing the gravity
well of a galaxy. But an alternative explanation for the effect
has been published by Mordehai Milgrom, under the name "MOND" or
Modified Newtonian Dynamics. The original paper is Mordehai
Milgrom “A modification of the Newtonian dyanamics as a possible
alternative to the hidden mass hypothesis”, Astrophysics. J.
270:365-370 (1983), a more recent article is
titled
MOND - A Pedagogical Review - M. Milgrom and a
search of the internet for the terms "MOND" and "Milgrom"
will provide plenty of hits. MOND assumes a characteristic
acceleration that modifies weak gravitational fields so as
to make them stronger. The characteristic
acceleration in MOND is about 1x10^{-10}m/sec2.

This paper makes the assumption that gravitation is a strictly geometric effect, rather than a force mediated by the exchange of particles such as gravitons. Under this assumption, gravity is not a force, but is a direct geometric consequence of the ether. Even quantizing in a flat proper time topology is difficult, but a curved topology, even as a perturbation, is very difficult and confusing. Rather than attempting such a program, it is much easier to assume a solution of the equations in the curved coordinates, and then to see what kind of potential is required to cancel the effect of the curved solution. This ensures that the problem is calculated in true coordinates, but without our actually having to solve the problem. Instead, we look for the amount of potential required to cause an acceleration that would cancel the apparent acceleration caused by the combination of curved coordinates and ether flow.

Since the coordinates are assumed attached to the ether, the unperturbed Hamiltonian will correspond to a particle subject to the ether. Not only is the ether curved, but in terms of the way we usually look at the world, the ether is "moving". For example, if the earth were stationary in an infinite empty ether sea (so we ignore the sun, stars etc.) then from our vantage point on the earth's surface ether would be constantly streaming down into the earth. This corresponds to the fact that when we throw small objects up, they come back down. If we want to make a calculation in ether stable coordinates, (or, more accurately, in assumed ether stable coordinates as in other varieties of gauge theories) we cannot use the constantly accelerating coordinates that are so natural for us surface bound creatures. Instead we must make a gauge choice that is at least theoretically compatible with a possible ether.

We know how fast the ether is "moving", that is, moving with respect to the accelerating coordinates that we prefer to use for galaxies, newspaper delivery routes, and the like. Einstein's equations give this rate, it is simply the gravitational acceleration of local frames of reference. As is usual with gauge effects, the approximate Lorentz symmetry eliminates our need to know the velocity of the ether, and the effect of the ether will turn out to be independent of the ether motion. The following calculation appears to be independent of an assumption of the ether, but this is not quite true, because the unperturbed Hamiltonian is implicitly assumed to be true to the ether. The absence of an explicit need to gauge the ether is due to my explicitly cancelling the ether out of the calculation by the way that the perturbation is set up.

In order to simplify the calculation, it also helps to ensure that only two of the velocity quantized wave functions need to be used. That is, the particle will be forced to travel at only one of two possible speeds. To do this, we have to make sure that the total velocity change is small, and the easiest way to do this is with a perturbation that only lasts a short time. In addition, I will only deal with one dimension as is suitable for very flat, but still flowing, ether. A generalization for the missing dimensions is obvious.

I will follow the notation of Morrison, Estle, and Lane's
Quantum States of Atoms, Molecules, and Solids, Prentice-Hall,
1976, pages 124-140, as this is the book I keep around (26 years) for
perturbation calculations, but any undergraduate book on quantum mechanics
should cover time dependent perturbation theory. So let
*H*^{(0)} be the unperturbed Hamiltonian, and
let the perturbation be a short pulse of length τ with an
acceleration λ in the positive *x* direction as follows:

(30) *H*^{(1)}(*x, t*)

= { 0 if *t* < -0,

= { -*m*λ*x*; if 0 < *t* < τ,

= { 0 if τ < *t*.

Classically, the above acceleration will increase the velocity (as compared to the unperturbed Hamiltonian), by λτ. In order to use perturbation theory on this problem, I need to have not only that λ be small, but also that τ be sufficiently small that the classical change in velocity is less than the quantization of velocity as imlied by equation (16). That is, I require:

(31) λ τ <<
√(2 *c*^{3}/*R*ω).

Assume that before the perturbation, the particle is in
an eigenstate |Ψ_{k}(*x, t*)> of the
unperturbed Hamiltonian, with its momentum in the positive
*x* direction. After the perturbation, the particle
will be in a mixture of the |Ψ_{k}> and
|Ψ_{k+1}> states. The resulting state will
not be an eigenstate, but we can compute the average
momentum and see what effect the perturbation had.
Write the state at time *t* in terms of a superposition
of these states as follows:

(32) Ψ(*x, t*) = α_{k}(*t*)
Ψ_{k}(*x, t*) + α_{k+1}(*t*)
Ψ_{k+1}(*x, t*)

If Newtonian mechanics applies, the change in momentum
is *m* λ, so the mixture will satisfy:

(33) < *p*(τ) >_{Newton} = *p*_{k}
+ *m* λ τ.

Perturbation theory gives the following approximations for the coefficients for the two momentum eigenstates that we are considering:

(34a) α_{k}(τ) =~ 1 - *i*
∫_{0}^{τ}
< Ψ_{k}(*x*) | *H*^{(1)}
| Ψ_{k}(*x*) > *dt* / * h*,

(34b) α

For sufficiently small τ the exponential in (34b) will become unity. This allows the time integrals to be performed giving, after substituing in equation (30):

(35a) α_{k}(τ) =~ 1 + *i*
< Ψ_{k}(*x*) | *m* λ τ *x*
| Ψ_{k}(*x*) > / * h*
= 1 +

(35b) α

By a change of coordinates, we can zero the (unperturbed) average position in (35a). This shows that to first order, as expected, only the coefficient (35b) is non zero. But (35b) can be used to compute the change to (35a) so as to make that coefficent compatible when computing the average momentum of the perturbed state. The squared magnitudes of the two coefficents must add up to one. Consequently, the average momentum in the final perturbed state is:

(36) < *p*(τ) > =
(1 - |α_{k+1}(τ)|^{2})*p*_{k} +
|α_{k+1}(τ)|^{2}*p*_{k+1}

= *p*_{k} +
|α_{k+1}(τ)|^{2}
(*p*_{k+1} - *p*_{k}).

It's not immediately obvious how to compute
the factor < Ψ_{k+1}(*x*) | *x*
| Ψ_{k}(*x*) > , but
it is clear that for non relativistic particles this
will approximately be a constant, and that it will be
proportional to *R*. Accordingly, substituting
*κ**R* for this factor, equation (36) gives
the momentum at time τ as:

(37) < *p*(τ) > = *p*_{k} +
*m*^{2} λ^{2}
τ^{2} κ^{2} *R*^{2}
(*p*_{k+1} - *p*_{k})
/ ~~h~~^{2}.

Comparing equation (37) with equation (33), it's clear that there is something odd going on here. Equation (33) is linear in λ, but equation (37) is bilinear. It appears that the quantum state takes a little while to accelerate, but keeps on accelerating. Of course this perturbation eventually fails, so the acceleration does not continue gaining strength.

Eventually the particle will be accelerated up to the next quantum velocity and the squared acceleration will cease. If we had a good solution to the equations, we could then compute the change in velocity, divide it by the time required to get to that next quantum velocity, and we would have the average acceleration. It would be nice if that result would match the classical equation. But in the light of the difficulty of solving these equations, we can model the effect by assuming that that the α coefficients will vary in time in a sinusoidal manner. That is, assume:

(38a) |α(τ)| = *cos*(*t*/T),

(38b) |α(τ)| = *sin*(*t*/T),

and select T to match the perturbation solution (37).
In this case, we get T = * h*; / (

(39) < *p*(*t*) > = *cos*^{2}(*t*
*m* λ κ *R* / * h*; )

According to our approximation, the time required to get
to *p*_{k+1} is T π/2, so therefore the
acceleration, averaged over the step in velocity, is the
ratio, which can be simplified with equation (16) and (29)
to get:

(40) < a >_{quantum} =
(*p*_{k+1} - *p*_{k}) / (*m*
T π/2)
= *m* λ √( κ^{2} *R* ω /
2 *c*).

One hopes that the various approximations and the constant κ cancel inside the square root. Assuming this, equation (40) shows that the quantum acceleration averages out to the classical acceleration.

This would be the end of the story, except that this equation
is for particles with *s* motion, and who therefore
suffer the passage of time. According to the theory of this
paper, these particles must eventually suffer wave function
collapse. If the potential is so flat that wave function
collapse occurs much faster than the particle can make it
to the next velocity quantization, then the particle will
be artificially kept in the previous velocity state, as
will be shown next.

To put this into the Copenhagen interpretation, if the
particle momentum is "measured" frequently enough, the
transitions between velocity eigenstates will be suppressed.
This will have the effect of decreasing the acceleration
felt by the particle. Since this calculation is being
made in the limit of very flat potentials, this effect
cannot be avoided, unless one assumes that particles
*never* get their wave functions collapsed.

It is my belief that wave function collapse is a continuous function of time. But I don't have an equation for it, so to model the effect of wave function collapse, I will instead assume a periodic Copenhagen type collapse, where the wave function is reduced with a probability according to the square of the wave function.

Accordingly, returning to equation (37), and assuming
wave function collapses occur every τ_{0}
seconds, we get that the average acceleration over this
period is, again using (16) and (29):

(41) < a >_{collapse} = *m*^{2}
λ^{2} τ_{0}
κ^{2} *R*^{2}
(*p*_{k+1} - *p*_{k})
/ (~~h~~^{2} *m* )

= λ^{2} √( τ_{0}^{2}
κ^{4} ω^{3} *R*^{3}
/ 8 *c*^{5})

= λ^{2} / a_{τ0},

where a_{τ0} has the units of acceleration. It
gives a characteristic rate of acceleration where
quantum effects of wave function collapse become noticeable
in the proper time topology.
This effect is independent of the proper time topology in
that any system with quantized velocities, and with periodic
wave function collapse, will exhibit the same behavior.

To connect this calculation to MOND, consider the case of a spaceship which is attempting to maintain a "stationary" (i.e. not orbiting) position far away from a massive galaxy. To do this, it must apply a force in opposition to gravity, that is, it must accelerate away from the center of the galaxy. If it fails to do so, it will slowly be dragged down into the gravity pit. How much force is required to maintain themselves against gravity?

Use ether coordinates. Since these coordinates are (slowly)
being sucked up by the galaxy, the spaceship will have to
constantly accelerate itself against these coordinates.
In order to remain where they are, they will have to
produce a constant acceleration. Suppose that they hook
up a very weak engine and produce the force *m* λ,
where *m* is their spaceship mass. According to the above
calculation, if λ is sufficiently small, the effect
of wave function collapse will reduce the actual acceleration
that they receive from their thrust. Not knowing of this
effect, they will perceive their problem as an unexpectedly
strong gravitational field. When they get their engine
running fast enough to keep themselves stationary, they
will record their apparent acceleration, λ, as the
local gradient in the gravitational field. This value
will be different from the number predicted by Newton
according to:

(42) λ = √(a_{τ0} a_{Newton}).

The above is the MOND equation for small gravitational accelerations.

But the implications of equation (42) run deeper than just the slowly accelerated rocket. Because the laws of physics are highly Lorentz invariant, the same supplemented gravitational acceleration (or more precisely, reduced acceleration against a very weak gravitational field) will be felt by all other observers, even those travelling at the speed of light. If we could correctly quantize the proper time topology, with wave function collapse, in a slightly curved space, the resulting particle tracks would act as if the gravitational force were increased according to (42). The topology itself tells me nothing about the calculation, but the result of the calculation is not in doubt. The effective gravitational force will have to be supplemented, otherwise the behavior of the thought experiment spaceship will appear to local observers, who fall with the ether, to be even more anomalous than the MOND effect itself. That is, such a free falling local observer would note a violation in conservation of momentum.

This effect where a repeated measurement of a system causes unexpected changes to the dynamics is also present in the "Quantum Zeno Effect". In both that case and this, it is important that there be a discrete set of eigenvalues. With the MOND calculation, the eigenvalues are for velocity, while in the QZE calculations, the eigenvalues are for energy. A good simple introduction to the QZE that will illustrate the similarities to the above calculation is in this thesis paper:

"*This is a result of the measurement feedback onto the system,
and this effect must be taken into account in any experiment
seeking to perform frequent observations of any sort of weak signal*."
Quantum measurement theory and the quantum Zeno effect,
Michael J Gagen, 2nd March 1985

The calculation of the previous section reminds me of a thought experiment that many smart people who have been well trained in relativity nevertheless get wrong: If a man standing on the earth's surface has two watches, and throws one vertically up into the air and then catches it, which of the two watches will have been slowed by time dilation? Most people will say the thrown watch is slowed, as it is the one that "had its velocity changed and went on a journey, just like the travelling twin", but in fact the held watch is the slow one. This can be verified by integrating out the Schwarzchild metric, which you might do if you are a graduate student studying for your preliminary exams, but that the held watch is the one that is accelerated, while the thrown watch is the one that travels along a geodesic, is easily verified by comparing the situation to the equivalent one where the man with the watches is travelling inside a space ship that is accelerating at one g. In this case, without the confusing presence of the earth's apparently immobile surface, the thrown watch is easily seen to be travelling on a geodesic, and therefore suffering a maximum experience of time between the two events "throw the watch", and "catch the watch".

Since the above thought experiment is one of my favorites, it's somewhat surprising that I had such difficulty realizing that the perturbation calculation above is an equivalent situation. I made the calculation months ago but assumed that there was a mistake as the indication seemed to be that small accelerations are squared in quantum mechanics, and therefore small gravitational accelerations should be similarly reduced in strength. Since the MOND effect is an increase in strength, I concluded that this was a bug in the calculation. I hope you enjoyed reading this almost as much as I enjoyed writing it.

Carl Brannen

September 16, 2003

For those who have difficulty seeing how it is that the Proper Time topology, which explicitly possesses a preferred frame of reference, could be equivalent to special relativity, which makes the assumption that no such frame exists, this appendix provides detailed calculations for time dilation and length contraction both in special relativity and in Proper Time topology.

Special Relativity Solution: The voyage requires 3/0.6 = 5 years
each way for a total of 10 years. This is the proper time
experienced on the Earth. The spaceship experiences a time
dilation of √(1 – 0.6^{2}) = 0.8, so the proper time
experienced on the spaceship is 10 x 0.8 = 8 years.

Proper Time Solution: The spaceship starts at the point (*x, y, z, s*) = (0, 0, 0, 0).
Align the *x* axis with the direction of travel. The velocity of the spaceship on the
outgoing voyage is therefore given by the vector (0.6, 0, 0, 0.8). The 0.8 value is required
to make the speed of the spaceship work out in total be 1. The spaceship’s position as a
function of the global time t is therefore:

(0, 0, 0, 0) + (0.6, 0, 0, 0.8) *t*_{1}

Setting this equal to (3, 0, 0, *s*_{1}) gives *t*_{1}, the global
coordinate time for the arrival of the spaceship at its destination, and *t*_{1}
is therefore 5 years. Note that the value of *s*_{1} is unspecified, as the
total length of the hidden dimension is negligible as compared to the many light years of
travel.
Since the proper time component of the velocity of the spaceship is
0.8, the total elapsed proper time on the outgoing voyage of the spaceship is therefore
0.8 x 5 = 4 years. Similarly, the return trip uses a velocity of (-0.6, 0, 0, 0.8) and
results in a coordinate time passage of 5 years and a proper time for the spaceship of another
4 years. The result is, of course, identical to the Special Relativity result.

Special Relativity Solution: The Lorentz contraction factor is
1/√(1 – (12/13)^{2}) = 13/5, so the proper length of the rod is
6m x 13/5 = 15.6 meters.

Since the Proper Time topology does have a preferred coordinate system, the question is not
as clear as it is in special relativity. But in any given coordinate system, the constancy
of the speed of light provides a technique for measuring length. Accordingly, the rod can
be measured in its own frame of reference by calculating the time required for light to
travel the length of the rod. Since proper time is a property of individual particles,
rather than dimensional objects such as rods, the length of the rod will have to be measured
by computing the time required for the light to travel down the rod, be reflected at the
end, and then travel back to the point of origin on the rod. The proper time experienced
by the end point of the rod during this flight will indicate (when multiplied by *c* = 1)
twice the length of the rod.

So let the rod begin at position (0, 0, 0, 0) through (6m, 0, 0, 0), and set the velocity vector for the rod to be (12/13, 0, 0, 5/13) so that it moves in the +*x* direction. The light
signal starts at (0, 0, 0, 0) and proceeds with a velocity vector of (1, 0, 0, 0) until it
meets with the other end of the bar at time *t*_{1}. The light direction is then
reversed, and it travels with velocity (-1, 0, 0, 0) until it meets up with the trailing end
of the bar at time *t*_{2}. The length of the bar, in the reference frame of the
bar, is then 1/2 the proper time experienced by the trailing end of the bar from 0 to
*t*_{2}. The equations for *t*_{1} and *t*_{2} are therefore:

(0, 0, 0, 0) + (1, 0, 0, 0) *t*_{1} == (6, 0, 0, 0) + (12/13, 0, 0, 5/13) *t*_{1},

(1, 0, 0, 0) *t*_{1} + (-1, 0, 0, 0)( *t*_{2}
- *t*_{1}) == (0, 0, 0, 0) + (12/13, 0, 0, 5/13) *t*_{2}.

Since our real world does not distinguish between the hidden “proper time” coordinate, the equalities need only be established for the first three coordinates.

The solution is *t*_{1} = 13 x 6 meters, and *t*_{2} = 2028/25 meters.
The proper time experienced on the trailing edge of the rod is, by time dilation, 5/13 of
*t*_{2}, which gives 156/5. Half of this is the proper length of the bar, which is
the same as the value given by special relativity. Therefore, both theories show the Lorentz
contraction of the bar to be the same.