1. Fluid Space Theory Offers a Solution to the Galaxy Rotation Problem
By John S. Huenefeld
huesoft@frontier.com
ABSTRACT
It is the purpose of this paper to offer an alternative to dark matter for resolving the
difference between observed rotation rates of galaxies and the rotation rates predicted by
gravitational theory. This shall be accomplished by recognizing an existing property of
ordinary matter that has, until now, been overlooked. The theory presented in this paper
is not a type of MOND (modified Newtonian dynamics) but instead a fully consistent
physical theory which will have implications far beyond the problem addressed in this
paper. It is called Fluid Space Theory. There is minimal need for references, as the math
presented is based on applications of special relativity, calculus, and differential equations
which may easily be found in any number of math and physics text books. In summary,
this paper proposes a formulation of gravitational fields consistent with general relativity,
plus an undiscovered property of ordinary matter in the form of a previously unrecognized
contraction field, and demonstrates that inclusion of this contraction field predicts rotation
curves in spiral galaxies similar to observations without the need for dark matter.
SECTION 1. THE CONCEPT OF VELOCITY FIELDS
In cosmology, the concept of universal expansion is widely accepted. In this theory,
the fabric of space-time expands, carrying galaxies outward like raisins in a rising loaf of
bread. At any given position inside this space-time an invisible expansion field may be
imagined as spheres of space-time moving outward with a velocity increasing in
proportion to the distance from the central point. This is a widely accepted and easily
understood example of a space-time velocity field.
There is another somewhat accepted velocity field. In the description of what is
happening at the event horizon of a black hole, the waterfall analogy is commonly used.
In this analogy, space-time is said to be falling into the black hole at the speed of light.
Like a swimmer upstream of a waterfall, anything caught in this flow will be swept down
into the black hole never to return. This will happen even while swimming at top speed
(which in this case is the speed of light). Once again, an accepted concept of a space-time
velocity field.
There is, however, a problem with the waterfall analogy which is subsequently
ignored. If space-time is flowing into the black hole at the speed of light at the event
horizon, it must also be flowing into a concentric sphere imagined just outside of the event
horizon at a velocity just under the speed of light. In fact this may be imagined to go on
through increasingly larger spheres at lower velocities on out to infinity. Also, if this is
true of black holes, why would it not be true for neutron stars, or normal stars, or large
planets. Gravity is gravity whether generated by a black hole or any other massive body.
2. If the waterfall analogy applies to black holes it must also apply to stars and planets. It
may then logically follow that all objects with the property of mass must be surrounded by
an inward velocity field.
This is a problem, and a line of thinking that physics professors have been steering
students away from since the time of Einstein. I ask the reader to indulge me and follow
this line of reasoning, as I characterize these inward velocity fields. The first step is to
establish the concept of space-time flux.
Let us begin with flat space-time known in GR as Minkowski space. In the tradition
of Albert Einstein's thought experiments, let us travel in mind to a region far from any
massive bodies, where there are no energetic fields, where parallel lines never meet, and
an object left alone will travel forever in a straight line at a constant velocity. Let us
imagine a glass box measuring several meters on a side in this space. Inside the box are
several objects and a human observer. A human observer is posted outside the box as
well. At present, all these items float weightless and motionless (see fig 1A).
Figure 1.
Inertial and accelerated systems.
While weightless, these objects could be traveling through space-time at any velocity
from zero to c (the speed of light). They would have no way to tell what that velocity is,
but whatever it is, they can tell it is not changing. We could then say that the velocity
field, or space-time flux, inside the box is constant. If we discard any unknown
background velocity, using the cross sectional area of the box we can compute the relative
flow of space-time through the box. We don‘t know an absolute flux, but a relative flux can
3. be described by equation (1). The volume flux will have units of meters cubed per second,
m3s-1. All further references to space-time flux may be considered as “relative flux.”
(1)
In Figure 1B, a force has been applied to the top of the box and now the observer
inside the box experiences an increasing space-time flux. Space is flowing through the box
at a constantly increasing rate and she experiences gravity. The change in space-time flux
with time can be expressed by equation (2). The change in space-time flux will have units
of m3 s-2, and these are the units that provide the experience of gravity.
(2)
Understanding that acceleration is the time derivative of velocity this equation may be
rewritten as equation (3).
(3)
In figure 1B, a force was required in order to create a field of changing velocity. The
equation can be expressed in terms of that force and the mass of the glass box and its
contents as in equation (4).
(4)
Now we shall return to the problem of a spherical inflow velocity field around a planet.
Imagine now that instead of being pulled trough space by a force, the glass box is sitting
on the surface of a planet. The astronaut inside the box will feel the same sensation of
gravity. There is a name for this, it is called the equivalence principle, named by Einstein
himself. The equivalence principle states that there is no distinction between the gravity
felt while standing on the surface of a gravitating body or while being pulled through
space in a box. This means that on the planet’s surface, the flux of space time must also be
constantly changing with time. We may then replace the term for area A in equation (4)
with the surface area of the planet, which has radius r to arrive at equation (5).
(5)
While the box is sitting on the surface of the planet, the force of gravity felt by the
observer inside remains constant. This is also true for the box being pulled through space,
as long as the force remains constant. Looking at the term V double dot, we see it has
units of m3s-2 This is very similar to the units of the gravitational constant G which has
units m3kg-1s-2. If V double dot is assumed to be proportional to the mass M of the planet,
we may replace the change in space-time flux term with a convenient constant, 4 pi G,
which is applied in proportion to the mass M of the planet and we get the very familiar
equation (6).
(6)
In this way, Newton’s equation for gravity may be derived on the basis of a space-time
inflow velocity field. Of course, there is a problem with this. If space-time is flowing into
the planet from all sides, where is it going? The planet should quickly fill up with space-
4. time and the flow will have to stop. Also the flow velocity at the surface will have to be
constantly increasing with time to create gravity, making the situation even worse. The
amount of space-time volume passing through any imagined sphere at diameter D must be
accounted for, as well as the second order term for ever increasing velocity, and this seems
impossible.
At this point, most people have thrown up their hands and walked away from this line
of thinking, but not me. There is an answer, and it is precisely by accounting for this “lost
volume” of space-time, that the cause of galaxy rotations faster than predicted by
Newton’s gravitation equation (dark matter) can be explained.
SECTION 2. THE APPLICATION OF RELATIVITY TO VELOCITY FIELDS
Understanding how inflow volumes may be accounted for came as a “happy idea.”
Time and space, said Einstein, are not the rigid and inflexible things that Newton thought
them to be. Time does not pass at the same rate in all references frames, nor are distances
the same. He gave us the equations below that can be used to predict the rate of time and
the lengths of objects based on their relative velocities. These equations of special
relativity known as Lorentz transformations, equations (7a) and (7b), will be valuable in
applying relativity theory to spatial inflows.
(7a)
(7b)
In these equations, variables marked prime are those in the moving reference frame
while the unmarked variables are in the rest reference frame of the observer.
The next thing to consider is that objects that we perceive as solid are actually, almost
completely made up of empty space, it is only the fields surrounding very tiny objects at
the heart of matter that makes them seem solid. So when we use the equations of special
relativity to compute the length of an object traveling near the speed of light, we are
actually computing the changes in the space that object occupies. Specifically, the
coordinate axis in a moving reference frame, aligned in the direction of motion, will appear
to be compressed to an observer in another reference frame. Any object placed into that
moving reference frame will also appear compressed along that axis.
Now let’s turn back to the notion of space-time falling through the surface of an
imaginary planet. Imagine a small cube of space-time as it moves toward the surface of
the planet. As the block of space-time falls inward, it moves faster and faster. The faster
it moves, the shorter it becomes in the direction of motion (the radial direction), and its
internal volume decreases. If it can fall fast enough to reach the speed of light, its volume
will vanish entirely. In this context, the notion of space-time flowing into matter is not so
absurd after all. The inflow of space-time vanishes as it becomes compressed with
increasing velocity.
Before developing the mathematics of an inflow field I wish to establish a couple of
conventions. First radial vectors are defined as positive outward and negative inward
5. from the central point of a flow field. Second, I will use the diameter rather than the
radius of a sphere in the equations. With spatial compression in the radial direction, the
term radius can lead to some confusion because distances in the radial direction are not
constant. The diameter however remains a consistent measure. Finally when applying
relativity to the flow field, one must consider both the view of an observer greatly
removed, on the outside of the flow, and the view of an element of space-time traveling
within the flow field. These two views can become very different.
Starting from Newton’s equation and substituting diameter for radius (D = 2r) and
using Newton’s force relation (F=ma) to express inward acceleration rather than the force
of gravity, and we set acceleration equal to the time rate change of velocity we get
equation (8).
(8)
Next by the chain rule we look for the velocity change with respect to the diameter,
realizing that the time change in diameter of a falling shell is equal to 2v. We may now
solve for the change in velocity with respect to the diameter dv/dD as show in equations (9)
through (12).
(9)
Therefore.
(10)
Integrating both sides we get.
(11)
(12)
We recognize equation (12) as the Newtonian formula for escape velocity from a
gravitating body. This is the unique inflow velocity profile that results in an inverse
distance squared acceleration profile. In this case, we are not considering a body falling
through Newtonian space but the fabric of space-time itself, falling toward a central point.
This is commonly referred to as sink flow. At any diameter in a gravitational field, space-
time falls inward at escape velocity.
If space-time is considered an incompressible fluid, such an inflow would be
impossible. It is easy to calculate that the flux of space-time changes as a function of
diameter D. However, if space-time is allowed to compress as it flows inward, there is a
chance to account for the change in flow volume with diameter. This may be done by
adjusting the radial length of the flow for the effects of relativity. The Lorentz
transformations of Special Relativity may be applied to correct for both spatial
compression and time dilation as shown in equations (13) and (14).
(13)
6. (14)
Substituting the value of v from equation (12) into equation (14) we get the relativity
corrected equation for inflow velocity as a function of the diameter as observed from a
reference frame outside the flow field, equation (15).
(15)
Equation (15) represents the velocity field as a function of diameter that will be
observed around any object that has the property of mass according to Fluid Space Theory.
Figure 2. is a plot of this function. In this graph v is the Newtonian form of the velocity
profile and v prime is how the velocity will appear to an observer outside the flow field
after accounting for spatial contraction and time dilation.
Figure 2.
Newtonian and relativistic velocity profiles.
There are a few things to note about this graph. As with all similar figures presented,
there are no scales associated with either axis. The graph has been scaled to illustrate the
shape of the function. We must think of the v curve as what an element within the flow
field will see while the v prime curve represents the view of that element from outside the
flow field. Consistent with cosmological expansion theory, the inflow v may become
superluminal (exceed the speed of light) while the function for v prime falls off to zero at a
short distance before D equals zero. When v exceeds c, the element within the flow will
vanish at some minimum diameter, as far as the outside observer is concerned. That
diameter can be found by solving for D when v prime equals zero as shown in equations
(16) and (17). (This is the same as setting the flow velocity v equal to the speed of light c).
7. (16)
(17)
We recognize this equation as the Schwarzschild diameter (twice the radius). This is
the diameter at which the inflow comes to a stop from the point of view outside the flow
field. Relativity may be applied to the Newtonian form for acceleration in the flow field
using the same technique to obtain equation (18), the observed acceleration, a prime.
(18)
Figure 3 is a plot of the flow acceleration as a function of the diameter. In this graph,
a is the Newtonian form of the acceleration field and a prime is the acceleration field as it
would appear to an observer outside the flow field.
Figure 3.
Newtonian and relativistic acceleration profiles.
Knowing that the acceleration is inward, it has been plotted on the positive axis in
figure 3 to keep the graphs consistent with convention. We see the acceleration increasing
as we move toward the central point, matching Newtonian gravity. Then, at small
diameters, the a prime curve reverses and slows until it becomes zero at the
Schwarzschild diameter. You must look closely at this graph to see how a prime follows
the Newtonian form but then drops away leaving a sharp peak at 1.5 times the minimum
diameter.
So it may be possible to account for vanishing volume in a velocity field, but how can
the constantly increasing velocity of inflow at a planet’s surface be accounted for? It so
8. happens that a sphere is actually a form of funnel. It is the most extreme form where the
angle of the vertex is opened all the way to 360 degrees. In a flow down a funnel the
velocity of a fluid will naturally increase toward the narrow end. In this way, an
acceleration field will be superimposed on the velocity field while the flow rate can remain
in a steady state. The imposed acceleration field will create gravity (it actually is gravity).
It is important to note at this point that the superimposed velocity and acceleration
fields described by equations (15) and (18) constitute a gravitational field completely
consistent with General Relativity. The fields are nearly identical to those specified by
Einstein’s field equation. These inflow fields produce gravity, curved space-time, and
gravitational time dilation. The differences between these fields and GR fields are subtle
and that is a subject for another paper.
So now, it has been shown that not only Newton’s equation can be derived from space-
time inflow fields, but a form of General Relativity is arrived at as well. This provides a
solid foundation for moving forward even though space-time is not usually treated this
way.
Figure 4. is a diagram of Fluid Space Theory funnel flow. It is useful for establishing
parameters for describing “lost volume” when setting up the equations of fluid space flows.
In this figure, the flow field outside view and inside view are superimposed.
To move ahead, it will be helpful to think in four dimensions. Four dimensional space-
time, is composed of the three familiar spatial dimensions x, y, z, and an additional
dimension tc which exists on the time axis. According to special relativity theory, for any
contraction on the x axis, there is an equivalent expansion on the tc axis. The tc axis must
also be considered perpendicular to all three spatial axes.
While three dimensional volume is not conserved in spatial inflows, four dimensional
volume is conserved. Four volume is defined simply as the product of the four dimensions
as in equation (19).
(19)
Under a velocity transformation four volume is unchanged as shown in equation (20).
(z=z’ and y=y’).
(20)
While contraction on the x axis might be noticed by the outside observer, expansion on
the tc axis is much harder to detect or comprehend and is generally overlooked.
Application of time dilation is almost exclusively reserved for high energy physics.
In Figure 4, space-time is imagined to be flowing down a funnel, which can also be
thought of as inflow from all directions into a sphere. The straight taper is what the
outside observer will see (assuming flat space) while the curved, hyperbolic funnel
represents what is actually going on for an element inside the flow field (curved space). At
an infinite diameter the curved funnel will be tangent to the straight funnel and the
compression will be zero. At Dmin (v=c), the curved funnel will be tangent to a line
Dmin/2 off the radial axis and the compression will become infinite (volume equal to zero).
9. Figure 4.
The two views of spatial in flow.
By the time an element in the flow field reaches an arbitrary diameter D, due to
spatial compression, it will have traveled an additional distance l down the curved funnel,
from its point of view, than what is observed from outside the flow field. The shaded
region lA represents the volume of space that has been compressed at any diameter D.
The flow or flux of space-time passing through a sphere of diameter D has been
established in equation (1), and is repeated here for convenience.
(1)
Substituting the value of v from equation (12) and the area for a sphere we get
equation (21).
(21)
This is the “true” volume of spatial flow as seen by elements embedded within the flow
field. The flux of space-time as observed from outside the flow field, accounting first for
spatial compression is shown in equation (22).
(22)
The amount of spatial compression taking place at any diameter D will be the
difference between the inside view and the outside view, that is the difference between the
uncompressed flow going down the hyperbolic funnel and the compressed flow observed
going down the straight funnel. This is shown in equation (23).
10. (23)
We may now apply time dilation to arrive at a complete expression for “lost flux” as a
function of diameter in equation (24).
(24)
Figure 5. is a plot of this function and both spatial compression volume dot and
volume dot prime are shown. What this graph shows is a bit surprising. The compression
rate volume prime starts at zero at Dmin and increases parabolically with diameter. The
lost volume curve without accounting for time dilation is similar and may be used at very
large diameters (that is the only reason it is shown).
The meaning of this function is that while the effects of relativity diminish as
diameter increases, because of the rate volumes increase with diameter, there is a
significant and ever increasing spatial effect. Any dark matter physicist should recognize
the shape of this curve. It is the shape of the gravitation curve for dark matter required to
flatten a galaxy rotation curve.
Figure 5.
Lost Flux as a function of diameter.
SECTION 3. THE SECOND (MISSING) COMPONENT OF GRAVITY
The concept of space-time flow fields will take some getting used to. This is not the
way physicists are trained to think. It may be time for the reader to ponder the first two
sections of this paper for a while. There are some important things to keep in mind. The
11. inflow velocity field by itself is invisible and can be superimposed over any other velocity
field without consequence. The inflow velocity field will not sweep objects toward the
center of the flow other than in conjunction with the acceleration field. It is the change
with respect to time in the velocity field that creates the acceleration field which we call
gravity. The acceleration field can be observed, and may also be superimposed over other
accelerations fields to build up an accumulated field which will behave according to the
distribution of masses within it. It is, however, the magnitude of the accumulated velocity
fields that determines the curvature of space-time.
In the background of the Fluid Space Theory gravitational field there is a second field.
The “lost flux” field, which represents a contraction of space-time surrounding all objects
that have the property of mass. This lost flux field constitutes the volume of space-time
which has been compressed by relative velocity and thereby shifted over to the tc axis. As
such it must be considered a separate field, orthogonal and acting independent from the
primary gravitational field. Therefore, the effects of this second field cannot be simply
added to gravity. It must be dealt with separately.
The lost flux field manifests as a contraction around matter in a spherical shell as a
function of diameter according to equation (24). Dividing equation (24). by the area of the
sphere yields what may be called the drift velocity.
(25)
This represents the velocity at which objects in the lost flux field will be swept toward
the center. It has been computed on the basis of the diameter so to find the radial drift
velocity the value must be cut in half to arrive at equation (26).
(26)
This function is plotted in Figure 6.
Figure 6.
12. This function has a similar form to the acceleration curve in Figure 3 but it is a
velocity curve, first order with time, while the acceleration curve is second order with
time. In order to account for the complete motion of a particle in a gravitational field both
equations (18) and (25) must be applied. Equation (18) will dominate out to very great
distances but eventually the drift velocity becomes equal to the gravitational acceleration
produced per unit time. After that, the drift velocity may become many times greater than
the gravitational acceleration.
The method I have employed to calculate the additional orbital velocity required to
overcome the inward drift, is to assume the drift is created by an acceleration which will
produce the same value as the drift velocity over a period of unit time. First we calculate
the acceleration required to produce the drift velocity per unit of time to arrive at equation
(27).
(27)
Next this term is combined with the normal gravitational acceleration to create a scale factor
as shown in equation (28).
(28)
The total orbital velocity is then computed using the scaled acceleration as in equation
(29).
(29)
To illustrate the long range effect of this newly revealed component of gravity I have
prepared a model of our solar system and a crude model of a galaxy based roughly on the
size of the Milky Way. In these models, normal gravity has been computed according to
equation (18) and the drift velocity has been computed according to equation (26). The
total adjusted orbital velocity is computed by applying the scale factor to the normal
gravitational acceleration. This is quite similar to the dark matter method of computing
additional gravity created by an assumed unseen mass. Mass and acceleration are in
direct proportion in the gravity equation, so the dark matter theorist scales up the mass
while Fluid Space Theory scales up the acceleration. While dark matter theorists must
assume unseen matter, the contraction field of Fluid Space Theory is deduced through
logic and reason.
Figure 7. is a plot of the orbital velocities for the planets in our solar system predicted
when both equations are applied. In this figure, orbital velocity (for circular orbits) in m s-
1 is plotted on the vertical axis while the horizontal axis has no scale, our solar system’s
features are simply listed in order from the inside out. As you can see, the predicted
values match well with observations. The new corrected values add a small and nearly
constant amount to the values predicted by standard gravity. However, this represents an
increasing percentage of the orbital velocity with increasing distance from the sun.
13. Figure 7.
There may have to be a re-evaluation of the value of the gravitational constant G.
Until now, G has been computed based on the assumption of a single component
gravitational field. In light of the additional contribution of the contraction or drift field,
G may have to be changed slightly from its current value in order to match observations.
This may actually help establish G with greater precision and could be the reason for
variations in the measurement of G carried out by different methods at different
distances. In addition, this may also help predict the orbits of Oort cloud bodies where the
contraction field contribution becomes more significant.
Figures 8. and 9. are based on the galaxy model. The model was created in an excel
spread sheet by breaking the galaxy into 16 primary zones 1,000 parsecs wide containing
galactic matter with four additional 1000 parsec wide zones containing diminishing
amounts of matter to fade out the galactic rim.
A super massive black hole of 2.6 million solar masses was placed at the center. Each
zone was represented by a concentric ring 1000 parsecs wide located outside the previous
zone. The galactic disk thickness was set to 600 parsecs at the core (central cylinder) with
tapering thickness down to 100 parsecs at the 16,000 parsec outer radius ring. The
remaining four rim rings tapered to 30 parsecs. Masses for each ring were calculated by
multiplying the volume of the ring by an estimated stellar density. The stellar densities
also diminish in magnitude from the core outward. The density in zone 1 was set high to
simulate a galactic bulge with the remaining zones having much lower densities. The
target mass was just over 20 billion solar masses (200 billion if you assume dark matter).
14. Figure 8.
Stellar orbital velocities in m s-1 predicted by combined fields. The blue line, velocity
from G, is the contribution from gravity alone. The red line, velocity from C (compression),
is the contribution from the compression field.
Figure 9.
Galactic mass distribution in solar masses.
15. Orbital velocities were calculated at the outside of each zone based on the
accumulated mass of all the zones inside. Because of the crudeness of the model, the plot
jumps up quickly on the left side near the core. A finer spacing of data points near the
core would smooth the curve. However, this model was only intended to test Fluid Space
Theory for the prediction of higher orbital velocities outside the core than predicted by
gravity alone. As you can, see it does that very well, predicting a quite flat total orbital
velocity curve all the way to the galactic rim.
The acceleration scale factors computed for each zone are listed in Table 1.
Zone Scale Factor Zone Scale Factor
1 3.04 11 8.41
2 3.75 12 8.79
3 4.40 13 9.16
4 5.01 14 9.50
5 5.59 15 9.82
6 6.13 16 10.13
7 6.63 17 10.42
8 7.11 18 10.70
9 7.57 19 10.97
10 8.00 20 11.23
Table 1.
From this simple model, acceleration scale factors reached values more than 10 times
that of gravity acting alone. The long range nature of the contraction field is also revealed
with scale factors climbing slowly from the galactic core and continuing to climb all the
way to the galactic rim, even while galactic mass content was tapering off. This
completely replicates the results of a dark matter halo, without the need to have any dark
matter at all.
SECTION 4. CONCLUSIONS
Let me start by saying that I did not create Fluid Space Theory to solve the galaxy
rotation problem. This is a recent and unexpected discovery, and a new application of a
well considered and documented (if not widely accepted) theory. I’ve been working on it
since the late 1990’s when the “happy idea” of how inflow volume is lost occurred to me.
For many years, as I explored the theory, all I seemed to have was an easier way to
explain and understand General Relativity. Each new conclusion and prediction I made
was already covered by GR. It was like finding little road signs along the way saying
“Albert Einstein was here.”
Extraordinary claims require extraordinary proof. I believe this result is
extraordinary and worthy of further consideration. It is significant because it is based on
a whole and self contained theory of the universe and it is not a patch or band-aid slapped
16. onto an existing theory in order to make it match observations. And at last, I have found a
place along the road without any sign from Albert Einstein.
Fluid Space Theory replicates many features of General Relativity, but it also departs
from GR in several important ways. In the form presented in this paper, Fluid Space
Theory is not fully generalized and I hope more refinements and investigation can produce
better results and a fully generalized theory may be published at a future date. I look
forward to working with physicists interested in exploring this line of reasoning.
The galactic contraction fields predicted will be in the form of a spherical halo and will
cause gravitational lensing exactly as do the dark matter halos they replace. In this case,
the contraction fields are predicted on the basis of the properties of known matter and
generally accepted properties of space-time. These fields will require a name, and I
humbly submit the term, Mannfields in memory of my mother and uncle who left this
world in 2014.
How this theory affects cosmology is not clear to me at the time of writing this paper.
Galactic contraction fields could certainly offset the rate of overall universal expansion
and may increase the predicted age of the universe. These fields, while diminishing in
magnitude outside a galaxy, will still continue to outstrip the effects of gravity and may
also change the dynamics of galaxy clusters.
The very long range influence of these fields has only been marginally investigated.
Computations of the vacuum energy curve for the Milky Way galaxy model above show
negative values just outside the galactic core which remain negative out to the rim and
beyond. This is consistent with a contracting space-time within galaxies. Interestingly at
about 90 parsecs, the vacuum energy turns positive again and remains so on outward.
This hints that Fluid Space Theory may also hold the key to explaining dark energy as an
effect of normal gravity as well.
Perhaps the most surprising thing about this prediction is that it is the result of
relativity acting at very large distances, and very low velocity. Relativity is not supposed
to be significant in these areas, so such a finding is quite unexpected. Unexpected
discoveries are sometimes the most satisfying and this actually gives me greater hope that
my proposal will prove out.
I have some small reservations about the acceleration scaling method applied. It
works well enough on the galactic scale, but I believe a better, more exact solution to
calculate orbital velocities in a contraction field may be found. I would not be surprised if
this has already been done by a mathematician working on first order attraction fields. I
will continue investigating an exact solution for the drift field and provide updates if
successful. Investigation of the action of the contraction field with a higher fidelity
galactic model, including dust, spiral arms, and actual mass distributions, is currently
beyond my means. I would gladly join such an effort if presented with the opportunity.
In summary, this paper proposes a new formulation of gravitational fields consistent
with general relativity, an undiscovered property of ordinary matter in the form of a
previously overlooked contraction field (the Mannfield), and demonstrates that this
contraction field predicts rotation curves in galaxies similar to observations without the
need for dark matter.
