Fluid Space Theory

Fluid Space Theory
“I Reject Your Physics and Substitute My Own”
This web site is to propose that gravity is the manifestation of fluid motions in a four
dimensional space-time-energy continuum. The reasons for this arise from philosophical
and technical problems with today’s accepted theory. First of all, the fact that dark energy,
and dark matter must be invented out of nowhere to make the standard model match
observations is a huge red flag that our understanding is fundamentally wrong. Second,
the separation of matter-energy from space-time in Einstein’s form of GR precludes
exploration of energized space-time which is the key to truly understanding our cosmos.
To follow this explanation of Fluid Space Theory, the reader must have a good knowledge
of modern physics, including Einstein’s Special Theory of Relativity and a basic
understanding of what General Relativity is.
I will not use tensor notation in any of my derivations. I freely admit that when I look
at a tensor equation I get no mental picture of what is going on inside the mathematics.
Einstein himself lamented the circumstance whereby progress in physics was attained by
increasing dependence on abstract mathematics, putting a greater distance between the
theorist and experimental observations of physical phenomena. For example, Maxwell’s
equations for electromagnetism were first written as differential equations, they were only
generalized more recently. Full generalization is not actually better in conveying
meaning. A generalized rhetorical question might be phrased “Do fauna deposit waste
among flora?” While very clean, compact, and general, it lacks all the character and
meaning of the original “Does a bear crap in the woods?” So, in the spirit of Maxwell, I
will stick to specific equations with specific meaning.
About equations. Math is the language of physics. If you are reading this you must be
interested in learning about physics. Learning the math is key to truly understanding
physics, so a student of physics should not shy away from equations. Would you want to
learn about China, or Japan, or Mexico from books that strictly avoided using Chinese,
Japanese, or Spanish words? In fact, it would be best to learn about those countries in the
language of those countries. So I will use equations without apology.
So, what does Fluid Space Theory do, and how is it different from Einstein’s general
relativity? The following lists show the what and the how, the following pages explain the
why.
What Fluid Space Theory does:
1. FST replicates Newtonian gravity, from small radii out to infinity.
2. FST replicates curved space and time dilation of Einstein’s general relativity.
3. FST reveals an undiscovered space-time contraction field around matter which
replicates the effects currently attributed to dark matter and dark energy.
4. FST eliminates the troublesome singularity inside a black hole and transitions
seamlessly into quantum theory.

How Fluid Space Theory is different from Einstein’s general relativity:
1. FST is simple and easy to understand. Nothing more than differential equations is
required, no tensor calculus is needed (unless you really want to use it).
2. FST recognizes the existence of relative velocity fields in space-time, fields that Einstein
should not have dismissed.
3. FST has none of the mysticism of general relativity. It makes curved space-time easy to
visualize.
4. FST unites matter and space-time into a single entity. No more “matter tells space-
time how to curve and space-time tells matter how to move.” In FST, matter is curved
space-time and it tells itself how to move.
Contents:
“I Reject Your Physics and Substitute My Own” ..........................................................1
Section 1: The Football Analogy...................................................................................3
Section 2.1: Velocity Fields and Gravity....................................................................10
Section 2.2: Is Space-Time Something or Nothing? ...................................................12
Section 2.3: Sink flows and acceleration fields. ..........................................................15
Section 3: Application of Relativity to Spatial Flows..................................................19
Section 4: How Fluid Space Theory Replaces Dark Matter and Dark Energy.............27
Section 4.1: Orbits in Gravity and Expanding or Contracting Space-time ..................28
Section 4.2: Defining the Mannfield...........................................................................29
Section 4.3: FST and the arrow of time......................................................................33
Section 5. The Case for Energized Space-Time...........................................................35
Section 6: Fluid Space Flow Geometry and Quantum Theory.....................................40
Section 7: Gravity and Electromagnetism..................................................................44

Section 1: The Football Analogy
The first time I recall thinking about Fluid Space Theory was around age 16 while
throwing a football (American football) around in the yard. I had recently begun studying
Newton’s laws and imagined that if the football field was out in space, when I threw the
ball it would continue in a straight line and constant velocity and I would never get it
back. In order to make the ball arc back to the ground, the field would need to have
rockets, which when fired would create artificial gravity. In the Newtonian perspective,
the football would continue in a straight line but the field would accelerate toward it
through space, eventually catching up when the football would “hit the ground.” In
reality, it is the ground, or field which hits the football.
I then tried to bring the Newtonian analogy to the surface of the Earth. I reasoned
that if the football field had to accelerate through space to catch up with the football,
something similar might be going on with gravity. I didn’t know it at the time but I was
trying to apply Albert Einstein’s principle of equivalence (more about that later).
I thought that in order to make the football fall back down, space would have to be
accelerating downward through the surface of the Earth. I was pretty excited about my
cleverness and thought I might be onto something until it occurred to me that the Earth is
round, and since gravity is the same everywhere, space would have to be passing through
the earth’s surface everywhere, all the way around, but where would it go? I concluded,
like many before me, that this was a silly idea and the analogy was flawed. But the
thought continued to nag at me as a problem that needed to be solved.
I am happy to report that I have in fact solved the problem, and the process of solving
it has revealed several fallacies incorporated as truths in modern physics and additional
blunders that Albert Einstein made.
I would like to begin with a quick review of Isaac Newton’s laws of motion, which I
would rather call his “laws of space”, because they tell us how things relate to space. I will
not cite any references, just do a web search for them and you will find many places and
phrasings from which you can choose.
1. An object in a state of uniform motion will remain in that state of motion until
acted upon by an external force. An object at rest will remain at rest until acted upon by
an external force.
2. The change in the state of motion of an object acted upon by an external force is
directly proportional to the magnitude of the acting force and inversely proportional to the
mass of the object. (acceleration a=f/m or force f=ma).
3. For every action there is an equal and opposite reaction.
These three laws gave birth to the industrial revolution and are the basis of all our
modern mechanical devices from skateboards to satellites and everything in between. It is
truly amazing how powerful three simple and correct statements which illustrate a true
understanding of nature can be.

Fluid Space Theory is based on considering space as something real, as a definable
stage upon which objects and forces may act. There is nothing inherently wrong about
thinking of space in this way. There are however, many scientific observations which
require an increasingly sophisticated concept of space in order to be explained, but fear
not, Fluid Space Theory is up to the task. In terms of Fluid Space Theory, Newton’s laws
tells us the following:
FST 1. Objects move through space without resistance, and conversely space moves
through objects without resistance. An object will not change its relationship with its
spatial reference frame unless acted on by an external force.
FST 2. An object acted upon by an external force will change its relationship with the
initial spatial reference frame in direct proportion to the magnitude of the force and
inverse proportion to the mass of the object. (a= f/m or f=ma).
FST 3. In order to change the relationship of an object to its resting spatial reference
frame, the acting object must change its relationship with its resting spatial reference
frame in equal proportion.
In short, where Isaac Newton saw forces acting between material bodies, Fluid Space
Theory sees reference frames in motion and material bodies changing their relationships
with those reference frames. Now we can revisit our rocket powered football field and
attempt to bring it down to Earth.
It is an unwritten rule in any sport you can name that it must be played in 1 Earth
normal gravity or 1G. The sizes of the fields and goals, the size and masses of the balls,
and the inflation pressures are all tailored to normal Earth gravity. If played under
different gravity, the game would not be the same at all. Imagine playing basketball on
the Moon, even a fourth grader would be able to do a slam dunk or take a shot from
anywhere on the court. It would be a completely different game. So our space football field
must have rockets strong enough to accelerate at 1 Earth normal gravity or 1G to play
Earth football.
Lets imagine a rocket powered football field out in space and also imagine that the
players are robots so we don’t have to worry about having an atmosphere or using a space
suit. It is a practice field, so to conserve fuel, the rockets are turned off between plays.
Also to keep the field small, it ends at the out of bounds lines and the coaches and trainers
float nearby equipped with rocket packs of their own so they can keep up.
For the first play, lets imagine that the rockets never come on. In this situation, when
the quarterback throws the ball it will fly in a straight line up and away from the field at a
constant velocity and one of the trainers with a rocket pack will have to fly out and
retrieve it. . See figure 1-1.

Figure 1-1.
For the next play, lets imagine that a coach is floating about 40 yards above the field
so he can watch from above. The ball is snapped, the field thrust rockets fire and gravity
appears on the field, the quarterback fades back and throws the ball to a receiver running
down the sideline. The receiver outpaces the defender, makes a catch and runs into the
end zone for the score. The play is over, the rockets turn off and gravity on the field goes
away. This all happens before the field passes the coach who has not fired his rocket pack.
Let’s take a look at what the floating coach sees from his point of view. See figure 1-2.
Figure 1-2.

The coach floating above will see the play start and the field begin to accelerate toward
him. When the ball is thrown, he will see it travel in a straight line at a constant velocity
just like it did last time. The ball will fly up above the field until the field catches up and
lifts the receiver into position to intercept the path of the football and make the catch.
Once the play is over, the coach will see the field fly past and will have to use his rockets
to match velocity.
To the players on the field, the field remains stationary and the ball flies in a arc as
shown on the top of figure 1-2. So why does the floating coach see the ball trace a straight
line while players on the field see it travel in a arc? To see how this works we can use
Newton’s first two laws and a little math to draw figure 1-3.
An equation representing a football at rest is simply its position relative to some
arbitrary starting point along a horizontal axis we will call the yard line. I use the
subscript 0 to indicate the position of the ball at time t=0. This is a pretty boring example
because x sub zero never changes. It is what we see when the official places the ball on a
yard line and we wait for the next play to start.
𝑦𝑎𝑟𝑑𝑙𝑖𝑛𝑒 = 𝑥0
Next, let’s give the football a constant velocity toward the opposing goal line. The
position of the football will now change at a constant rate so the position can be calculated
by multiplying the velocity by the amount of time that passes.
𝑦𝑎𝑟𝑑𝑙𝑖𝑛𝑒 = 𝑣0 𝑡 + 𝑥0
So far we have only used Newton’s first law. It is time to employ the second law. If
the mass of our the football is constant and the force applied to it is constant, the
acceleration will also be constant. We can write the following equation to find the yard
line of the football if a constant force is applied to it.
𝑦𝑎𝑟𝑑𝑙𝑖𝑛𝑒 =
1
2
𝑎𝑡2
+ 𝑣0 𝑡 + 𝑥0
Let’s say that while floating in weightlessness above our outer space football field, a
small rocket is attached to the football that will accelerate it at 2 yards per second each
second toward the opposing goal line and that it is above the 5 yard line moving 2 yards
per second toward the opposing goal line when the rocket fires. The equation becomes as
follows once we plug in the values for a, v zero, and x zero.
𝑦𝑎𝑟𝑑𝑙𝑖𝑛𝑒 =
1
2
2𝑡2
+ 2𝑡 + 5
Just plug the amount of time since the clock is started into the equation and you can
find the yard line the football will be at. This is great for one dimension, but football is
played in space and space has three dimensions. The football can not only move back and
forth between the goal lines, it can move sideways and up and down as well. Traditionally
these directions are labeled x, y, and z. We can use x for the direction between goals, y for

the direction side to side and z for the direction up and down. Now we just use three
identical equations, one for each direction and plug in the values for acceleration, velocity,
and position that apply in each direction.
We can simplify the math by setting the starting point of the throw as yard line zero (x
sub zero becomes zero). We can also have the quarterback throw the ball straight down
field so we don’t have to bother with the side to side, or y direction motion at all (y velocity
is zero). Also, knowing that the rockets attached to the field are pointed straight down, we
will only have to deal with acceleration in the z direction ( ax and ay are zero).
Let’s say that in the pass play, the quarterback throws the ball sixty yards and it
takes six seconds to get there, so the velocity down field in the x direction is ten yards per
second. Now, to make the math easy let’s say the field rockets increase the upward
velocity of the football field at two yards per second each second (Earth gravity is more
like 10 yards per second each second but using 2 makes the math easier). let’s also say
that the quarterback gives the football an upward velocity of six yards per second. Figure
1-3 is a chart of what the coach above the field sees by remaining stationary relative to the
Newtonian reference system at the start of the play and we can write the following
equation for the position of the ball and field during the play.
𝑓𝑜𝑟 𝑡ℎ𝑒 𝑏𝑎𝑙𝑙, 𝑥 = 10 𝑡 𝑎𝑛𝑑 𝑧 = 6𝑡
𝑓𝑜𝑟 𝑡ℎ𝑒 𝑓𝑖𝑒𝑙𝑑, 𝑧 =
1
2
2𝑡2
𝑜𝑟 𝑧 = 𝑡2
Figure 1-3.

We can plot the path of the ball on a graph as in figure 1-1 and see that the ball
travels in a straight line moving up six yards and downfield ten yards each second. We
can also plot the position of the field showing that it will be 1 yard above its starting point
at 1 second, 4 yards at 2 seconds, 9 yards at 3 seconds, 16 at 4, 25 at 5 and 36 at 6, and we
see that the field catches up to the ball at six seconds when the ball is sixty yards down
field.
To see what the players on the field see, take another piece of graph paper and lay it
over the top of figure 1-1. For each second, move the field level z=0 line up to the position
of the field and mark the position of the ball for that second. When finished you will see
that the points form a curve as show in figure 1-3.
This is what I was thinking about on that day throwing the football around in my
yard. The instant the football is thrown, it enters a state of freefall. If you were riding on
it, you would feel weightless, just like an astronaut in space. Out in space, with a rocket
powered field, the ball actually travels in a straight line and the field accelerates upward
to catch up with it. If gravity as felt on a rocket powered football field in space is the same
as gravity felt on the Earth, then space has to be imagined as accelerating downward
through the surface of the Earth. This is a problem for a lot of people, because a lot of
space would have to vanish, and rather than solve that problem, they will say “you just
shouldn’t think about space that way”. I can think about it that way, I do think about it
that way, and I have solved the problem of the vanishing space!
Before going on I want to show that it is possible to see both points of view either from
in space with a rocket powered field or on the surface of a planet. Seeing the point of view
from the field in Figure 1-3 is easy, you just stand on the sideline of the practice field and
watch. Seeing the field as the floating coach did in figure 1-3 takes a little more work.
While floating in space with no rockets firing, you are in the condition of Newton’s first
law, or otherwise known as freefall. In order to experience this on the surface of a planet
you must jump from a tower or airplane or otherwise be launched into the air, and for a
while as you fall, before air resistance builds up, you will experience weightlessness or
freefall.
To view the play from a Newtonian freefall reference frame, the coach must jump from
a tower above the field at the moment the ball is thrown. If the planet has gravitational
acceleration of 2 yards per second each second, he will be 1 yard lower after the first
second, 4 lower after second number two, 9 after the third, etc. He will fall the same as
the field rose in the outer space example. See Figure 1-4.

Figure 1-4.
Now if you take a sheet of graph paper and lay it over figure 1-4 with the coach
starting 36 yards above the field and mark the position of the ball at t equals zero, and
then move the sheet down to put the coach where he would be at the end of each second
and mark the position of the ball as it travels through its arc, you will see that from the
coach’s point of view, the ball moves in a straight line at a constant velocity. In case you
were worried, there is a giant air pillow for the coach to land on.
From a Newtonian point of view both cases work either out in space or on the surface
of a planet. It cannot be denied that on a planet with gravity, inertial reference frames
are constantly falling through the surface. All you have to do is drop or throw something
to see it in action. Somehow, the Earth is gobbling up falling reference frames. How can
this be happening? In the next section I will take a more structured and mathematical
look at what is going on.

Section 2.1: Velocity Fields and Gravity.
Understanding gravity is central to understanding relativity theory. In his 1916
paper, Albert Einstein strove to cover all experiences of gravity in a general equation. But
this lumping of all experiences of gravity as being exactly the same might have been
hasty. After all there are only three ways to experience gravity in this universe. Each is
quite distinct and two of them are easily understood using Newton’s laws. Figure 2.1-1
shows each of the three possible ways to experience gravity.
Figure 2.1-1
First, while being pulled through flat space in a box, or chest as Einstein called it, by a
constant force, gravity is created by linear acceleration. Second, if standing on the rim of a
large turning wheel in flat space, gravity is created by centripetal acceleration. The third
way is to stand on the surface of a massive body, like a planet, and we are told that gravity
is created by the curvature of space-time surrounding the planet. Einstein had a gut
feeling that these were all manifestations of the same physical law.
What Einstein thought these situations all have in common is curved space-time. Was
he correct in this, or is it something else that they have in common? To establish a way to
tell when space-time was curved, Einstein imagined constructing a lattice work of
identical rods and clocks. The rods are all exactly the same length and the clocks all run
at the same rate and may be synchronized to show the same time. I will add to this that
the clocks are also equipped with a pressure pad on the base so that each time one is set
down, it will display its weight.
Let’s look at situation 1 and have our astronaut start building a 3D lattice of rods and
clocks. The box is accelerating through flat space, so there is no initial curvature to the
local space-time as he sets out. As he builds in the plane of the floor of his box, the rods
line up easily and the clocks run at the same rate. Next he builds upward. In his frame of
reference, the floor and the ceiling are not moving, as he builds upward the rods remain
the same length vertically as they were horizontally. Each clock he places on the shelf at
the top of a rod shows the same weight as the one on the rod above and below. When he
reaches the ceiling, he discovers a hatch leading to an identical box just above the one he
started in. He can continue to build upward box after box and the rods in the vertical
direction and the weights of the clocks will never change no matter how far he goes.

It is easy to see what is going on from the point of view of someone floating weightless
nearby, watching the stack of boxes accelerate past. Every box has the same relative
velocity to the observer no matter where it is in the stack, and by the laws of relativity,
clocks moving at the same relative velocity to an observer will run at the same rate. This
grid work is not revealing any signs of curved space.
For the second situation, lets imagine two space wheels out in a vast area of flat space-
time. The first wheel will serve as the reference wheel. On it, no matter where our
astronaut goes she feels weightless. She may also see stars at a great distance appear to
be fixed in the sky. The hubs of the two wheels are aligned on a common axis and she
makes her way over to the turning wheel. On this wheel the stars describe slow circles as
the wheel rotates and as she progresses outward to the rim, she notices gravity becoming
increasingly strong.
Back on the first wheel she sets up a grid of rods and clocks with no problems. Each
clock shows zero weight and runs at the same rate as the others. Every rod remains the
same length, and at any radius from the hub, the circumference comes out to be exactly 2
pi r. She repeats this process on the turning wheel. The first thing she notices is that only
the clock at the center shows zero weight, while all the others show increasing weights as
she goes farther from the hub. Each clock around the rim at the same radius weighs the
exactly same as all the others at that radius and runs at the same rate. As she progresses
farther from the center, the force of gravity becomes quite intense and she notices that
clocks that far away from the hub are beginning to run more slowly than those closer to
the center. She also discovers that the rods aligned around the rim are starting to add up
to more than 2 pi r. This rotating reference frame is showing the signs of curved space-
time, even though it is floating in a region of flat space time and the space-time of the
wheel next to it remains flat.
From the point of view of an astronaut on the first wheel it is easy to see what is going
on. The astronauts and elements on the turning wheel are moving faster and faster the
farther they are away from the hub. As the velocity gets higher, the effects of relativity
start to show up. Rods aligned in the direction of motion appear shortened and clocks run
more slowly. Space-time is being curved according to the laws of relativity.
Now let’s take a look at the third situation. In this case, let’s put our astronaut in the
middle of a tall tower erected on the planet’s surface. He starts building a grid work of
clocks and rods up and down the tower. Clocks at the same level on the tower run at the
same rate and weigh the same amount. This could go on outward to create a spherical
shell around the planet if he continued to build laterally. As he builds upward, he notices
that the clocks are showing less weight and running a bit faster than the clocks lower
down. As he builds downward, he notices that the weight of the clocks is increasing and
they are running more slowly than ones higher up. He also notices that while the vertical
rods appear to him to be the same length, as he builds downward, he is not getting as
much closer to the planet’s surface as he expected. This reference frame is also showing
the tell tale signs of curved space-time.

Only two of the three situations above have curved space-time and Einstein is on
shaky ground (along with his followers) if he tries to claim that situation one has curved
space-time. These three situations do have something in common but it is not curved
space-time. What they all have in common is that the astronaut experiencing gravity is
constantly changing velocity compared to a reference frame in free fall. We will see how
this works in the rest of section 2.
Section 2.2: Is Space-Time Something or Nothing?
Albert Einstein strove to regard space-time as having no material realness. That is,
material objects move through it, and exist within it but space-time has no existence itself.
Ether theories abounded in his day and he strove to distance himself from them because
no ether theory ever worked. But despite his best efforts, the realness of space-time keeps
popping up like mole hills in his well tended lawn. He proposed curved space-time, but
how can something that doesn’t exist be curved? He almost predicted the expansion of
the universe and later embraced the theory even though the concept has some ether like
concepts.
The concept of universal expansion is quite widely accepted today. 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 a central point, creating a velocity field that spans the
universe. 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 real waterfall, if he can’t outpace the flow, he will be swept
down into the black hole never to return. This will happen even to objects moving at the
speed of light. Once again, an accepted concept of a space-time velocity field and not
much different than my football analogy, where reference frames are constantly falling
into the Earth.
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 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 or
any gravitating body? Gravity is gravity whether generated by a black hole or by any
other massive body. If the waterfall analogy applies to black holes it must also apply to
stars and planets, even atoms. It then logically follows that all objects with the property
of mass must be surrounded by an inward velocity flow field.

This is a problem, and a line of thinking that physics professors have been steering
students away from since the time of Einstein. It sounds too much like an ether theory
and they avoid the subject. 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 General Relativity (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 body, 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
2.2-1A).
Figure 2.2-1.
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 (Newton’s or FST first law). We could
then say that the velocity field, or space-time flux, (the amount of space-time passing
through 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 be described by equation (1).
The volume flux will have units of meters cubed per second, m3s-1. All further references
on this website to space-time flux may be considered as “relative flux.”
∀̇ = 𝑣𝐴 (1)

In Figure 2.2-1B, 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 describe accelerated flow and produce the sensation
of gravity.
∀̈ =
d
dt
(𝑣𝐴) (2)
Understanding that acceleration (a) is the time derivative of velocity this equation
may be rewritten as equation (3).
∀̈ = a𝐴 𝑜𝑟 𝑎 = ∀̈ /𝐴 (3)
In figure 2.2-1B, a force was required in order to create a field of changing velocity, or
a space-time acceleration field. The equation can be expressed in terms of that force and
the mass of the glass box and its contents as in equation (4).
𝐹 = 𝑚𝑎 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 𝐹 = 𝑚 ∀̈ /𝐴 (4)
It is worth noting here that equation 4 is the gravitational equivalent of Faraday’s law
for electromagnetism. It can be read, the gravitational force is equal to the time rate of
change of spatial flux through a given area acting on a given mass.
Figure 2.2-2.
Inertial and accelerated systems.
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 observer is standing
on a tower on the surface of a planet. The astronaut inside the box will feel the same

sensation of zero gravity while falling toward the planet as she felt while floating in space.
The astronaut being pulled by a force or standing on the tower will feel the same sensation
of positive 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 locally between the
gravity felt while standing on the surface of a massive body or while being pulled through
space by a force. 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).
F = m∀̈ /4πr2
(5)
While the observer is sitting on the surface of the planet, the force of gravity felt
remains constant. This is also true for the observer being pulled through space by a
constant force. 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, V double dot, 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).
𝑖𝑓 ∀̈ = 𝑀4𝜋𝐺 𝑡ℎ𝑒𝑛 𝐹 =
𝑚𝑀4𝜋𝐺
4𝜋𝑟2
𝑜𝑟 𝐹 =
𝑚𝑀𝐺
𝑟2
(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-
time and the flow will come to a stop. Also, it would seem that the flow velocity at the
surface would 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 radius r 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), and changes in the observed rate of
expansion of the universe (dark energy) can be explained.
Section 2.3: Sink flows and acceleration fields.
This section will show and explain how gravity is created by space time flux in a sink
flow field. To begin with we need to understand space-time flux, or any kind of flux for
that matter.

Figure 2.3-1
Einstein tried to relate velocities only between one material object and another and
not to space itself. This is quite reasonable because the is no handle on empty space that
you can pin a velocity to. But when comparing the reference frames of two moving objects,
one is comparing the relative motion of the spaces they occupy. For the astronaut floating
weightless, watching the box get pulled away by the rope, he sees the box move away
through the space-time of his reference frame and can calculate a change relative space-
time flux through the box, as done above and figure out how much gravity the astronaut in
the box is feeling.
Figure 2.3-1 shows four pipes of equal area and length. If a fluid of any kind flows
down these pipes at velocity v, the flux will be the same in all the pipes, and it will be
equal to the velocity of the flow times the area of the tube. Flux has the dimensions of
length cubed per second.
Figure 2.3-2

Next, the tubes are fitted with tapering ends. The inlet area of each tube is the same,
but the area of the outlet is smallest for the top tube and largest for the bottom tube. If
the flux is the same down each tube, the velocity of the fluid when exiting any tube will be
higher than it was upon entering. This is known as a nozzle, and is used to accelerate
fluid flows. The smaller the exit, the higher the velocity. The tube at the top will have the
highest exit velocity. If you know the amount of flux (flow volume per second) and the
area of the exit, the velocity can be easily calculated.
Let’s take a closer look at the tapered section of a tube. In a straight section of tube
with a constant area, the velocity remains constant. For space-time flux this represents
an inertial reference system. It is only in the tapered section of the tube where the area is
changing that the velocity is also changing. Changing velocity is known as acceleration.
Within the tapered section, an acceleration field exists within the fluid flow. In Fluid
Space Theory, an acceleration field of space-time flux represents an accelerated reference
frame, or a gravity field.
Going back to the astronaut in the glass box of figure 2.2-1A, it is as if she is in one of
the constant area pipes where space-time is moving past at a constant velocity. When the
cable attached to the box is pulled, as in figure 2.2-1 B, it is like moving her to an area
where the flow velocity of space-time is constantly changing. From her point of view, it is
placing her into an acceleration field within space-time. This can be imagined as either
increasing the flow velocity at a steady rate down a straight section of pipe, or by moving
her into a tapered section of pipe with a constant space-time flux.
FST law number 1 says that space flows through objects without resistance. So there
is no material pipe that can contain or funnel a flow of space-time. The only thing that
can resist the passage of space-time is other space-time. If space-time could be magically
removed inside a small sphere, the surrounding space-time would rush in to replace it. If
this is kept up on a continuing basis, a self funneling inflow field will form.
In the study of fluids, a flow field that originates from, or converges into, a central
point is called source or sink flow. If one of the tapered sections of pipe is separated from
the straight pipe sections and then packed in with a bunch of other tapered pipes around a
central point, it can be seen how sink flow is no different from flow in a tapered pipe. See
figure 2.3-3.
Figure 2.3-3

So in figure 2.2-2B, while the astronaut is standing on the tower, above the planet, he
is standing in a tapered flow field, or acceleration zone, also known as a gravity field.
(Remember, in Fluid Space Theory, an acceleration field in space-time is a gravitational
field.)
So at least a part of the mystery of the vanishing space-time has now been solved. By
visualizing sink flow of space-time into the Earth, the flow can be in a steady state, the
flux at any radius can be constant over time. There is no need for an increasing amount of
flux to create gravity as in figure 2.2-1 B. The taper of the flow creates the acceleration
field, so while standing in this field we feel gravity even though the space-time flux is
constant.

Section 3: Application of Relativity to Spatial Flows
At the end of section 2, space-time around a massive body was described as a fluid
flowing down a funnel. Funnel flow of a three dimensional fluid, however, will not have
the same acceleration field as gravity. To get it right, space-time must be considered as a
four dimensional fluid, behaving according to the laws of relativity. After all, Einstein’s
theories of relativity are based on the four dimensional nature of space-time.
In our normal understanding of length in three dimensions, we can find the distance
between any two points given their coordinates using the Pythagorean theorem.
𝐿 = �∆𝑥2 + ∆𝑦2 + ∆𝑧2 𝑜𝑟 𝐿2
= ∆𝑥2
+ ∆𝑦2
+ ∆𝑧2
To introduce the fourth dimension of time, the first thing that must be done is to
express time in units of length so we use the term tc ( time multiplied by the speed of
light) to give it the units of length.
𝐿 = �∆𝑥2 + ∆𝑦2 + ∆𝑧2 − ∆𝑡𝑐2 𝑜𝑟 𝐿2
= ∆𝑥2
+ ∆𝑦2
+ ∆𝑧2
− ∆𝑡𝑐2
The Pythagorean theorem works with any number of dimensions and in this case we
are using a four dimensional axis, x, y, z, and tc, each axis is perpendicular to all the
others. You may have noticed that the tc term is subtracted rather than added. This is
called a Lorentz metric, and it is done to make the calculated space-time intervals match
observations. So not only is the tc axis perpendicular to x, y, and z, but distances along it
are measured in reverse.
This is where the Lorentz transformations Albert Einstein used in his special theory of
relativity come from. 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. In 1905, 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 will be valuable in applying relativity theory to spatial flows.
𝑙′
= 𝑙�1 − 𝑣2/𝑐2
𝑡′
= 𝑡/�1 − 𝑣2/𝑐2
By understanding that lengths perpendicular to the direction of motion are not
contracted, we can write an equation for the volume ∀ of an object as it approaches the
speed of light. Where volume ∀ equals length times width times height and ∀ prime
equals length prime times width times height.
∀= 𝑙𝑤ℎ 𝑎𝑛𝑑 ∀′
= 𝑙′
𝑤′ℎ′ 𝑤ℎ𝑒𝑟𝑒 (𝑤 = 𝑤′ 𝑎𝑛𝑑 ℎ = ℎ′)
∀′
= ∀�1 − 𝑣2/𝑐2 (7)
Variables marked prime (l') are those in the moving reference frame while unmarked
variables are in the reference system of the observer. Above, the letter l stands for length

and t means time and ∀ means volume, where v is the relative velocity of the moving
reference frame and c is the speed of light. If the prime and normal quantities are known
for two reference systems in relative motion, the equations may be inverted to solve for the
relative velocity as shown below.
𝑣 = 𝑐�1 − 𝑙′2
/𝑙2
𝑣 = 𝑐�1 − ∀′2
/∀2
𝑣 = 𝑐�1 − 𝑡2/𝑡′2
Equation 7 tells us that the observed volume of an object in relative motion is less
than the volume of that same object when observed at rest. So how does an object change
when accelerated to a high relative velocity?
Let us consider what we know of “tangible objects”. If we say that an object is “moving
through space,” we may also be saying that space is “moving through the object.” We
know that ordinary objects are composed of tiny molecules arrayed in space, that those are
in turn composed of even smaller atoms, and that there is space between them. Thanks to
the work of Rutherford and Bohr, we also know that atoms are themselves mostly empty
space where tiny fuzzy electrons whirl about a dense and tiny nucleus. This nucleus is in
turn occupied by a host of even smaller fuzzy things (protons and neutrons), and there is
space between them. They are in turn composed of even smaller fuzzier things, and there
is space between those (quarks or strings).
Thus, an ordinary object would present no greater impediment to the passage of space-
time than the planets would prevent space-time from passing through our solar system.
Objects that we perceive as solid are actually, almost completely made up of empty space,
it is only the fields surrounding these very tiny particles that makes them seem solid. So
when we use the equations of special relativity to compute contraction of the length of an
object traveling near the speed of light, we are actually computing the changes in the
space the object occupies, not the object itself.
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. So it is the space that is changing, not the object. Therefore in the study of sink
flows of space-time we must understand that the 3 dimensional volume of the flow is not
conserved under a velocity transform.
Now let’s return to the notion of space-time falling through the surface of our
imaginary planet. As the inertial reference frame falls inward relative to the rest frame of
the planet, 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. When it
reaches the speed of light, its volume will vanish entirely. At this point, space-time may
continue to fall endlessly inward without ever getting any closer to the central point. This
is called the event horizon and space-time beyond this horizon is bent so much that it may
be imagined as flowing off perpendicularly to our universe. 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 velocity and eventually all length in the direction
of motion is shifted over to the tc axis.
Before developing the mathematics of an inflow field I must establish a couple of
conventions. First vectors are defined as positive outward and negative inward toward
the central point. Second, I will be careful using the radius of a sphere in the equations
because with spatial compression in the radial direction, using the radius can lead to some
confusion. 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
traveling within the flow field. These two views can become very different. From
equations 4 and 6 we can use the face that a=f/m and come up with the following.
−
𝐺𝑀
𝑟2
= 𝑎 = 𝑑𝑣/𝑑𝑡
First we set the acceleration equal to the time rate change of velocity
−
𝐺𝑀
𝑟2
= �
𝑑𝑣
𝑑𝑟
� �
𝑑𝑟
𝑑𝑡
� =
𝑑𝑣
𝑑𝑟
𝑣
Next by the chain rule we look for the velocity change with respect to the radius,
realizing that the time change in radius of a falling shell is equal to v. We may now solve
for dv/dr.
−
𝐺𝑀
𝑟2
=
𝑑𝑣
𝑑𝑟
𝑣
− �
𝐺𝑀
𝑟2
� 𝑑𝑟 = (𝑣)𝑑𝑣
Integrating both sides we get.
2𝐺𝑀
𝑟
= 𝑣2
𝑣 = �2𝐺𝑀/𝑟 (8)
We recognize this equation as the Newtonian formula for escape velocity from a
gravitating body. In this case, we are not considering a body falling through Newtonian
space but space-time itself falling toward a central point. At any radius in a gravitational
field, space-time falls inward or outward at escape velocity. The velocity of the flow field
must be adjusted for the effects of relativity as would be seen by an observer outside the
flow field. Applying the equations of Special Relativity from above, to correct for both
spatial compression and time dilation we get.
𝑣 = 𝑙/𝑡
𝑣′
=
𝑙′
𝑡′
=
𝑙�1 − 𝑣2/𝑐2
𝑡/�1 − 𝑣2/𝑐2
= 𝑣 �1 −
𝑣2
𝑐2
�

𝑣′
(𝑟) = �2𝐺𝑀/𝑟(1 − 2𝐺𝑀/𝑟𝑐2
) (9)
Figure 3-1 is a plot of flow velocity as a function of the radius. In this graph v is the
Newtonian form of an element of space-time fluid within the flow (equation 8) and v prime
is how the velocity of this same element would appear to an observer outside the flow field
after accounting for spatial contraction and time dilation (equation 9).
Figure 3-1
There are a few things to note about this graph. First, consistent with cosmological
expansion theory, the Newtonian inflow may become superluminal (exceed the speed of
light). If it does, it will vanish as far as the outside observer is concerned, at some
minimum radius. That radius can be found by solving for r when v prime equals zero.
(This is the same as setting v equal to c).
𝑆𝑒𝑡
2𝐺𝑀
𝑟𝑐2
= 1
𝑟 𝑚𝑖𝑛 = 2𝐺𝑀/𝑐2 (10)
We recognize this equation as the equation for the Schwarzschild radius. And this is
the radius at which the inflow comes to a stop from the point of view of the outside
observer (v’=0).
We now apply special relativity to the Newtonian form for acceleration in the flow field
using the same treatment we did above with velocity to obtain the equation for the
observed acceleration, a prime.
𝑎′
(𝑟) = −
𝐺𝑀
𝑟2
�1 −
2𝐺𝑀
𝑟𝑐2
� (11)

Figure 3-2 is a plot of flow acceleration as a function of the radius. In this graph, a is
the Newtonian form of the field acceleration of an element of space-time fluid in freefall
and a prime is how the field acceleration of this same element would appear to an observer
outside the flow field.
Figure 3-2
Knowing that the acceleration is inward, it has been plotted on the positive axis in
figure 5 to keep the graphs consistent with convention. We see the acceleration increasing
as we move toward the central point but then reversing and slowing until it becomes zero
at the Schwarzschild radius, as observed from outside the flow. You must look closely at
this graph to see how a’ follows the Newtonian form but then drops away leaving a sharp
peak at 1.5 times the minimum radius. The shape of a’ as a function of radius is also
clearly what is known as an energy well.
Finally, using a similar application of Special Relativity, I plot V double dot prime as a
function of radius in figure 3-3. Remember that V double dot is proportional to the
gravitational constant.
It turns out that the Newtonian form of V double dot as a function of radius is a
constant, but when time dilation and spatial contraction are accounted for, an outside
observer will see a dramatic fall in the rate of change in spatial flux and thus an apparent
change in the gravitational constant at small radii. This is an important distinction
between Fluid Space theory and General Relativity. When V double dot vanishes, there is
no longer any force left to compress space down any smaller than r min. This leaves an
infinitely long corridor of space time at r min moving off at the speed of light
perpendicular to our familiar three spatial dimensions and avoids creating a troublesome
singularity.

Figure 3-3
∀̈ ( 𝑟) = 𝐴𝑎 = 4𝜋𝑟2
(𝐺𝑀/𝑟2
) = 4𝜋𝐺𝑀
∀̈ ′
(r) = 4𝜋𝐺𝑀 �1 −
2𝐺𝑀
𝑟𝑐2
�
(12)
Figure 3-4 is useful for establishing parameters when setting up the equations of fluid
space flows. The outside flow view and inside flow view are superimposed. It is also
helpful to think in four dimensions (if you can). In four dimensional space-time, we still
have the three spatial dimensions x, y, z, and an additional dimension tc which exists on
the time axis. For any contraction on the x axis, there is an equivalent expansion on the tc
axis. The tc axis is also considered perpendicular to all three spatial axes. While three
dimensional volume is not conserved in spatial flows, four dimensional volume is
conserved. Four volume is defined as the product of the four dimensions as follows.
∀4= 𝑧 ∗ 𝑦 ∗ 𝑥 ∗ 𝑡𝑐 𝑎𝑛𝑑 ∀4′ = 𝑧′
∗ 𝑦′
∗ 𝑥′
∗ 𝑡′𝑐
Under a velocity transformation four volume is unchanged (z=z’ and y=y’).
∀4
′
= 𝑧′ ∗ 𝑦′ ∗ 𝑥′
�1 − 𝑣2/𝑐2 ∗
𝑡′
𝑐
�1 − 𝑣2/𝑐2
= ∀4
While contraction on the X axis might be noticed by the outside observer, expansion on
the tc axis would be much harder to detect or comprehend and would generally be
invisible.

Figure 3-4
In Figure 3-4, the straight taper is what the outside observer will see looking at a
spherical flow field and assuming flat space-time, while the curved, hyperbolic funnel is
what an element inside the flow field will see as it enters the curved space at the heart of
the flow field. At an infinite radius the curved funnel will be tangent to the straight
funnel and the value of l will be zero. At r min (v=c), the curved funnel will be tangent to a
line r min off the radial axis and the value of l will become infinite.
By the time an element in the flow field reaches any arbitrary radius r, it will have
traveled an additional distance l down the curved funnel farther than what is observed
from outside the flow field. Remember, the curved funnel represents uncompressed flow
while the straight funnel will have compressed flow. The shaded area lA represents the
volume of space that has been compressed up to that point at any radius r.
∀ 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑒𝑑= ∀ − ∀′
= ∀ �1 − �1 −
4𝐺𝑀
𝐷𝑐2
� 𝑤ℎ𝑒𝑟𝑒 ∀= 𝑙𝐴
Substituting and dividing by unit time we get the volume compression rate as a
function of r.
∀̇ 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑒𝑑= 𝑙𝐴/𝑡 = 4𝜋𝑟2
�2𝐺𝑀/𝑟 �1 − �1 −
2𝐺𝑀
𝑟𝑐2
�
Finally, accounting for time dilation, we substitute t’ for t.

∀′̇ 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑒𝑑 = 𝑙𝐴/𝑡′ = 4𝜋𝑟2
�2𝐺𝑀/𝑟 �1 − �1 −
2𝐺𝑀
𝑟𝑐2
��1 − 2𝐺𝑀/𝑟𝑐2
(13)
Figure 3-5 is a plot of this function and what it shows is a bit surprising. The
compression rate is zero at r min where space becomes infinitely compressed and no
further compression is possible, as expected. However the function has a form similar to
Flamm’s Paraboloid and increases continuously with the radius. This means that while
the effects of relativity diminish with distance, due to the large volumes involved at very
large distances, there is a significant, and ever increasing spatial contraction (or
expansion) effect surrounding a body of normal matter.
Figure 3-5
We now have all the tools needed to show how inflow fields of space-time create
particles, gravity, and explain the large scale motions of galaxies and the expanding
universe.

Section 4: How Fluid Space Theory Replaces Dark Matter and Dark Energy.
A primary distinction between General Relativity and Fluid Space Theory is that
Fluid Space Theory does not predict singularities. While both predict black holes with
event horizons, General Relativity says that objects passing inside are crushed down to a
singularity at the center. Fluid Space Theory says that objects passing over the horizon
enter a narrow, infinitely long, corridor of space-time and that the event horizon encloses a
discontinuity in space-time. At the center of every inflow field there is a bubble or domain
of finite size that is beyond the reach of any space-time coordinate system, not a
singularity. In Figure 4-1, the distance along the vertical axis represents the amount of
spatial compression at any radius while the red cylinder represents a spatial discontinuity
inside the radius r min.
Figure 4-1
In the preceding sections, three different phenomena responsible for causing motions
of celestial objects have been discussed. The first is what we know as normal gravity as
defined by Newton, defined by Einstein in general relativity, and again by Fluid Space
Theory. The second is expansion of the universe, the rate of which is simply defined by a
constant H times the distance from the observer. Universal expansion is a linear function
that starts at zero in the observers location and goes to infinity at an infinite distance.
These two are well know components of the standard model. The third phenomena is a
space-time contraction field surrounding normal matter, (the Mannfield) arrived at by
following the waterfall analogy to its logical conclusion, and it is unique to Fluid Space
Theory. Proponents of MOND (Modified Newtonian Dynamics) have proposed a third
component to gravity but it is not founded on any philosophical basis other than inserting
it out of thin air to make the math match observations.
Each of these three phenomena dominate over a particular range. Normal gravity is
by far the dominant force from very small radii out to the range of the Kuiper belt. The
Mannfield becomes significant somewhere around the Oort cloud, and rises as the
dominant force at intra galactic distances. The Mannfield remains dominant out to
around 100 parsecs from the center of a galaxy the size of the Milky Way. Beyond that,

universal expansion becomes the most significant of the forces, acting at distances on the
order of galaxy clusters and larger. So how does this work?
Section 4.1: Orbits in Gravity and Expanding or Contracting Space-time
In ideal flat space-time, considering two bodies, a large central mass and a small
orbiting satellite, the gravitational field will extend to infinity where the orbital velocity of
the satellite will be zero. At any closer distance, the satellite must have a tangential
velocity to prevent it from falling toward the central mass. In a stable orbit, that velocity
increases the distance from the central mass at exactly the same rate the object is drawn
inward.
If an expansion rate is imposed on the space-time, additional distance will be created
between the satellite and the central mass over time due to spatial expansion. Therefore,
less tangential velocity will be required to maintain a stable orbit. The amount of
expansion increases in proportion to the distance between the mass and the satellite and
eventually there will be a point where the rate of expansion will exactly balance the rate
the satellite is drawn inward by gravity. At this point the satellite will require no
tangential velocity. In this way, expanding space-time has the effect of changing infinite
gravity fields into finite gravity fields.
Conversely, if a space-time contraction field is imposed on the space-time, the satellite
will require additional tangential velocity to maintain a stable orbit, as it is drawn inward
by both gravity and the contraction field. As described in the opening paragraphs of this
section, FST contraction fields have a range about five times the size of a galaxy after
which expansion takes over.
Observations show that stars within galaxies have higher orbital velocities than can
be accounted for by gravity alone, leading to the invention of dark matter. These higher
orbital velocities could be better explained by the existence of a space-time contraction
fields as described above, not invented, but arrived at by reasoning. Observations also
show that the expansion of the universe appears to be accelerating, leading to the
invention of dark energy. Once again this may be explained by FST arrived at by reason
rather than invention, as follows.
As developed by FST, a galaxy will be surrounded by a spherical contraction halo that
extends beyond its rim before tapering off into intergalactic space which is dominated by
expansion. This is a profound change for cosmology and could drastically change the
estimated age of the universe and even overturn the Big Bang theory.
Considering a region of the universe containing a cluster of galaxies, the overall
expansion of the region will be the difference between the amount of expanding space
between the galaxies and the amount of contracting space within the galaxies. If the
galaxies in the cluster are close enough together, the region will contract overall. As the
galaxies in the cluster are spread further apart, the volume of expanding space-time will
become greater and the observed expansion of the region will be less affected by the
contraction halos surrounding the galaxies. The farther the galaxies are spread, the
greater the overall expansion will become by proportion. This will lead to an observed

acceleration in the expansion rate of the region as the galaxies move farther apart. No
dark energy required.
Thus in FST, there is no need for dark matter or dark energy, the rotation of galaxies
and the acceleration of expansion of the universe may be accounted for by the properties of
ordinary matter.
Section 4.2: Defining the Mannfield.
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. As fluid space-time flows inward into matter, volume is
continuously lost or transferred over to the tc axis, creating an ongoing space-time
contraction field. 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. Gravity remains active in the remaining space-time, but objects also
move toward the field center due to the fact that the space between has simply vanished.
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 space-time contraction around matter in a spherical
shell as a function of radius according to equation (13). Dividing equation (13). by the area
of the sphere yields what may be called the drift velocity. It represents the velocity a shell
of radius r will be shrinking due to the loss of space-time within the shell.
𝑣 𝐷𝑟𝑖𝑓𝑡 = �2𝐺𝑀/𝑟 (1 − �1 − 2𝐺𝑀/𝑟𝑐2)�1 − 2𝐺𝑀/𝑟𝑐2 (14)
This represents the velocity at which objects in the lost flux field will be swept toward
the center. This function is plotted in Figure 4-2.
Figure 4-2

This function has a similar form to the acceleration curve in Figure 3-2 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 (11) and (14) must be applied. Equation (11) 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.
At galactic scales, the acceleration due to gravity, and the drift velocity due to space-
time contraction become very, very small, however, the distances and volumes involved
become very, very large. Normally the effects of relativity at low velocities are neglected
and thrown out. In this case, in order to reveal the presence of the lost flux field, they
must be taken into account.
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 over a period of time to arrive at
equation (15).
𝑣 𝐷𝑟 = 𝑎 𝐷∆𝑡 𝑤ℎ𝑒𝑟𝑒 ∆𝑡 = 𝑁𝑠𝑒𝑐𝑜𝑛𝑑𝑠 (15)
Next this term is combined with the normal gravitational acceleration to create a scale factor
as shown in equation (16). The value of N is established based on observational data.
𝑆 =
𝑎 𝐺 + 𝑎 𝐷 𝑁
𝑎 𝐺
𝑜𝑟 𝑆 = �1 +
𝑎 𝐷 𝑁
𝑎 𝐺
� (16)
The total orbital velocity is then computed using the scaled acceleration as in equation
(17).
𝑣 𝑇 = � 𝑎 𝐺 𝑟𝑆 (17)
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 (11) and the drift velocity has been computed according to equation (14). 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 and expressed in the form of an equation.
Figure 4-3 is a plot of the orbital velocities for the planets in our solar system
predicted for Newtonian gravity, FST gravity and when the contraction field is 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 FST values exactly match Newtonian values

and with observations. The new contraction field corrected values add a small amount to
the orbital velocity values predicted by standard gravity. This increase in orbital velocity
begins around Jupiter and increases with distance from the sun.
Figure 4-3
There may have to be a re-evaluation of the value of the gravitational constant G and
calculated masses of the sun and planets. 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 predict the
orbits of Oort cloud bodies will be faster than expected for normal gravity where the
contraction field contribution becomes more significant.
Figures 4-4 and 4-5 are based on the galaxy model. The model was created in an excel
spread sheet by breaking the galaxy into a core plus 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 to 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 around 20 billion solar masses (not including any dark matter).
Figure 4-4
Stellar orbital velocity totals are in m s-1 predicted by the combined fields. The blue
line, velocity from G, is the contribution from gravity alone. The red line, velocity from C
(contraction), is the contribution from the contraction field.
Figure 4-5
Galactic mass distribution in solar masses.

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 4-1.
Zone Scale Factor Zone Scale Factor
1 4.01 11 11.69
2 5.03 12 12.24
3 5.97 13 12.76
4 6.84 14 13.25
5 7.66 15 13.72
6 8.43 16 14.16
7 9.15 17 14.58
8 9.84 18 14.97
9 10.49 19 15.36
10 11.10 20 15.73
Table 4-1
From this simple model, acceleration scale factors reached values more than 15 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 4.3: FST and the arrow of time.
Up until now FST development has focused on inflow fields rather than outflow
fields. The math works equally well for both and in either case, the acceleration term is
inward giving both inflow and outflow fields normal gravity fields. It is an important test
for any theory that if the arrow of time is reversed the system should retrace its history. If
reversing the arrow of time caused a planet to explode or fall out of orbit, there would be a
problem. Having inward gravity around matter in either case, FST passes the arrow of
time test.
Figure 4-6 shows the four possible cases of inflow, outflow, and forward and backward
time. Due to symmetry, these break down to two cases. Inflow going forward in time is
identical to outflow going backward in time and vise versa.
The two cases may be identified as matter and antimatter. If an inflow field were to
meet an outflow field, both moving the same direction in time, they would annihilate each
other and release the stored field energy.

Figure 4-5
While both types of flow fields have normal inward gravitational fields, the Mannfield
(lost flux field) of each will be reversed. Matter will have a contracting Mannfield while
antimatter will be surrounded by an expanding Mannfield. In an early universe filled
with clouds of hydrogen and anti hydrogen, The Mannfield would work together with
gravity to clump normal matter together while working against gravity within the anti
matter cloud. Thus we would see matter coming together, forming stars and galaxies,
while we would see antimatter disperse into intergalactic space.
If the arrow of time is reversed for a star or planet made of either matter or anti
matter, gravity remains inward, they don’t explode and they retrace their orbital paths in
reverse. The same is not true for galaxies. If the arrow of time is reversed for a galaxy,
which is held together by a contracting Mannfield, the reversing the arrow of time will
give the galaxy an expanding Mannfield. The reversal of the Mannfield will rip the galaxy
apart. Galaxies fail the reversal of time test in FST. So is this the end for Fluid Space
theory? Not at all.
Going back to an early universe filled with clouds of hydrogen and anti hydrogen, if
the arrow of time is reversed, we will see the antimatter moving backward in time, with a
contracting Mannfield, clump together to form stars and galaxies while we see the normal
matter moving backward in time, with an expanding Mannfield disperse into intergalactic
space. Since antimatter moving backward in time is the same as normal matter moving
forward in time, regardless of the direction the arrow of time points, we will see a universe
filled with contracting normal matter galaxies within an expanding volume of antimatter.
FST not only passes the arrow of time test, it answers the arrow of time question and
explains why we see the universe the way we do. It is seen the only way it can be seen.

Section 5. The Case for Energized Space-Time
𝐸 = 𝑚𝑐2
Most physics texts will say that Einstein’s famous equation above shows the
equivalence between energy and matter. What it tells me is that there is no such thing as
matter at all. Everything in the universe that we call matter is actually some stable form
of condensed energy. GR says that space-time tells matter how to move and matter tells
space-time how to bend, but in GR, the two are entirely separate. I propose that space-
time can become energized through spatial contraction and time dilation, due to unseen
velocity flow fields. In this light, everything in the universe is made of Space-Time-Energy
and these may be the only ingredients needed to make everything we observe.
At the end of Section 1, we crept near to some concepts for energized space. I would
like to formalize those now. To begin, I will define spatial and temporal strains
mathematically. Strains, by definition, have a value between zero and one.
Figure 5-1
Figure 5-1 shows a unit cube of space-time which undergoes a velocity change in the x
direction. The unit cube becomes compressed, or spatially strained. A spatial strain is
defined as follows.
𝜖 𝑠𝑝𝑎𝑐𝑒 =
𝑥 − 𝑥′
𝑥
=
𝑥 − 𝑥�1 −
𝑣2
𝑐2
𝑥
= 1 − �1 − 𝑣2/𝑐2
For every spatial strain there is a proportional perpendicular temporal strain on the
invisible tc axis. A vibrating spatial strain will produce a perpendicular temporal strain
equal to and in phase with the spatial vibration. This is the second parallel between Fluid

Space Theory and electromagnetism. Let’s take another look at the four dimensional
transform.
𝑧 ∗ 𝑦 ∗ 𝑥 ∗ 𝑡𝑐 = 𝑧′
∗ 𝑦′
∗ 𝑥′
∗ 𝑡′
𝑐 𝑤ℎ𝑒𝑟𝑒 𝑧 = 𝑧′
𝑎𝑛𝑑 𝑦 = 𝑦′
and
𝑥′
= 𝑥�1 − 𝑣2/𝑐2 𝑡′
= 𝑡/�1 − 𝑣2/𝑐2
According to these formulas, space can only be compressed and time can only be
stretched. A meter will never be seen longer than one seen in the observers rest reference
frame and a second will never be shorter than one seen in the observers rest reference
frame. (It would be great if this were not true because then all of the cool stuff in science
fiction would be possible, such as warp drive, and worm holes that actually go somewhere).
This requires a different definition for a temporal strain in order to produce a value
between zero and one.
𝜖 𝑠𝑝𝑎𝑐𝑒 =
𝑥 − 𝑥′
𝑥
𝑓𝑜𝑟 𝑡𝑖𝑚𝑒 𝜖𝑡𝑖𝑚𝑒 = (𝑡′
− 𝑡)/𝑡′
𝜖𝑡𝑖𝑚𝑒 =
𝑡′
− 𝑡
𝑡′
=
⎝
⎛ 𝑡
�1 −
𝑣2
𝑐2⎠
⎞ − 𝑡
𝑡
�1 −
𝑣2
𝑐2
= 1 − �1 −
𝑣2
𝑐2
= 𝜖 𝑠𝑝𝑎𝑐𝑒
Figure 5-1 also shows a standard force deflection plot with a linear elastic constant.
The slope of the line is the modulus and the area under the line is the energy required to
cause the deflection. Now if we only knew the elastic modulus of space-time we could
compute the energy of a fluid space flow field. Casting about for a possibility, I realized
that c to the fourth power divided by G has the units of force (kg-m-s-2). Thus I propose
the following relationship. I have included a constant K in case it is actually some
multiple of my guess.
𝑓𝑜𝑟𝑐𝑒
(𝑢𝑛𝑖𝑡 𝑎𝑒𝑟𝑎)
=
𝐾𝑐4
𝐺𝐴
= 𝑌𝑜𝑢𝑛𝑔′
𝑠 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑠𝑝𝑎𝑐𝑒 − 𝑡𝑖𝑚𝑒
Strain energy U is commonly defined in engineering as follows. (E is Young’s
modulus).
𝑈 = �
1
2
�∀𝜎𝜖 = �
1
2
�∀𝐸𝜖2
It is important to understand that U as defined above represents the energy
accumulated between a strain of zero and the strain at a given deflection. As applied to a
fluid space inflow field it represents the energy accumulated as space-time falls from

infinity to any given radius r. We know the strain as a function of velocity and we know
velocity as a function of radius, so first we define U as a function of velocity as in equation
18. Substituting the proposed values and using the volume of a sphere of radius r for the
volume, yields the following.
𝑈(𝑣) = �
1
2
� ∀ �
𝐾𝑐4
𝐺𝐴
��1 − �1 −
𝑣2
𝑐2
�
2
= �
𝐾𝑐4
∀
2𝐺𝐴
� �1 − �1 −
𝑣2
𝑐2
�
2
(18)
To express energy U as a function of radius we set v2 equal to 2GM/r as before.
𝑈( 𝑟) = �
4𝜋𝑟3
𝐾𝑐4
24𝜋𝑟2 𝐺
� �1 − �1 −
2𝐺𝑀
𝑟𝑐2
�
2
= �
𝑟𝐾𝑐4
6𝐺
� �1 − �1 −
2𝐺𝑀
𝑟𝑐2
�
2
(19)
In the case of an inflow field, the value of U at a given r represents the amount of
energy required to compress space time surrounding a gravitating body from infinity to
any given radius r. At r equals infinity, the energy is zero. The value of U increases as r
is reduced from infinity to r min. When v is set equal to the speed of light c, or r is set
equal to r min, 2GM/c2, we get the following.
𝑈(𝑟 𝑚𝑖𝑛) =
𝐾2𝐺𝑀𝑐4
6𝐺𝑐2
(1) 𝑜𝑟
𝐾𝑀𝑐2
3
(20)
Now if we assign a value of 3 to the constant K we find the following.
𝑖𝑓 𝐾 = 3, 𝑡ℎ𝑒𝑛 𝑈(𝑟 𝑚𝑖𝑛) =
𝐾𝑀𝑐2
3
= 𝑀𝑐2
(21)
We have arrived now at Einstein’s matter energy equivalence equation by an entirely
different route. This tells us that the total energy of the space-time contraction field
around a gravitating body is equal to the rest mass times the speed of light squared. What
does this mean? It means that there is no such thing as matter separate from space time.
Matter is energized space-time. Matter doesn’t tell space-time how to curve, matter is
curved space-time!
This confirms my choice for the elastic modulus of space-time and more. The
interpretation of this for Fluid Space Theory is that when sufficient energy is concentrated
in a small enough volume, a discontinuity in space-time will pop into existence. Thus
laying down a theoretical basis for the existence of “quantum foam”.
This also provides a method of calculating the vacuum energy in space-time created by
a gravitational field. Equation 19 gives the amount of energy in a flowing space-time field
outward from any given radius. It can be expressed as follows.

𝑈( 𝑟) =
𝑟𝑐4
2𝐺
𝜖2
𝑤ℎ𝑒𝑟𝑒 𝜖 = �1 − �1 −
2𝐺𝑀
𝑟𝑐2
� 𝑎𝑛𝑑 𝑣 = �2𝐺𝑀/𝑟
(22)
Figure 5-2
Figure 5-2 is a plot of the total field energy as a function of inflow velocity. It can be
seen that the energy content of flowing space-time remains very low until velocities reach
about half the speed of light with the bulk of energy content coming between .8c and 1.0 c.
This happens at small radii, indicating that the field energy of an object we would
previously have thought of as matter is concentrated near the heart of the flow field.
Figure 5-3
Figure 5-3 is a plot of the field energy as a function of radius. It can be seen that
almost all of the energy is concentrated close to the minimum radius.
In equation 22, U has the dimension of Joules. The energy density or vacuum energy
VE at any radius can be found using the shell method. The field energy at a larger radius
is subtracted from the field energy at a smaller radius and the remainder is divided by the
volume of the shell formed between the two radii as follows.

𝑉𝐸( 𝑟1) ≈ (𝑈( 𝑟1) − 𝑈(𝑟2))/(4𝜋𝑟1
2
∗ (𝑟2 − 𝑟1)) (23)
The volume of the shell has been approximated by multiplying the area at radius r1
one by the thickness of the shell (r2 – r1). This assumes that r2 is only slightly larger than
r1. VE in equation 23 has units of Joules per cubic meter.
Unlike calculating the field energy surrounding a single central mass, when using the
shell method in the galaxy model, with distributed stars, the mass within the shell also
has to be taken into account. The field energy at r1 is calculated without the mass inside
the shell and the field energy at r2 is calculated using the mass inside r1 plus the mass
within the shell. This results in a negative vacuum energy, consistent with a contracting
space-time. Figure 5-4 shows the galaxy rotation curves with vacuum energy.
Figure 5-4

Section 6: Fluid Space Flow Geometry and Quantum Theory.
Quantum theory has some features that don’t seem to mix well with our everyday
notions of space and time. First of all, we consider empty space as a continuum with no
structure to speak of. It is assumed that there is no volume of space or time that cannot
be further divided. In the quantum world, things come in lumps or quanta of larger or
smaller size and there are gaps above and below these quantum states where nothing is
allowed.
Second, quantum theory has superposition of states. Any system we can observe, such
as an atom, or a molecule, or a cat in a box with a capsule of poison, has a finite number of
states in which it can exist. Until we observe the system, we don’t know what state it
occupies. Superposition says that the unobserved system actually occupies all possible
states and it is the act of observation that causes the system to resolve itself into a single
state. In quantum physics this is known as collapse of the wave function. Each possible
state has a particular probability attached to it and with enough observations, eventually
all possible states may be observed.
Part of the problem in relating quantum theory to our everyday world is that we
consider ourselves as 3 dimensional beings living in a 3 dimensional world. As Einstein
showed us, and as I have illustrated in earlier sections, space-time is four dimensional.
We need to accept that we live in a four dimensional universe. What kind of creatures and
objects would you expect to occupy a four dimensional universe? Four dimensional of
course. We must embrace this and understand that we ourselves and our surroundings
are in fact four dimensional.
Consider the following thought experiment. Two boys are passing the time at a space
train station by tossing a rugby ball across the tracks to each other. Space trains travel at
just below the speed of light and they run on a strict schedule. Some trains stop at the
station and some express trains speed right through it. If an express train were to strike
the ball while it is over the tracks, the station and everyone in it would be annihilated in a
tremendous explosion. But since nothing is scheduled to pass through any time soon,
station security allows the boys to play.
Since the space trains move at nearly the speed of light there is no warning of their
arrival. Any signal from the train could only arrive fractions of a second before the train
itself, thus the adherence to a strict schedule. Unfortunately, as the boys toss the ball
across the tracks, an unscheduled maintenance drone passes through the station at nearly
the speed of light. Luckily, the maintenance car is built as a cylindrical cage with an open
truss framework. It is empty on the inside and it passes around the rugby ball while it is
over the tracks.
As the maintenance car passes through the station, for an instant, the boys see their
rugby ball inside the cage of the speeding car. To them, the ball, being in their own
reference frame, appears normal. The car however appears contracted along the direction
of the track to a fraction if its normal length.
If you only consider the objects, as Einstein did, there is no problem with seeing an

uncompressed ball surrounded by a compressed train car inside an uncompressed station.
If, as FST proposes, it is the space-time the object occupies that is compressed, you might
ask how can the space-time of the station and the ball be uncompressed at the same time
the space-time of the rail car is compressed.
The answer lies in embracing the full potential of four dimensions. When we look at a
volume of empty space, we naturally put that space into our own reference frame and
consider it to be motionless. However, we must understand that any given volume of
space-time is capable of allowing passage of an object at any allowed velocity, in any
direction, at any time. Borrowing from quantum theory, FST says that all possible states
of space-time exist at all times within any given volume. It is only the observed passage of
an object that causes the space-time to resolve into any particular state.
Given this new understanding of space-time, let’s look at the geometry of relativistic
source/sink flows and how it can relate to quantum particles.
The earlier section covered the simple case of a non rotating, smooth inflow object or a
black hole. Figure 6-1 shows how a FST black hole is conceptually different from a black
hole for mainstream physics.
Figure 6-1
Both black holes have a funnel shape and transition from essentially flat space into a
region of highly curved space. Both describe a gravitational field. Both have an event
horizon which divides normal space from complex space (more about that later). Going
beyond the event horizon, the traditional black hole has a region of superluminal velocity
that continues down to a singularity. Beyond the event horizon of the FST black hole, the
velocity may or may not be superluminal and there is no more tapering down, instead
there is a space-time discontinuity of finite size.
Another profound difference between a FST black hole and the traditional black hole
is that while everything the makes up the traditional black hole is concentrated far below
the event horizon in a singularity, everything that makes up the FST black hole (energy,

matter, momentum, entropy, etc) lies above the event horizon. Inside the event horizon of
the FST black hole lies a domain that is not part of our universe. Our space-time
coordinates don’t go there. It is a bubble of something else, a discontinuity. If anything
lies inside, what it is cannot be known.
As presently defined, a FST black hole can be made from any amount of energy and
have a rmin of any size. It is also smooth round and not rotating. If the inflow of space-
time is not smooth, a number of possible vibrations could take place. Rather than level
out at velocity c, irregularities in the flow may cause the core of the flow stream to become
superluminal.
As sown in Figure 3-2, the acceleration curve at rmin (flow velocity =c) acceleration is
zero, meaning that gravity has stopped and there is no more accelerating force. There is
nothing to drive further acceleration above c or cause any further reduction in the size of
the flow stream below rmin. If the curve is continued beyond rmin, the acceleration becomes
negative, which would result in push back against the flow.
Figure 6-2
In superluminal flow, below rmin, the term under the radical in the velocity equation
becomes negative and the solution becomes a combination of real and imaginary numbers,
a complex number. Following the curve in figure 3-2, when the velocity becomes
superluminal acceleration becomes negative, creating a repulsive force to drive the
velocity back below c. At this point the flow stream may oscillate between sub and
superluminal velocity, or between real and complex space-time.
From the point of view inside the flow field at the event horizon, two spatial
dimensions (up/down and left/right) are restricted to very tiny values. The
inward/outward dimension, however, is infinite. From the point of view outside the flow
field, time at the event horizon has come to a virtual stop. In this way, a four dimensional
flow field transitions into an essentially one dimensional object. This one dimensional

object will be very stiff and could behave like a piano string. See figure 6-2 for possible
vibration modes of a FST object. Thus, at the heart of a fluid space inflow field, there is
found a complex harmonic oscillator. This oscillation would best be described by a wave
function involving complex numbers.
𝜑( 𝑥) = 𝑒 𝑖(𝑘𝑥−𝜔𝑡)
We may now proceed to develop quantum and string theory in harmonious connection
with a gravitational object.

Section 7: Gravity and Electromagnetism.
A well known parallel between gravity and electromagnetism is Coulombs law for
static charges.
𝐶𝑜𝑢𝑙𝑜𝑚𝑏′
𝑠 𝐿𝑎𝑤 𝐹 = 𝑘 𝑒 �
𝑞1 𝑞2
𝑟2
�
Which has an identical form to Newton’s equation for gravity.
𝑁𝑒𝑤𝑡𝑜𝑛′
𝑠 𝐿𝑎𝑤 𝐹 = 𝐺 �
𝑚1 𝑚2
𝑟2
�
I believe this is no coincidence and potentially enlightening. In Coulomb’s equation ke
plays the role of G and charge plays the role of mass. The SI units for ke are kilogram
meters cubed per second squared Coulombs squared. Whereas G has units of meters
cubed per kilogram second squared. Both constants have a time change of spatial flux
term, m3/s2. This suggests that charge may also be a manifestation of an energized spatial
flow.
If the time change of spatial flux term is removed we are left with kg/C2 from ke and
1/kg from G. If there were a mass energy equivalent for charge like E=qc2 or an energy
conversion factor between coulombs and kilograms, ke would reduce to the exact same
units as G. Let’s look for that conversion factor. Assume that the matter energy
equivalence may be related to a similar charge energy equivalence through a conversion
factor H.
𝐸 = 𝑚𝑐2
≈ 𝐻𝑞𝑐2
𝑤ℎ𝑒𝑟𝑒 𝐻 = 𝑚/𝑞
First we find a set of charges and masses such that:
𝐹 = 𝐺 �
𝑚1 𝑚2
𝑟2
� = 𝑘 𝑒 �
𝑞1 𝑞2
𝑟2
� 𝑎𝑛𝑑
𝑚1
𝑞1
=
𝑚2
𝑞2
Dividing out the r’s and substituting H for m1/q1 and m2/q2 we get.
𝑘 𝑒
𝐺
= 𝐻2
Punching in the numbers yields. (This H is not Plank’s constant.)
𝐻 = 1.160453123 ∗ 1010
𝑘𝑔/𝐶𝑜𝑢𝑙𝑜𝑚𝑏
This tells us that it would take 1.16 times 1010 kilograms of matter to create the same
gravitational force as the electric force produced by 1 coulomb of charge. If the energy
equivalence equation holds, then there is far more energy tied up in the form of charge in
the universe than there is in matter. This energy, if present, does not create gravity nor
show itself in terms of mass and momentum in the material elements it is attached to and
only acts on other elements of charge. This all seems very unlikely. While the value of H

is probably valid in the comparison of forces, it may not be useful in terms of calculating
energy content. It is more likely that electromagnetism has a different space-time flow
pattern involving another dimensional axis, replacing the x, y, z, or tc with a new axis
perpendicular to the one it replaces. The spatial distortion energy associated with the new
axis could be quite different from that of the tc axis, involving an entirely different
modulus. Since we see charge acting in our x, y, and z dimensions, it must be the tc axis
that has been replaced. This would also prevent electric charge from interacting with time
and therefore gravity.
It may be helpful to think of a gravitational field as a four dimensional object in
rotation about a plane or pair of planes. Visualization of such an object is difficult,
however a 4D hyper-sphere rotating about the z-y and x-tc planes could appear in 3D
space as an inflow/outflow field. Another pairing of planes might account for
electromagnetic fields.
It is important keep in mind that these objects are fluid, not made of elastic
membranes or solids. As fluid forms, they will take the shape necessary to keep all forces
in balance and be flexible and self-correcting when acted upon by an outside force. This
may be viewed similarly to the way charge distribution inside a conductor flows to always
keep the electric potential equal to zero.
Figure 7-1
Figure 7-1 shows a 3Sphere intersecting a plane. If the surface of this sphere is
projected onto the plane and the sphere is rotating at a constant rate about the vertical
axis, a two dimensional being living on the plane will see a circle in which area magically
appears on one side of the circle, travels across the circle and vanishes on the other side.

Figure 7-2
The left image of Figure 7-2 shows a bounding volume of a 4Sphere around the time
axis. The right image is a close-up of the center, showing that the passage does not go to
zero but has a minimum radius Rmin. The surfaces in these images represent volumes in
which the dimension normal to the surface has been compressed to zero and the green axis
in the figure is the time axis. Imagine that the torus is rolling along the time axis like a
smoke ring. If the surface (volume) of the torus is projected onto the central plane
(volume), a 3D being living in that volume will see space magically appear at the edge and
accelerate toward the center where it will slow and then vanish at Rmin. This is exactly
what we see in the gravity field of a material object. A FST inflow field can be viewed as
an energized hyper-sphere rolling along the time axis.
Only one bounding volume is shown in Figure 7-2 but there would be an infinite
number of them ranging in size from Rmin to infinity, all rolling along the time axis, each
representing a radius/velocity profile proportional to the central mass. This rolling smoke
ring form is shared by all material objects and is nested in space-time with all other
objects as in a fractal pattern encompassing the entire universe.
In a doughnut shaped universe (like the one in Figure 7-2 or like the one stolen from
Homer Simpson by Stephen Hawking), The “Big Bang” would not be an event but a
permanent feature of the universe where recycled energy is spewed out continuously on
one side, travels around the torus and is eventually swallowed up on the other end only to
be spewed out once more (no need for inflation).

Figure 7-3
The left image of Figure 7-3 shows a bounding volume of a 4 sphere similar to the one
in Figure 7-2, however this time it is rolling on a spatial plane xy, xz, or yz. The right
image shows how, if projected to the central plane, a 3D being living in that volume will
see a dipole field with space-time coming out one end and going into the other.
Furthermore if two of these objects are brought together, like ends will repel and unlike
ends will attract, in short, a magnetic field.
As it turns out, there is another plane which may be considered as existing
perpendicular to the three spatial axes and the tc axis. It is the complex plane. The
complex plane has several advantages. First, it may be used to provide additional flow
field interactions without invoking another dimension. Second it has an established role
in electromagnetic theory (not to mention quantum mechanics). And third, complex
numbers only interact with other complex numbers, similar to the way EM/charge only
interacts with other EM/charge.
As shown earlier, gravity is the 3D manifestation of a rotating 4 sphere involving the
three spatial dimensions and time with conservation of four volume and invariant under
the Lorentz transformations. Let’s call these 3s,1t fields.
The remaining possibilities are a four sphere involving the three spatial dimensions
and one complex dimension (3s,1i), or a four sphere involving two spatial dimensions, one
complex dimension and time (2s,1i,1t).
Let’s consider the first case. A 3s,1i field would be invariant in time and would
manifest in the three spatial dimensions where we live as an inverse distance square
monopole field. Additionally it would come with two solutions either +i or –i. This sounds
like a good candidate for electric charge. The mathematical description will involve a
combination of Lorentz transformations for real space and possibly Mobius transforms for
the complex spaces. (I will need help here with the math.)

The second case (2s,1i,1t) will be a time dependant field which would manifest in 3D
real space as a dipole field with the solutions for +i and –i simply flipping the polarity.
This is a good candidate for magnetism.
These fields will be limited in how they can attach, or stem from the 3s,1t spatial
inflow fields described in the earlier sections. These attachments or stemmings can only
happen at the heart of the inflow at the critical energy levels where the flow field oscillates
between real and complex space.
Let’s take a look at 3s,1i space and apply the same reasoning we used for Minkowski
space (3s,1t). While three dimensional volume is not conserved in spatial flows, four
dimensional volume is conserved. Four volume is defined as follows for 3s,1i space.
∀4= 𝑥1 ∗ 𝑥2 ∗ 𝑥3 ∗ ∓𝑖𝑥4 𝑎𝑛𝑑 ∀′
4= 𝑥1
′
∗ 𝑥2
′
∗ 𝑥3
′
∗ ∓𝑖𝑥4
′
Under a Lorentz transformation four volume is unchanged (x1=x1’ and x2=x2’).
∀4
′
= 𝑥1′ ∗ 𝑥2′ ∗ 𝑥3
′
�1 − 2𝐺𝑀/𝑟𝑐2 ∗
∓𝑖𝑥4
′
�1 − 2𝐺𝑀/𝑟𝑐2
= ∀4
The terms under the radical in the above equation are still expressed in terms of
gravitational fields. Let’s see if they can be expressed in terms of electromagnetism. We
will make the following substitutions.
𝜖0 𝜇0 =
1
𝑐2
𝑎𝑛𝑑 𝐺 =
𝑘 𝑒
𝐻2
𝑎𝑛𝑑 𝑈𝑒/𝑐2
= 𝐸/𝑐2
= 𝑀
Thus
2𝐺𝑀
𝑟𝑐2
=
2( 𝜖0 𝜇0)2
𝑘 𝑒 𝑈𝑒
𝑟𝐻2
And now we have:
∀4
′
= 𝑥1
′
∗ 𝑥2
′
∗ �𝑥3
′�1 −
2( 𝜖0 𝜇0)2 𝑘 𝑒 𝑈𝑒
𝑟𝐻2
� ∗
⎝
⎛
∓𝑖𝑥4
′
�1 −
2( 𝜖0 𝜇0)2 𝑘 𝑒 𝑈𝑒
𝑟𝐻2 ⎠
⎞ = ∀4
We can now retrace all the reasoning applied so far to gravitational flow fields on
these complex flow fields. In this case rather than looking at how the spatial dimensions
change with time, we look at how they change with respect to the complex dimension x4.
We will arrive at a pair of monopole, inverse distance square fields, with opposite signs.
These will manifest in space-time whenever they become attached to a space-time flow
field as described earlier, sharing the spatial dimensions but invariant or immune to the
effects of time. In particle physics, charge is always found in conjunction with a particle
that has the property of mass. But some particles that have mass do not carry charge.

For this reason I present the following. I postulate that 3s,1i fields can only exist when
attached to a 3s,1t field but a 3s,1t field can exist without a 3s,1i field (or with two
opposite superimposed 3s,1i fields). This is the basis of electric charge and how it is linked
to gravity. All of current EM theory descends directly without change but with a new
basis founded on spatial flow fields.
My primary question at this point is if and how these complex 3s,1i flow fields may be
used to influence 3s,1t flow fields.
(C) Hugh Mannfield 2017

Fluid Space Theory

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