The Two or One Dimensional Galaxy
The Modified Einstein Field Equation (MEFE)
The Cosmic Segmentation

The "Dark Matter" be nothing but the "Light".
by Iori Onestone Fujita Grossfeld
--- This is a hypothesis. ---
from June 11, 2014
Music of Sacred Temperament - Music of Intellect - Music of Desire
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Genesis 1:3 And God said, Let there be light: and there was light.
Genesis 1:4 And God saw the light, that [it was] good: and God divided the light from the darkness.
April 10, 2006, August 13, 2008, November 19 2008, January 03 2009, November 28 2009
June 01, 2014

←←←Gravity→→→ ↑↓Light Emission↑↓

The background music is Arnold Schoenberg's "6 Kleine Klavierstucke op.19" arranged by me

The spiral galaxy flat rotation problem

In the disk portion of a spiral galaxy, outside of the central bulge, the average orbital speed of an object in rotation at a given radius should have decreased inversely with the square root of the radius of the orbit, based on Newton's laws. Actual observations of the rotation velocity curve of spirals, however, tell us a different fact, which is that the speed is nearly a constant function of radius. This phenomenon is called the "flattening of galaxies' rotation curve".

Calculation based on the ordinary three dimensional universal gravity equation;
GM(<r)m / r2 = mv2 / r → v = ( GM(<r) / r )1/2
M(<r) is for all the matter inside the radius of the orbit.
If v = const, then M(<r) ∝ r ,
so the mass of the Galaxy is thought to be bigger and bigger. The most widely accepted approach to explaining this problem requires the existence of dark matter which could be ten times or more heavier than the total mass of visible stars inside of the galaxy.

For the two dimensional universal gravity equation; It can be called "New2ton's equation".

F can be transformed into F = G'Mm / r
Here G' = the two dimensional universal gravity constant
F = G'Mm / r → G'Mm / r = mv2 / r
v = const = ( G'M )1/2
M ≈ M(<r) ∝ v2 = constant for each galaxy
When the gravity is described as F = G'Mm / r , the Potential would be U(r) = − G'M ln( r ).
When r → ∞, U(r) → ∞. It seems to be very awkward, but the space around the spiral galaxy is not authentic 2 dimensional, it might be quasi authentic 2 dimensional.
So let's change F = G'Mm / r a little bit to, for example, F = G'Mm / r1.00001 .
By this change, r → ∞ leads U(r) → 0, happily.

From this we learn that the force law is
Inversely proportional to r, if r >> d/2 ,
Inversely proportional to r2, if r << d/2 ,
Something a bit complicated if r and d are about the same size.

But inside of the slab and r << d/2 ,
then around there it looks like an ordinary 3d space.

Look at the small green ball in the right.
Take one of the two objects; a bigger red ball.
Draw a surface surrounding one of the objects
that lies at a constant distance r from the first object;
This surface is the green ball.
Measure the surface area.
Then if the surface is nice and symmetric,
the strength of the force is inversely proportional to that area!

In a world with three spatial dimensions of infinite extent,
the dependence on the distance r of the force between the two red balls
is inversely proportional to the area of the sphere
that is centered on one ball and passes through the other.

There in left are objects in red a distance r away.
Let’s draw a surface that contains all the points
that are a distance L away from the center red object: that’s a sphere, of radius r.
That’s the greenish sphere.
What’s the area of the sphere? 2 π r d.
What’s the force law? It is inversely proportional to r.
This is true for both Newton’s law of gravity and Coulomb’s law of electrical forces.
It’s a purely geometric effect.

inspired by Theoretical Physicist Dr. Matt Strassler's
"Extra Dimensions & Newton’s Gravity [June 20, 2012]"

Of course I am not glad to change Newton's law for galaxies alone. But it is reasonable to consider the possibility of the universal gravity equation for the two dimensional space with certain depth like a pizza dough. Stars like the Sun are swaying up and down confined by the steep gradiant of the gravity field of the surfaces of the disk. Inside of this pizza dough, the space is ordinarily three dimensional while the distance of objects is relatively short, for example, within the solar system.

One trajectory of a matter has both the nearest and the furthest point from the center of the galaxy. And the circle tangent velocity against the center of the galaxy is to be constant. Then where has been the potential energy? We can find it in the vertical movement. When the trajectory comes the nearer to the center of the galaxy, it will sway up and down the more rapidly.

new - The Grand Hypothesis No.00 - new
The Law of Universal Gravitation and Separation

First of all, I would like to present this fundamental working hypothesis. It is the universal force which unifies gravity and separator into one. No one ever knows about the separator force. So now, I give a definition of the law of universal separation at first.
Fs = − Sp Ea Eb / r2 F ; the separation force
Ea ; Energy which belongs to the point a
Eb ; Energy which belongs to the point b
r ; the distance between a and b
Sp ; the separation constant
This force will not be detected on the Earth. It can be negligible even in the solar system. But it will work in the galactic scale.
Next step is to unify Gravitation and Separation into one law.
Fg = G Ma Mb / r2 -----------------Then,

Fg+s = G Ma Mb / r2 − Sp Ea Eb / r2

And assume that Sp = G / c4, because E = mc2.
Fg+s = G Ma Mb / r2 − (G / c4) Ea Eb / r2
One step forward by using the complex number formula.
F = G ( Ma + i Ea / c2 ) ( Mb + i Eb / c2 ) / r2
F = G Ma Mb / r2 − (G / c4) Ea Eb / r2 + i ( G Ea Mb / ( r2 c2 ) + G Ma Eb / ( r2 c2 ))

The real part is Re( F ) = Fg+s, but I don't know how to deal with the imaginary part ; Im( F ) = G Ea Mb / ( r2 c2 ) + G Ma Eb / ( r2 c2 ).

So an existing substance is to be described as S = M + i E / c2 .
In a certain independent area, if M + E / c2 = constant ( in other words when M desceases by ΔM, E will increases by ΔE = ΔM c2 ), then abs( S ) = will be minimum when M = E / c2, because abs( S ) = root( M2 + E2 / c4 ).
The Void is like the White Mountain which is increasing its height.
The Void is a very large region of space with containing few, or no, galaxies. It is the empty space between galaxy filaments. There is nothing but the light, or energy. And the Void is like the White Mountain. Unlike White holes which are thought to be similar to black holes except white holes are ejecting matter verses black holes are absorbing matter, the White Mountain, the Void, is a mountain growing up at the light speed. It means that the peek of the mauntain or the center of the Void ejects an expansion of space at the light speed. So the light from outside can not reach to the center of the Void. Then the void looks like a concave lens which diffuses the light, while the strong gravity acts as a convex lens.

Lights which have fallen down from the Void Mountain, in other words, the positive separational potential, would have gotten too much potential energy to stay as photons. When thier wavelengths become shorter than Planck length lp, photons would turn into matters which soon activate gravity forces in the negative potential valleys.
The Cosmic Segmentation instead of the Cosmic Inflation
In physical cosmology, cosmic segmentation is a new idea that the nascent universe passed through a phase of exponential segmentation and extension that was driven by a separation force of the light emission energy. As a direct consequence of this segmentation, all of the observable universe originated in a small causally-connected region. Segmentation may answer the classic questions of the big bang cosmology: why does the universe appear flat, homogeneous and isotropic in accordance with the cosmological principle when one would expect, on the basis of the physics of the big bang, a highly curved, inhomogeneous universe. Segmentation may also explain the origin of the large-scale structure of the cosmos like voids and walls.
Segmentation has stopped when the potential of the separation field became so weak that no more "cell membranes" could be made. In these "cell membranes", there are galaxies and stars, in other word, matters.

Fm = m am Fm is Force for mass. m is mass and am is acceleration for mass.
Fe = e ae Fe is Force for energy. e is energy and ae is "acceleration" for energy.
But light can not be accellerated over c or
light speed is constant in the vaccum space.
So "acceleration" should be in a different way.
Fe cΔt = ΔE = ΣhΔνthen Fe = ΔE / cΔt = Σ(h/c) (Δν/Δt) = Σ(h/c) ν'
ν = frequency and h = Planck's constant
e = Σh ν finally Fe = e ν'/(cν)
This means ae = ν'/(cν) , which is the frequency acceleration.

new - The Grand Hypothesis No.01 - new
The Modified Einstein Field Equation (MEFE)

The extremely small world and the extremely big world are governed by the complex number mathematics.
Only the ordinary world can live with the real number mathematics easily.

Einstein original Rμν - 1/2 ⋅ gμν ⋅ R = 8πG/c4 ⋅ Tμν
Modified version Rμν - 1/2 ⋅ gμν ⋅ R = 8πG/c4 ⋅ (Tmμν + i⋅Teμν)
------Tmμν is the stress tensor by matter.
------Teμν is the energy tensor by emission energy.
The original metric: ds2 = dx2 + dy2 + dz2 - c2dt2
The modified metric: (dsσ ei σ)2 = (dxχ ei χ)2 + (dyψ ei ψ)2 + (dzω ei ω)2 + c2(i⋅dtτ ei τ)2
(dsσ ei σ)2 = (dxχ ei χ)2 + (dyψ ei ψ)2 + (dzω ei ω)2 + c2(dtτ ei (τ - π/2))2

σ = χ = ψ = ω = τ = 0 then The modified metric = The original metric
ds2 = dx2 + dy2 + dz2 - c2dt2

σ = χ = ψ = τ = 0 and ω = π/4 then The real part of the modified metric = The two dimensional metric
Real(ds2) = dx2 + dy2 - c2dt2 ----and---- Im(ds2) = dz2

σ = χ = τ = 0 and ψ = ω = π/4 then The real part of the modified metric = The one dimensional metric
Real(ds2) = dx2 - c2dt2 ----and---- Im(ds2) = dy2 + dz2

---new--- The Modified Schwarzscchild Metric : MSM ---new---
This Schwarzschild solution has two singularities at r = 0 and r = 2GM/c2 = rs. When r apprachs rs, one of the metric components, ****dr2 goes up to the infinity. The singularity at r = rs is said to be an illusion caused by the bad choice of coordinates or coordinate conditions. But the theory should be independent from the choice of cordinates, which is the core of the Relativity.
Instead of M, I place M + iE/c2 ( assume that E won't be zero). Then the singularity at r0 will vanish .

The General Hypothesis No.1 for the Two Dimensional Galaxy
A portion of each spiral galaxy, outside of its central bulge, is crushed or curled up in a 2 dimensional disk with a certain depth. In a great structure like one galaxy, the three dimensional gravitational space must be transformed or shrinked into a disk in the outskirt area of the galaxy where the ordinary three dimensional universal gravity equation would not work for the relation between stars and the galaxy.
"Kepler's third law" for the spherical symmetric space;
(3) The squares of the planets' periods of revolution are proportional to the cubes of the planets' mean distances from the sun.
It can be explained for the two dimensional space like;
(3') The squares of the stars' periods of revolution are proportional also to the squares of the planets' mean distances from the center of the galaxy.
Then I can explain that stars in the galaxy can move around the center of the galaxy with the same velocity, not related to the distances from the center. And now I don't have to assume the existence of the "Dark Matter" against the galaxy rotation problem.
According to the Einstein Theory;
Matter warps the continuum according to the amount (or mass) which exists in any locality. Then all motion along world lines in the vicinity must follow the warp.If world lines must follow warps in space, beams of light passing near the mass should be deflected with a certain angle.
But here for me, the light can't be deflected one-sidedly. It should also affect back the gravity field. Beams of light will loosen the gravity field to the orthogonal direction slightly. And huge amount of beams of light or emissions will shove away the gravity field in a thin-depth disk.
NGC 891, Spiral Galaxy in Andromeda, is a fine edge-on spiral with a faint but beautiful "dust" lane along its equator. The dark band across the middle is said to be a lane of "dust" which absorbs light. People say that this "dust" can be the "Dark Matter". But actually in this band the gravitational field is so condensed that beams of light are pushed away to the upper or lower side of the disk. In other words there is no light emissioned from this band. So the "Dark Matter" has nothing to do with this dark edge.

This is the first, and brightest lens system discovered in 1995 with the Hubble telescope. The light cames from a quasar about 11 billion light-years distant. The elliptical lensing galaxy is located 7 billion light-years away. A two dimensional galaxy concentrate gravity vectors on the plane containing the disk of its galaxy. The upper image and the lower image have different distances from the center of the galaxy depending on the inclination of it.

The General Hypothesis No.2 with Schrödinger equation
for the Two and One Dimensional Galaxy

Schrödinger equation is;
For the galaxy I have to consider two things; the Gravity Potential Distribution and the Emission of light Potential Distribution.
The spherical harmonics are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present.
Visible ShapeMajor Gravity+Major Emission
elliptical galaxy
e.g. NGC4881
Three Dimension
GM(<r)m/r2 = mv2/r

gravitationally unstable
spiral galaxy
e.g. NGC4414
Two Dimension
G'M(<r)m/r = mv2/r
barred spiral galaxy
One Dimension
G''M(<r)m = mv2/r
Meat is Matter. Pan is Light.

A portion of each barred spiral galaxy, outside of its small central bulge, is sharpened in a one dimensional cylinder with a certain diameter. And each barred spiral galaxy itself is rotating around its center of cylinder.
For the one dimensional universal gravity equation; It can be called "New1ton's equation".
F can be transformed into F = G"M(<r)m ; ( if >r then F = constant )
Here G" = the one dimensional universal gravity constant
F = G"M(<r)m → G"M(<r)m = mv2 / r
M(<r) = 2M'r ( where <r ) → v2 = 2G"M'r2 → v = (2G"M')1/2r
Here M' is distributed uniformly in the cylinder.
This means that in a barred spiral galaxy the velocity fields are consistent with rigid body rotaition. Arms are made of overflows from the cylinder ends while the strong emission deprives the mass of the cylinder and makes it shorter in length little by little. A shorter and a little bit lighter cylinder should have a slower rotation speed than overflowed stars. Overflowed stars are going ahead of the cylinder rotaition with the velocity of
v = (2G"M')1/2r , where there is only a weak three and two dimensional gravity potential and where are a lot of interstellar gas and materials which interfere thier movings. Most of them take the same elliptical orbit. While traveling halfway around, they are duffused and slow down, and then a small amount of remainings are absorbed into the another end of the cylinder with a tremendous speed because the gravity is so strong there that I can't see the fallings. Observations reveal that the velocity of rotation of barred spiral galaxy is somehow flat or a little increasing by the distance from the center. It is because the observed velocity is the average of stars both in the cylinder and in arms.

The General Hypothesis No.3 "without a help from your dark friends"
The space between galaxies, intergalactic space, is filled with a tenuous plasma with an average density less than one atom per cubic meter. This plasma is a form of Emission. And galaxies are arranged into sheets and filaments surrounding vast empty voids. Galaxies have "two or one dimensional" gravitational spreads which will work as "surface tension" for forming large scale bubble structures without a help from "dark matter or dark energy".

The Additional Hypothesis No.1
"Newton's apple was not perfectly round."
A black hole has such powerful gravity that it is absorbing any matter that comes too close. Any matter in the near space is crushed to infinite density and is supposed to disappear forever. Even light can not escape the fierce gravitational pull, so they are thought to be remain black and invisible. Schwarzschild calculated how compressed a star would have to be for its gravity to trap light. This is called the Schwarzschild radius which is the radius of the event horizon of a non-rotating black hole.
Extragalactic jets are thought to be from a black hole's accretion disk, not from a black hole itself. In the other case, narrow jets of particles at relativistic speeds spout out up and down, vertically or one direction.
But the gravitational potential is not formed perfectly spherically.
α|Y00|2|Y11|2 will make a gravitational potential figure like a cut end of an apple with two hollows up and down. Then the Schwarzschild radius sphere will have two holes from which Extragalactic or narrow jets come out.

The Additional Hypothesis No.2
The Weight of galaxies
The gravity vectors created within the buldge should be concentrated and then diffused only into the disk of the spiral galaxy. The disk has a thickness which decreases visibly depends on the radius. But invisible thickness should be constant. So the thickness d is set to be the visible thickness at the boundary of the buldge and the disk. If an effective radius of the buldge is ro, then ro = d / 2.
(G Mbuldge m / ro2 ) (ro / r) = m v2 / r
G' = G / ro : The Gravitational Constant for Two dimensional Galaxy
G' M = G Mbuldge / ro = v2
Mbuldge = v2 ro / G = v2 d / (2G)

Let's calculate the Weight of our Galaxy.

M(sun) = Ms = 1.9891x1030 kg
The thickness of our Galaxy = d
d = 15000 light year = 1.5 x 104 x 9.46 x 1015 m = 1.42 x 1020 m
v = 220 x 103 m/s
G = 6.7 x 10-11 m3s-2kg-1
Mbuldge = v2 ro/G = v2 d / (2G)
Mbuldge = (220 x 103)2 x 1.42 x 1020 / (2 x 6.7 x 10-11 )
Mbuldge = 5.1 x 1040 (kg)
Mbuldge / Ms = 2.5 x 1010
The weight of the buldge of our Galaxy is calculated as 25 billions of that of our Sun by this method.

The Additional Hypothesis No.3
"Why is the Sun a near-perfect sphere?"
The oblateness of the Sun is estimated at about 9 millionths which is nearly equal zero, while that of the Earth is about 0.35%. But the Sun is self-rotating at the period of 27 days 6 hours 36 minutes at the equator. Then the centrifugal force there is not zero but 1.8x10-5 of its surface gravity.
(the centrifugal force)/(its surface gravity) = (mRω2)/(GMm / R2) = (R3ω2)/(GM)
(the centrifugal force)/(its surface gravity) = 0.000018 = 0.0018%
G = 6.7x10-11m3s-2kg-1 , M = 1.9891x1030kg , R = 6.96x108m , ω = 2π/T = 2π/(2.357x106)s-1

I think that there is the equivalent gravity concentration around the equator whose increment is almost equal to the centrifugal force at the equator, to keep the Sun near-perfect round. Instead the solar photosphere is about 1 degree hotter near the poles than it is near mid latitudes according to the NOAO Newsletter 52 December 1997, which means for me that the emission of the Sun shifts subtly to the poles and pushes the gravity to the equator. The gravity concentration makes the light away from the equator. Then the equator looks a little bit darker. The more important is that the gravity concentration makes the solar system stable and many planets' orbital motions stay in one plane. Of course I have to mention about the fact that the axis of the Sun's self-rotation tilts 7 degree against the vertical axis of the solar system plane. This makes an unbalance between the centrifugal force and its surface gravity, which will causes the births of sun spots mainly between latitude 5 degrees and 20 in both the northern and the southern hemisphere.

The extreme phenomenon of a rapidly rotating star is "Gravity Darkening". Vega, located 25 light-years away from Earth, makes a full rotation about its axis once every 12.5 hours, while the Sun takes 27 days for one rotation. Vega, by its speedy rotation, bulges 23 percent fatter at its equator.
A traditional explanation says, "The darkening occurs because the gravity darkening star is colder at its equator than at its poles. Its equatorial bulge diminishes the pull of gravity at the equator, which causes the temperature there to decrease." On the contrary, it can be called a mini spiral galaxy where beams of light are pushed away to the upper or lower side of the equator which then looks cooler and darker than at its poles.

The Additional Hypothesis No.4
"Is there no difference of appearance of Mass from a different angle?"
Hypothesis ; M( t ) = ∫R m( φ, θ, t )dφdθ
m( φ, θ, t ) ; an appearance of Mass from a certain direction
Etotal = ∫R ( m c2 + Eactive )dφdθ = constant
Eactive = Eactive( φ, θ, t )
ΔEtotal = m c2 + Eactive
m = ( ΔEtotal − Eactive ) / c2 = ( ΔEtotal − Eactive( φ, θ, t ) ) / c2 = m( φ, θ, t )
Eactive( φ, θ, t ) ) = Energy of Emissions which is unevenly distributed.

The Additional Hypothesis No.5
The Two Dimensional Friedmann Equation modified by Iori Fujita

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Iori Fujita ;
April 11, 2006; April 16, 2006; June 13, 2006; June 27, 2006; Nov. 06, 2006; May 16, 2007; May 08, 2008

Iori Fujita