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 - 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 r ≠ 0 will vanish .

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.

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 Shape	Major Gravity	+	Major Emission
elliptical galaxy e.g. NGC4881 Three Dimension GM(<r)m/r2 = mv2/r gravitationally unstable			|Y00|2+|Y00|2
spiral galaxy e.g. NGC4414 Two Dimension G'M(<r)m/r = mv2/r			|Y11|2+|Y10|2
barred spiral galaxy e.g.NGC1300 One Dimension G''M(<r)m = mv2/r			|Y10|2+|Y11|2

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 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".

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+β|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 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 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.

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

then
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 Two Dimensional Friedmann Equation modified by Iori Fujita

A new interpretation is needed for the Bullet Cluster.

It will show the unnecessity of Dark Matter.

Before: (MAgalaxy + MAgas + MAdark) : (MBgalaxy + MBgas + MBdark) ~ 10 : 1 MAgalaxy : MAgas : MAdark ~ MBgalaxy : MBgas : MBdark ~ 1 : 5 : 30 MAgalaxy : (MAgas + MAdark) ~ MBgalaxy : (MBgas + MBdark )~ 1 : 35 Gas works against galaxies gravitationally like dark matter. ↓ ↓ ↓ ↓ ↓ ↓ ↓ After: MAgalaxy : MAdark ~ MBgalaxy : MBdark ~ 1 : 30 Then there is a smaller gravitational force (31/36 ; -14%) to keep galaxies as they were. So, galaxies should have been flying away.

These two galaxy clusters A & B collided recently (150 million years ago). B is the Bullet Cluster. It is said that this Bullet Cluster provides the best current evidence for the nature of dark matter. One important fact is that almost all galaxies and stars keep staying inside of their own original galaxy-clusters. Only their gas lumps collided and heated up, and were emitting X-ray, but were stripped off between two galaxy clusters. Then, some questions arise. Q1: Time has already passed 150 million years. How these two gas lumps can keep their shapes without the gravity of their original galaxy clusters? Q2: Why didn't the gas of these two gas lumps spread away because of their heated-up high pressure? Of course, these two gas lumps have their own masses. But they are not enough to keep them stay as they were. Q3: The mass of gas is about one-sixth of its whole galaxy mass. One-sixth is still big. Even though, why are there no indications of masses of the gas through the investigation of the gravitational lensing? Q4: Why is not the loss or gain kinetic energy or momentum of each gas lump equivalent? MBgas and 'db' are smaller than MAgas and 'da'. 'da' must be much smaller, or 'db' should be bigger. A1 & A2 : There should be no black hole in the center of gas lumps. So, there should be made a self gravitational constraint by emiting high energy X-lay. And X-lay like light surrounding the gas lump has the repulsive force would compress the gas lump. A3: The repulsive force of X-lay would cancel out the gravitational force by the mass of gas. A4: The kinetic energy Agas had obtained from Bgas was EA ~ ~MAgas*(da/T)2/2. The kinetic energy Bgas had lost was EB ~ ~MBgas*(db/T)2/2. EA~~EB, then db = square(MAgas/MBgas)*da = square(10/1)*da ~~ 3*da db should have been bigger than 3*da. According to this image, db looks smaller than da. It is still strange. The galaxy cluster B must have been slowed down to lose its kinetic energy. Then there had been no dark matter in the galaxy cluster B.

Iori Fujita