# 3D Complex Model of Universe

Dr Weiguang HUANG

## Abstract

Universe is modeling to a 5-dimensional model of f(x-space, y-space, z-space, time, mass) function. For one dimensional space, it is reduced to the 3D complex model, and the model showed the algebraic meaning, geometric meaning and physical meaning. It can be reflected on a plane to release relationships of four physical elements: time, space, mass, and energy, and show dark mass and energy. They are symmetry and conservation: space and energy are symmetrical along mass-time axis, and the space-time conservation, space-mass conservation and space-energy conservation, regardless of the object speed and space dimension.

1. Introduction

The Special Theory of Relativity (STR) first pointed out that observers of any two different coordinates who described "an event" such as time and space would get different results, which represented a breakthrough of the knowledge of space-time in human history. It has opened out the relation among space, time and motion. The Relativity Theory shows that space-time is curve.

Zhang Junhao and Chen Xiang [1-6] argued that space-time is not curve by their gravitational theory on the flat space-time (Minkovsky’s space-time) as the special relativistic gravitational theory.

Dr. Cui Silong [7] proposed the Theory of Analytical Space-Time with two hypotheses as principles: (I) the area of space-time is invariant (Principle of a string), and (II) any two coordinates with relative speed would deflect each other. The theory completes Lorentz transformation with a factor of two-dimensional or multi-dimensional rotation, obtains a new expression of astro-object precession angular speed, gives two forecasts, demonstrates Schrödinger equation with space-time rotation and concludes a space-time wave panorama for Newtonian space, Relativistic space, quantum space and black-holes. The theory will unify the foundations of Special Relativity, General Relativity and Quantum Mechanics. He gave two forecasts: (1) 0.71c space-time light cone vertex, and (2) Deflection of space-time results in double refraction of light. From his first forecast, he concludes that “anything that has relative speed 0.71c to us is invisible even though it moves in front of our eyes. The object can appear again from its rear side when relative speed u > 0.71c. This forms a phenomenon of light cone whose vertex point is 0.71c “. From his second forecast, he concluded that light from a moving system would produce a phenomenon of double refraction. Light will split into two rays: one is ordinary ray co and the other is extraordinary ray ce. co spreads with the same speed in all directions and follows the law of refraction whereas ce goes with a speed that is changeable in different directions and varies on the relative speed of a moving system and does not follow the law of refraction. Unfortunately, both forecasts are wrong because there are mathematical errors in his mathematical deduction.

In this paper, we will set up a 5-dimensional rotated model of f(x-space, y-space, z-space, time, mass) function to represent universe. For one dimensional space, it is reduced to the 3D complex model. It releases relationships of four physical elements: time, space, mass, and energy, and shows dark mass and energy. In this model, it will prove that space-time conservation, space-mass conservation and space-energy conservation, regardless of the object speed and space dimension, and show that space-time should be rotated. We will point out Dr. Cui’s mathematical errors in his mathematical deduction.

2. Plane of f(Time, Space)

2.1 Plane of f(Time, One-dimensional Space)

Let us introduce the Theory of Analytical Space-Time [7].

Definition: Given two right-angled coordinates (S') and (S), (S') is the moving coordinate and (S) is the observing coordinate. l' and t' in (S') indicate length and time upon the condition that (S') is in a stationary state relative to (S). If there is a relative motion between (S') and (S), we, being in (S), measure l ' and t' in (S'). The result of measurement is l and t, so l and t are all measured data [7].

Two hypotheses for the Space-Time theory [7]:

(I) Principle of invariant space-time area (Principle of a string)

Product of length l' and time t' in (S') and product of l and t in (S) are called space-time area S' and S respectively. The space-time area is invariant whether there is a relative motion between (S') and (S) or not. For any (l', t'), it must meet the equation:  l' t' = l t.

(II) Principle of space-time deflection

If a moving coordinate (S') leaves or approaches the observing coordinate (S) with speed u (or u'), (S') deflects (S) from the direction of u (or u'), and the angle q of deflection results from the relative motion and its sine is proportional to relative speed u.

Therefore

sinq = u/c or sinq = u'/c'

where c is speed of light.

Formula (1-1), (1-2), (1-3) and (1-4) in ref. [7] are as follows:

l = l' cosq                                                      (1)

t = t'/ cosq                                                     (2)

l = l’ Ö (1- u2/c2)                                           (3)

t = t’ / Ö(1- u2/c2)                                          (4)

These equations (1)-(4) are the basic equations of the Special Theory of Relativity. They showed that there is a definite meaning of the contraction factor: the deflection factor of space-time. It is the rotation of space-time that causes the contraction of a moving ruler and the delay of a moving clock.

2.2. Plane of f(Time, Two-dimensional Space)

Dr Cui showed Theory of Analytical Space-Time in one-dimensional space only. Let us extend it to two-dimensional space, i.e. area.

For any shape of an object, by double integration, its area A in (S) is defined as

A = òò dx dy                                                     (5)

Similarly, the area A’ in (S’) is defined as

A’ = òò dx’ dy’                                     (6)

Substitute eq. (1) into eq. (5), then it becomes

A = òò d(x’ cosq ) d(y’ cosq )

= cos2q   òò dx’ dy’                                      (7)

Substitution of eq. (7) with eq. (6) leads to

A = A’ cos2q                                                   (8)

l = k ÖA = k ÖA’ cosq

where k is a constant for a given shape of an object. If the shape of an object is square, its area A is l 2, then k = 1. If its shape is circle, its area is p/4 d2, then k = Öp/2. For a given speed, the value of cos2q  is a constant. This is relation between the area A in the observing coordinate (S) and the area A’ in the moving coordinate (S’). It proves that the area of the moving object appears smaller due to rotation of space-time, but its shape is unchanged although its size becomes small. This shows that its space should be rotated, instead of curved, otherwise its shape should be changed. As we known, when the object goes away far, it becomes smaller, but its shape does not change.

By the way, we point out Dr Cui’s mathematical errors in his mathematical deduction of two forecasts [7]:

1. In his first forecast (1.4.1) 0.71c space-time light cone vertex, he predicts that “anything that has relative speed 0.71c to us is invisible even though it moves in front of our eyes.” It is wrong. Because he set x' = y' = c't', which means that the object speed is light speed, as x' = y' = c't' and x = ut leads to u = c, so it should be q = 90 degree instead of q = 45 degree. If q = 45 degree for speed 0.71c, then A = A’ cos2 45° = A’/2, its area reduces half, instead of disappear.

2. In his second forecast (1.4.2) Deflection of space-time results in double refraction of light, he set c = c’. It means that all light speeds are the same. It is obvious conflict with his conclusion of ce < c in his formula (1-32).

2.3. Plane of f(Time, Three-dimensional Space)

Let us expand the above model to three-dimensional space, i.e. volume.

For any shape of an object, by triple integration, its volume V in (S) and V’ in (S’) are defined as

V = òòò dx dy dz                                               (9)

V’ = òòò dx’ dy’ dz’                                          (10)

Substitute eq. (1) into eq. (9), then it becomes

V = òòò d(x’ cosq ) d(y’ cosq ) d(z’ cosq )

=  cos3q  òòò dx’ dy’ dz’                               (11)

Substitution of eq. (11) with eq. (10) leads to

V = V’ cos3q                                                   (12)

l = k 3ÖV = k 3ÖV’ cosq

where k is a constant for a given shape of an object. If the shape of an object is cubic, its volume V is l 3, then k = 1. If its shape is sphere, its volume is p/6 d3, then k = 3Ö(p/6). For a given speed, the value of cos3q  is a constant. This is relation between the volume V in the observing coordinate (S) and the volume V in the moving coordinate (S). It proves that the volume of the moving object appears smaller due to deflection of its space-time, but its shape is unchanged. This shows again that its space should be rotated, instead of curved, otherwise its shape should be changed although its size becomes small.

2.4 Space-Time Conservation

Square root of eq. (8) leads to

Ö(A/A’) = cosq

Cubic root of eq. (12) leads to

3Ö(V/V’) = cosq

Combination of these equations with eq. (1) and (2) leads to

l/l’ = Ö(A/A’) = 3Ö(V/V’) = t’/t = cosq                                           (13)

then

l t = l’ t’ = k ÖA t = k ÖA’ t’ = k 3ÖV t = k 3ÖV’ t’                  (14)

where l for one-dimensional space, ÖA for two-dimensional space, and 3ÖV for three-dimensional space.

It shows that products of length, square root of area, or cubic root of volume with time are the same, regardless to the object speed and space dimension. We call it as space-time conservation.

If the shape of an object is square, its area A is l 2, then from eq. (14), ÖA t = ÖA’ t’ becomes to lt = l’t. Therefore, one-dimensional space is a special case of two-dimensional space. If the shape is cubic, its volume V is l 3, then from eq. (14), 3ÖV t = 3ÖV’ t’ becomes to lt = l’t. Therefore, one-dimensional space also is a special case of three-dimensional space. These show again that the shape of object is unchanged although its size, area and volume are reduced.

We can separate eq. (14) into

l t = l’ t’                                                          (15)

Product of eq. (2) and (8) leads to

At = A’t cosq                                                  (16)

Product of eq. (2) and (12) leads to

Vt = V’t cos2q                                                            (17)

When u = Ö3/2 c = 0.866c or q = 60 degree, then t = 2t’, l = l’/2, A = A’/4, and V = V’/8. We call this speed as the speed of double time, the speed of half length, the speed of quarter area, and the speed of one-eighth volume.

3. Plane of f(Mass, Space)

If t and t’ in two angled coordinates (S) and (S’) are replaced with mass m and m’, then a plane of f(time, space) becomes a plane of f(mass, space), so eq. (2) and (4) become:

m = m’ / cosq                                                 (18)

= m’ / Ö(1- u2/c2)

It proves that the moving mass appear heavier due to deflection of space-mass.  This is well-known mass equation in the relativity theory.

Multination of it by eq. (1) leads to

ml = m’ l’                                                        (19)

For 2-d and 3-d space, similarly, we get relations similar to eq. (14):

l m = l’ m’ = k ÖA m = k ÖA’ m’ = k 3ÖV m = k 3ÖV’ m’                    (20)

It shows that products of length, square root of area, or cubic root of volume with mass are the same, regardless to the object speed and space dimension. We call it as space-mass conservation.

The densities in (S) and (S’) are defined as

D = m/V                                                          (21)

D’ = m’/V’                                                      (22)

Combination of eq. (12), (20), and (22) into eq. (21) leads to

D = D’ / cos4q                                                             (23)

It proves that a moving object appears to compression due to deflection of space-mass. This is well known in the relativity theory.

When u = Ö3/2 c = 0.866c or q = 60 degree, then m = 2m’ and D = 16D’. We call this speed as the speed of double mass, and the speed of 16x density.

4. Plane of f(Energy, Space)

As relationship between energy and mass by Einstein: E=m cc, if mass is replaced with energy in above formula to get Plane of f(Energy, Space).

If t and t’ in two angled coordinates (S) and (S’) are replaced with energy E and E’, then a plane of f(time, space) becomes a plane of f(energy, space), so eq. (2) and (4) become:

E = E’ / cosq                                                   (24)

= E’ / Ö(1- u2/c2)

It proves that the moving object appear more energy due to deflection of energy-mass.  This is well-known energy equation in the relativity theory.

Multination of eq. (24) by eq. (1) leads to

El = E’ l’                                                         (25)

For 2-d and 3-d space, similarly, we get relations similar to eq. (14):

l E = l’ E’ = k ÖA E = k ÖA’ E’ = k 3ÖV E = k 3ÖV’ E’                       (26)

It shows that products of length, square root of area, or cubic root of volume with energy are the same, regardless to the object speed and space dimension. We call it as space-energy conservation.

In order to simply discussion, from now on we are talking space in one dimension until we specially point out.

5. Three-Dimensional Model of f(Time, Space, Mass)

As we known, real number is one dimension, it can be expanded into two dimensions by complex number. On the same way, complex number (x+y i) is two dimensions, it can be expand into three dimensions by a hidden number h for f(x,y,z)=(x+y i+z h). Its analytical function of 3D complex number is f(x,y,z)=x 1 +y i + z h, where the x dimension is real number, the y dimension is imaginary number i, and the z dimension is hidden number h. If x=y=z=1, then f(x,y,z)=1+i+h, it is three dimensions of complex number. If x=y=1 and z=0, then it is reduced to two dimensions x+y i.

5.1 Algebraic  Meaning

For example, an equation x^2+1=0, its solutions are x=i and –i. On the same way, for an equation |x|+1=0, its solution are x=h and –h.

5.2 Geometric Meaning

If the 2-d model of complex number is added by third dimension of hidden number as third dimensional coordinate, the model becomes to three dimensions of complex function f(x,y,z)=f(Re,Im,H). For y(x)=|x|+1, if a curve of |x|+1 is turn upside down, then rotated by 90 degree with the y-axis, the curve had two crosses with the z-axis, which crosses are  solutions of h and –h.

5.3 Physical Meaning

If the 2-d model of f(time, space) is added by third dimension of mass as third dimensional coordinate, the model becomes to three dimensions of f(time, space, mass), where time is x-coordinate, space is y-coordinate, and mass is z-coordinate. It is called as the three-dimensional complex model of f(time, space, mass). The three-dimensional model of f(time, space, mass) can be shown as following Figure 1:

y

x

z

Im

Re

H

Space

Time

Mass

Figure 1.  The 3D model of f(time, space, mass)

As we known, there are 95% hidden mass in universe, where h<0 is for hidden or dark mass, and for photon, mass h=0, so photon is two dimension.

When an object is moving, the moving coordinate (S') deflects the observing coordinate (S) in the coordinate, the angle q of deflection results from the relative motion, and its cosine is proportional to relative speed u in eq. (1). So the model rotated, it is a 3-D rotated model on Table 1.

Its analytical function is a 3D function: f(x,y,z) = f(t,s,m) = t1 + si + mh.

Table 1. The 3D model of f(time, space, mass)

 Dimension Coordinate Direction Dimensional Rotation Time x east t = t'/ cosq Space y north l = l' cosq = k ÖA = k ÖA’ cosq  = k 3ÖV = k 3ÖV’ cosq Mass z West-south m = m’ / cosq Energy z West-south E = E’ / cosq

Table 2. Relationship of 4 physical elements in the 3D model of f(time, space, mass)

 Plane Coordinate Position Slope Conservation (time, space) f(x, y) (Upper, right) v = dl/dt l t = l’ t’ = kÖA t = kÖA’ t’ = k 3ÖV t = k 3ÖV’ t’ (mass, space) f(z, y) (Upper, left) 1/D = dV/dm l m = l’ m’ = kÖA m = kÖA’ m’ = k 3ÖV m = k 3ÖV’ m’ (energy,space) f(z, y) (Upper, left) v2 = dE/dm E/m = E’/m’ (time, energy) f(x, z) (Down, right) p = dE/dt p = E/t = E’/ t’= p’

The model proves that the contraction of a moving ruler, the delay of a moving clock, and heaver of moving mass, because the moving coordinate (S') deflects the observing coordinate (S), instead of space-time to curve. Not only do we realize that space, time and mass have been changed, but also the model of f(time, space, mass) actually deflects them all.

The planes release relationships of four physical elements: time, space, mass, and energy on Table 1 and 2. The x-y plane is the well-known plane of f(time, space), a slope on the plane is a speed v = dl/dt. Similarly, the z-y plane is a plane of f(mass, space), its slope is 1/density, 1/D = dV/dm;  and is also a plane of f(energy, space), its slope is v2 = dE/dm. The x-z plane is a plane of f(time, energy), its slope is a power p = dE/dt. From previous sections it is shown that space and energy are symmetrical along mass-time axis.

6. Three-Dimensional Model of f(Time, Space, Energy)

If mass is replaced with energy in the above model, we can get a model of f(time, space, energy) in three dimensions, where time is x-coordinate, space is y-coordinate, and energy is z-coordinate. Because by principle of mass-energy equivalence, a relationship of energy E and mass is E = mc2, then the mode of f(time, space, mass) becomes to the model of f(time, space, energy) by replacement of mass with E/c2. Therefore, the model of f(time, space, energy) is similar to the model of f(time, space, mass), where h<0 is for hidden or dark energy.

If space is one-dimensional, the model is f(t, l, E), e.g. a star starts to run with acceleration a, then at time t, its speed v is v = at and distant l is l = vt = at2 = f(t2), it is a function of t square on a plane of f(time, space); its kinetic energy E = 0.5mv2 = 0.5ma2 t2 = f(t2),  it also is a function of t square on a plane of f(time, energy). If we scale E with 0.5ma, then E/(0.5ma) = at2. This is the same as l. So it shows that space and energy are symmetrical along time axis.

7. Five-Dimensional Model of f(x-space, y-space, z-space, Time, Mass)

If space is one-dimensional, the model is f(time, length, mass) or f(t, l, m); if space is two-dimensional, the model is f(time, area, mass) or f(t, A, m); if space is three-dimensional, the model is f(time, volume, mass) or (t, V, m). If space is 3-dimentional, then above model is expanding to 5-dimensional model of f(x-space, y-space, z-space, time, mass) function. Universe can be represented in this 5-D model. For example, for photon, mass h=0; and for hidden or dark mass, mass h< 0.

8. Conclusions

Universe is modeling to a 5-dimensional model of f(x-space, y-space, z-space, time, mass) function. For one dimensional space, it is reduced to the 3D complex model, and the model showed the algebraic meaning, geometric meaning and physical meaning. It can be reflected on a plane to release relationships of four physical elements: time, space, mass, and energy, and show dark mass and energy. They are symmetry and conservation: space and energy are symmetrical along mass-time axis, and the space-time conservation, space-mass conservation and space-energy conservation, regardless of the object speed and space dimension. It proves that moving object appears heavier and compressed, its length, area and volume appear to contraction, and moving clock appears to run slower in the model. It indicates that its space-time should be rotated, instead of curved; otherwise its shape should be changed although its size become small.

## 9. References

[1] Zhang Junhao and Chen Xiang, International Journal of Theoretical Physics, 29, 579, (1990).

[2] Zhang Junhao and Chen Xiang, International Journal of Theoretical Physics, 29, 599, (1990).

[3] Zhang Junhao and Chen Xiang, International Journal of Theoretical Physics, 30, 1091, (1991).

[4] Zhang Junhao and Chen Xiang, International Journal of Theoretical Physics, 32, 609, (1993).

[5] Zhang Junhao and Chen Xiang, International Journal of Theoretical Physics, 34, 429, (1995).

[6] Zhang Junhao, Physics Essays, 10, 1, (1997).

[7] Cui Silong, Theory of Analytical Space-Time, APS, 5, Mar 04, (2000),

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