Dr Weiguang HUANG
Universe is modeling to a 6-dimensional model of f(x-space, y-space, z-space, time, mass, energy) function that is reflected on a plane to releases relationships of four physical elements: time, space, 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 c_{o} and the other is extraordinary ray c_{e}. c_{o} spreads with the same speed in all directions and follows the law of refraction whereas c_{e} 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 6-dimensional rotated model of f(x-space, y-space, z-space, time, mass, energy) function to represent anything in universe, and to release relationships of four physical elements: time, space, mass, and energy. In the 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 not be curved. 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- u^{2}/c^{2}) (3)
t = t¡¯ / Ö(1- u^{2}/c^{2}) (4)
These equations are the basic equations of the Special Theory of Relativity. We know 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.
Dr Cui showed Theory of Analytical Space-Time in one-dimensional space only. Let us extend it to two-dimensional space, i.e. area.
2.2 Plane of f(Time, Two-dimensional Space)
For any shape of an object, by double integration, its area A in (S) is defined as
A = ¨°¨° dx dy (5)
Similarly, the area A¡¯ in (S¡¯) is defined as
A¡¯ = ¨°¨° dx¡¯ dy¡¯ (6)
Substitute eq. (1) into eq. (5), then it becomes
A = ¨°¨° d(x¡¯ cosq ) d(y¡¯ cosq )
= cos^{2}q ¨°¨° dx¡¯ dy¡¯ (7)
Substitution
of eq. (7) with eq. (6) leads to
A = A¡¯ cos^{2}q (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 d^{2}, then k = Öp/2. For a given speed, the value of cos^{2}q 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. This shows that the space should be rotated, instead of curved, otherwise its shape should be changed. If the object speed is light speed, the rotated angle is 90 degree, then its area becomes 0. It means that the object disappears, so we cannot see the object even the object move in front of our eyes. It is so-called black hole.
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¡¯ cos^{2} 45¡ã = 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 c_{e}_{ }< 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 = ¨°¨°¨° dx dy dz (9)
V¡¯ = ¨°¨°¨° dx¡¯ dy¡¯ dz¡¯ (10)
Substitute eq. (1) into eq. (9), then it becomes
V = ¨°¨°¨° d(x¡¯ cosq ) d(y¡¯ cosq ) d(z¡¯ cosq )
= cos^{3}q ¨°¨°¨° dx¡¯ dy¡¯ dz¡¯ (11)
Substitution
of eq. (11) with eq. (10) leads to
V = V¡¯ cos^{3}q (12)
l = k ^{3}ÖV = k ^{3}ÖV¡¯ 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 d^{3},
then k = ^{3}Ö(p/6). For
a given speed, the value of cos^{3}q 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 space-time, but its shape is
unchanged. This shows again that the space should be rotated, instead of
curved, otherwise its shape should be changed. If the object speed is light
speed, the rotated angle is 90 degree, then its volume
becomes 0. It shows again that the object disappears, so we cannot see the
object even the object move in front of our eyes. It is so-called black hole.
2.4 Space-Time Conservation
Square root of eq. (8) leads to
Ö(A/A¡¯)
= cosq
Cubic root of eq. (12) leads to
^{3}Ö(V/V¡¯) = cosq
Combination of these equations with eq. (1) and (2) leads to
l/l¡¯ = Ö(A/A¡¯) = ^{3}Ö(V/V¡¯) = t¡¯/t = cosq (13)
then
l t = l¡¯ t¡¯ = k ÖA t = k ÖA¡¯ t¡¯ = k ^{3}ÖV t = k ^{3}ÖV¡¯ t¡¯ (14)
where l for one-dimensional space, ÖA for two-dimensional space, and ^{3}ÖV 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), ^{3}ÖV t = ^{3}ÖV¡¯ 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¡¯ cos^{2}q (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 right-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- u^{2}/c^{2})
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 ^{3}ÖV m = k ^{3}ÖV¡¯ 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¡¯ / cos^{4}q (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)
If t and t¡¯ in two right-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- u^{2}/c^{2})
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 ^{3}ÖV E = k ^{3}ÖV¡¯ 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.
5.
Three-Dimensional Model of f(Time, Space, Mass)
If the 2-d model of f(time, space) is added by third dimension of mass into three-dimensional coordinate, we can get a model of f(time, space, mass) in three dimensions, where time is x-coordinate, space is y-coordinate, and mass is z-coordinate. 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 tree-dimensional, the model is f(time, volume, mass) or (t, V, m).
When an object is moving, the moving coordinate (S') deflects the observing coordinate (S) in the z-coordinate, the angle q of deflection results from the relative motion, and its sine is proportional to relative speed u. It is called as the tree-dimensional deflected model of f(time, space, mass).
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.
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 = mc^{2}, then the mode of f(time, space, mass) becomes to the model of f(time, space, energy) by replacement of mass with E/c^{2}. Therefore, the model of f(time, space, energy) is similar to the model of f(time, space, mass),
If space is one-dimensional, the model is f(t, l, E), e.g. a car starts to run with acceleration a, then at time t, its speed v is v = at and distant l is l = vt = at^{2} = f(t^{2}), it is a function of t square on a plane of f(time, space); its kinetic energy E = 0.5mv^{2} = 0.5ma^{2} t^{2} = f(t^{2}), 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) = at^{2}. This is the same as l. So it shows that space and energy are symmetrical along time axis.
7.
Three-Dimensional Model of f(Mass, Space, Energy)
If time is replaced with mass in the above model, we can get a model of f(mass, space, energy) in three dimensions, where mass is x-coordinate, space is y-coordinate, and energy is z-coordinate.
If space is tree-dimensional, the model is f(mass, volume, energy) or f(m, V, E), e.g. a baby starts to grow up, then a relation between its body volume V and mass m is V = m/D, where D is density, and its rest state energy E is E = mc^{2}, it is a straight line on a plane of f(mass, energy), and it also is a straight line on a plane of f(mass, volume). So it shows that space and energy are symmetrical along mass axis.
8.
Four-Dimensional Model of f(Time, Space, Mass, Energy)
If the 3-d model of f(time, space, mass) is added by fourth dimension of energy into four-dimensional coordinate, we can get a model of f(time, space, mass, energy) or f(t, s, m, E) in four dimensions, where time is x-coordinate, space is y-coordinate, mass is z-coordinate, and energy is j-coordinate in fourth dimension. Its analytical function is a 4-D function: f(t,s,m,E) = tx + sy + mz + Ej. As the model rotated, it is a 4-D rotated model on Table 1. The four-dimensional model of f(time, space, mass, energy) can be reflected on a flat plane as following Figure 1:
Space
Mass Time
Energy
Figure 1.
The reflected plane of the 4-D model of f(time,
space, mass, energy)
Table 1. The 4-D rotated model of (time, space, mass,
energy)
Dimension |
Coordinate |
Direction |
Dimensional
Rotation |
Time |
x |
east |
t = t'/ cosq |
Space |
y |
north |
l = l' cosq = k ÖA = k ÖA¡¯ cosq = k ^{3}ÖV = k ^{3}ÖV¡¯ cosq |
Mass |
z |
west |
m = m¡¯ / cosq |
Energy |
j |
south |
E = E¡¯ / cosq |
Table 2. Relationship of 4 physical elements in the
model of (time, space, mass, energy)
Plane |
Coordinate |
Position |
Slope |
Conservation |
(time, space) |
(x, y) |
upper right |
v = dl/dt |
l t = l¡¯ t¡¯ = kÖA t = kÖA¡¯ t¡¯ = k ^{3}ÖV t = k ^{3}ÖV¡¯ t¡¯ |
(mass, space) |
(z, y) |
upper left |
1/D = dV/dm |
l m = l¡¯ m¡¯ = kÖA m = kÖA¡¯ m¡¯ = k ^{3}ÖV m = k ^{3}ÖV¡¯ m¡¯ |
(mass, energy) |
(z, j) |
down left |
v^{2} = dE/dm |
E/m = E¡¯/m¡¯ |
(time, energy) |
(x, j) |
down right |
p = dE/dt |
p = E/t = E¡¯/
t¡¯= p¡¯ |
The upper right is the well-known plane of f(time, space), a slope on the plane is a speed v = dl/dt.
Similarly, the upper left is a plane of (mass, space), its slope is 1/density,
1/D = dV/dm. The down left is a plane of f(mass, energy), its slope is v^{2} = dE/dm. The down right
is a plane of (time, energy), its slope is a power p = dE/dt. The plane releases relationships of
four physical elements: time, space, mass, and energy on Table 1 and 2. From previous sections it is shown that space
and energy are symmetrical along mass-time axis.
8.
Six-Dimensional Model of f(x-space, y-space, z-space, Time, Mass, Energy)
If space is 3-dimentional, then above model is expanding to 6-dimensional model of f(x-space, y-space, z-space, time, mass, energy) function. Anything can be represented in this 6-D model. For example, for photon, mass=0; for black hole, mass < 0.
9.
Conclusions
Universe is modeling to a 6-dimensional model of f(x-space, y-space, z-space, time, mass, energy) function, which can be reflected on a plane to releases relationships of four physical elements: time, space, 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 objects appear heavier and compressed, their length, area and volume appear to contraction, and moving clocks appear to run slower in the model. It indicates that space-time should not be curved; otherwise the shape of the object should be changed.
[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),
https://authors.aps.org/eprint/files/2000/Mar/aps2000mar04_005/tast.htm
4/3/2000