Dr Weiguang HUANG
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
z
Im
H
Space
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’ |
|
|
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.
[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