Chapter 2
Setting up of the set of equations of the rotary antigravitation engine
Abstract
It can be inferred from the theory of relativity that a rotating body in a certain situation can bring about the effect of inertial frame dragging. Based on this principle a rotary engine can be designed. It can produce usable and controllable effect of inertial frame dragging, and can be used in the fields like space flight.
2.1
The sine-curve-like geodesic of a body in inertial motion
Suppose a well-balanced body called Body B is in inertial motion along the x-axis. According to the general theory of relativity, since Body B has mass, the space around it is curved.
Because of Lorentz contraction, the space curvatures both in front of and at the back of Body B are larger. According to the general theory of relativity, the geodesic of Body B will be bent: first it will leave the x-axis, go round the large-curvature area, and then get near the x-axis. When Body B moves along this geodesic, a new area where the curvatures are larger forms. So the geodesic will dodge this new area and, when arriving at the x-axis, will go round the large-curvature area via the other side of the x-axis.
Thus the geodesic of Body B looks like a sine curve. Since the energy of Body B is constant, the geodesic should be a sine curve.
Let the sine-curve-like geodesic be
y = A sin ( ωx + φo ) , (1)
where
A is the amplitude, ω
is the angular frequency, and φo is the initial phase.
Then there should be the following relation:
A / c = Δt , ( 2 )
where
c is the speed of light, and Δt
is the smallest range of uncertainty in the time that an observer takes when he or she
receives information from Body B .
Suppose B is a particle of μ . Uncertainty relation gives
E Δt = h / (4 π) , ( 3 )
where
E is the energy of the particle of μ
, and h is Planck's constant.
From Eqs. (2) and (3), the amplitude can be described by
A = c h / ( 4 π E) . ( 4 )
Substitute
E=mc2 in Eq. (4), and
Eq. (4) becomes
A = h / (4 π m c) . ( 5 )
From
the point of view of relativity, the particle of μ
observes the same physical law of space-time as a body made up of any other
materials. Hence Eq. (5) holds true for any Body B .
Let
the wavelength of the sine curve described by Eq. (1) be l
. The experiment on the diffraction of electrons demonstrates that
l
=
h
/ (m v)
.
( 6 )
From a mathematical point of view, the wavelength is
l = 2 π / ω . ( 7 )
From Eqs. (6) and (7), the following can be obtained:
h / (m v) = 2 π / ω ,
i.e.
ω = 2 π m v / h . ( 8 )
Substitution of Eqs. (5) and (8) in Eq.(1) yields
y = [ h / ( 4 π m c ) ] s i n [( 2 π m v / h ) x + φ o ] . ( 9 )
Eq. (9) is the geodesic of a well-balanced body in inertial motion.
Let the average value of the y-component velocity of Body B, when this component velocity is larger than zero, be Vy
. Let the average value of the y-component velocity of Body
B, when
φ
o = 0 , 0 ≤
x
Then there is
Vy = V1 .
Write v in Eq. (9) as vx .
It can be known from Eqs. (9) and (6) that when
φ
o = 0 , x = ( 1
/
4 ) l
, x is
x = h / (4 m vx ) ,
and
y is
y = [ h / (4 π m c) ] sin ( 2 π / 4 ) ,
y = h / (4 π m c) .
Hence it follows that
y / x = vx / (π c) ,
V1 / vx = vx / (π c) ,
V1 = vx2 / (π c) . ( 10 )
Since Vy = V1 , Vy is
Vy = vx2 / (π c) . ( 11 )
2.2
Postulates, definitions and corollaries
2.2.1 The interaction between the moving body and its own curved space-time, which is stated above, shows that the curved space is not empty. Hence Postulate 1 is obtained:
Postulate 1. The carrier of the space-time point is a kind of matter.
Definition 1. The matter stated in Postulate 1 is called the gravitational field matter, or simply, the gfm.
2.2.2
Postulate 2. The distribution of the gravitational field matter is described by the curvature
of space-time.
It can be known from the first paragraph of Section 2.1 that the distribution of the gfm of a body in inertial motion is always changing. A body and its gfm interact with each other and are relatively independent of each other. Hence Corollary 1 is obtained:
Corollary
1. Gravitational field matter is fluid.
Definition 2. The moving gravitational field matter is called the antigravitational field.
The motion of gfm drags spacetime. Hence Corollary 2 is obtained:
Corollary 2. The gfm in motion has the effect of inertial frame dragging. Under certain conditions, this effect makes the inertial frame move with the gfm.
Definition 3. The effect of inertial frame dragging of the moving gravitational field matter is called antigravitation.
Motion and gravitation are both inseparable attributes of matter. Hence Corollary 3 is obtained:
Corollary 3. Matter has antigravitational field.
2.2.3 A body moving along a sine curve causes its own gfm to move with it, forming the gfm current. The gfm current causes the gfm waves in the local gfm of the universe. The gfm current moves along its own geodesic and gfm waves have the motion of interference, diffraction, refraction, transmission, reflection and so on. The gfm current has the wave-particle duality.
A body can be dragged by the gfm waves in their interference, diffraction, refraction, transmission, reflection and so on. Hence Postulate 3 is obtained:
Postulate 3. The gfm waves are the material base of the de Broglie waves.
2.3 Rotary antigravitation engine and its set of equations
According to the above principles, a rotary antigravitation engine can be designed. It is made up of the rotation device, the sealing device, the disturbance device and the carrying device (see Chapter 1 and Chapter 3 ). The sealing device and the carrying device should be smooth, balanced, and simple-shaped, so that their disturbance to the gfm current can be reduced.
When the rotating body of the engine rotates, at every moment its gfm runs backward and forward along the sine curve in a plane parallel to the y-axis. The disturbance device disturbs and scatters the gfm whose y-component velocity directs towards the back, while the gfm whose y-component velocity directs towards the front drags the rotating body and carries it forward.
On the one hand, since Lorentz contraction is a kinematical effect and is not affected by forces, there is only superposition of antigravitation and other kinematical effects. There is no superposition of antigravitation and force, neither is there superposition of antigravitation and gravitation. On the other hand, if a path stretches between two rows of ponds, then the path will not run zigzag but will go straight ahead. Similarly, if the rotating body receives too large a force in the front or back direction, the antigravitation in the front or back direction of the antigravitation engine will not appear. So antigravitation has the nature of "whole or none".
Antigravitation is not force, but it brings about acceleration. Therefore, through
acceleration the antigravitation can be compared with a force.
Let the shaft of the
rotating body is the y-axis. When
the equivalent particle
of mass, at Point A (see Chapter 1), of the
rotating body rotates for a distance equal to the wavelength l
, Vy increases once, just as the speed of a bicycle
increases once each time a boy pedals the bicycle, with only his right foot, along a smooth
and slightly
downward road.
Δvy = Vy (Δx / l) . (12)
When Eq. (6) is substituted in Eq. (12), one obtains
Δvy = Vy Δx m vx / h .
From
Eq. (11),
Δvy = [ vx2 / (π c) ] (Δx m vx / h) .
Since Δx = vx Δt , the equation for Δvy can be rewritten as follows:
Δvy = [ vx2 / (π c) ] (vx Δt m vx / h) ,
i.e.
Δvy = Δt m vx4 / (π c h) , ( 13 )
and Eq. (13) can be rewritten as
Δvy / Δt = m vx4 / (π c h) ,
in other words,
a = m vx4 / (π c h) . ( 14 )
Since there is
vx = 2 π r / T (15)
and the antigravitational acceleration has the nature of "whole or none", from Eqs. (14) and (15) the following can be derived:
a = 16 π3 m r4 / (c h T4 ) ,
when | 16 π3 m r4 / (c h T4 ) | > | ∑a' | ；
a = 0 ,
when | 16 π3 m r4 / (c h T4 ) | ≤ | ∑a' | ； (16)
where m is the mass of the particle or the mass of the ball of particles which moves freely for a long distance and in good order in the rotating body. This is because in the rotating body the antigravitational acceleration is produced by the particles which move freely for a long distance and in good order.
The
Set of Equations (16) can be called the set of equations of the rotary antigravitation
engine, or simply, the set of equations of the antigravitation engine.
When the rotating body is made of the ordinary metal, m in Eqs. (16) is me , which is the mass of the electron, r is the radius of gyration of the metal part of the rotating body, T is the period of the rotation of the end point (hereafter called Point A) of r , a is the antigravitational acceleration which points to the front and which is produced by the electrons of the metal part of the rotating body, and a is also the antigravitational acceleration of both the gfm waves, and the de Broglie waves, of the gfm current of the metal part of the rotating body, |Sa'| is the absolute value of the resultant acceleration which is along the front direction of the rotating body and which is obtained by the electrons at point A and which is other than antigravitational acceleration, π is pi, c is the speed of light, and h is Planck's constant.
Because of the macroscopic quantum effect, when being controlled by the gfm current, the boat is in uncertain spacetime, and hence it moves now fast, now slow, now forward, now backward, and sometimes it stops for a while.
Antigravitation has the macroscopic quantum phenomenon.
Chapter 1 An introduction to some antigravitation engine experiments that everyone can make
Chapter 2 The setting up of the set of equations of the antigravitation engine
Chapter 3 Know-how of the antigravitational mechanical experiment and range of application
Chapter
4 Data analysis (to verify the macroscopic quantum mechanical phenomenon)
Chapter 5 Data analysis (mainly to verify Eq. (1) in Chapter 1)
Chapter 6 A new state of matter: foggoid state