A quick look the context suggests many topics, which do not seem to be related. It looks a bid chaotic and any systematic approach seems lacking. It will be remedied here.
The main topic of this book is the precession shift of the polar axis for Earth. Although the sun wheel drive is equally important and also related to the precession shift. The third important subject is an oversight in the proof of mass energies of 22% dark matter and 78% dark energy together with baron matter for our cosmos. It is the external balance of energy leading to the proof that the gravitons out side the homogeneous 3D-space of quantum gravity have phase velocities of twice the speed of light.
The problem with the shift of the polar axis was that the author had not a clue how to relate the shift in terms of physics to the definition of the Einstein gravitons. Some feel had to be developed. So there were preliminaries.
Firstly it was necessary to prove that the precession angle of 60° stayed constant, if applied to Newton’s law of physics. See sections:
Introduction about Einstein gravitons
The many dimensional forms for Newton’s law of gravity
The physics of the Einstein gravitons for Newton’s law of gravity
Secondly, nothing was understood about the mechanism of precession for the gravitons. It all worked out fine in energy balances, but what were the implications? This lead to the discussions and derived expressions in the sections:
The all-around rotation of empty space
The storage of the flywheel energy for the universe
The sense of rotation of empty space
Notes about conjugated matter and gravity
Thirdly, and for most part it appeared to be a blind alley, that were the long wave quantum properties of the gravitons being the cause of their extremely low energies. Obviously it was thought that, like the behavior of the electron around the proton, the orbital angular momentum of the graviton could be coupled to the self-rotation of Earth, also expressed in gravitons. These are the sections:
A nonlinear harmonic string theory for Einstein gravitons.
A quantum string theory of Einstein gravitons in a conserved gravity field.
In conclusion the above three points were leading up before the two main topics could be tackled and these were properly defined in terms of physics, which is chapter 2. The last topic is the proof of the other universe by the external energy balance. It is mainly chapter 3
Between thirty and forty years ago Denaerde mentioned in one of our discussions that the Apocalypse will be initiated by mankind by some kind of mind over matter collectively but not by individuals themselves, generating waves of paralysing or catatonic fear. The already long overdue latent tilt or shift of the Earth rotation axis sets on causing havoc, firstly internally in the magma core and layers, upsetting the Earth crust and destroying all life on Earth, mostly at the dry lands and causing ultimate tsunamis.
Since no one knew how the Apocalypse , the final end of Judging day, may be caused, it is an interesting hypothesis and for people with a scientific background it may explain perfectly how life can be extinguished. On the other hand a shift of the polar axis of Earth cannot be according to Newton’s law of physics, which of course is confirmed later by GRT, a far more fundamental theory. But the magic word was the latent shift could be caused by the gravity of the sun. Some unknown phenomena in the quantum gravity of the sun could have in store the explanation for the paradox in Newton’s laws.
However the usual theories of quantum gravity are extremely complex and difficult to grasp and only a few know how it works. Further not any scientific literature mentioned that quantum gravity could cause such a disaster. In that respect it was clear that only a new and different theory could explain the mechanism of quantum gravity causing some additional tilt of the polar axis and in general for any macroscopic object orbiting a star.
The question may arise why was this topic never published before? The author cannot speak for Denaerde. But Denaerde knew as every one else with common sense that such a speculation would be swept aside if there is no clear foundation for such a statement. Similarly it is the reason why the author publishes his findings now a new theory gives a possible explanation.
The outcome of the first crude estimates determine a shift for the polar axis of Earth for every 50 or 70 million years, but also 10 or 20 million years are found according to the geological time scale of Earth. No end or stopping is possible or available for this mechanism. It is inherent due to the drive of gravitons generating also the spin of the sun due to fusion. Thus one should not too much worry about this latent mechanism. May be, was it not that Denaerde seemed to be convinced that the melting of the ice caps and flattening of the Earth sphere, also played a role. In the optics of Denaerde, the time scale was much more frequent. And for what it is worth the statement was that the predestination of the human race was attuned to the potential of the shift. To make things perfectly clear, the effect of flattening etc., are not accounted for in study here. In our calculations the precession shifts are based on the inertia momentum of a perfect sphere for Earth. The possible influence of above mentioned crucial details is left to a more eager young generation of scientists to unravel this mystery.
Another aspect never mentioned explicitly but now very clear is that all intelligent races in our cosmos are subjected to deal during their planetary bound in the evolution of their race, with an angle of the rotation axis of their planet with the plane of the eclipse. They should find away to cope with it. It is a universal feature valid everywhere for all planets round a star with or without carrying life.
INTRODUCTION ABOUT EINSTEIN GRAVITONS
As a start for the book, Physics 3, we begin with a summary of the properties of the Einstein gravitons. It seems with the successes so far booked, a good understanding as a start is needed.
Einstein gravitons are a means of assistance to determine the energy or momentum of empty space. The gravitons are defined according to Einstein’s principle of equivalence. A local force of gravity, g, at a distance of a mass is made equal to a constant of acceleration, a, which is expressed in an energy, labour, equating the product of a force times the acting length, s, of the force. The length s can be put in relation with the radius R from the centre of mass. Then is valid:
R a = ½ v² Now a equates with: g = G M / R²
The first application for the Einstein gravitons is, if a is an outward directed acceleration and it is equated to the g-force, then there is only one radius where g and a are equal. This acceleration a is the acceleration constant of the empty space. It turns out that such a constant is specific for one kind of atom, often here the H-atom. The H-atom is the fundamental building block of the universe at the onset and its mediating mass as a representation of its graviton energy of empty space can be calculated explicitly. The vacuum constant of acceleration has a value of about 1.63 10exp(-10) m/sec².
Is the graviton of empty space accelerated over a long distance till the c velocity then the maximum energy of the graviton can be expressed by applying the uncertainty relation:
s = h / (mH c) = RH , the Hubble distance for the universe.
So the length of s is quantified in a probability wave. If s is a length of labour for less energy of the graviton then it is supposed to be a groups velocity, which should be composed from the gravitons of c-velocity. Consequently there is a compression of energy in empty space. The energy density of empty space increases with lower velocities than c. For a velocity v more gravitons with an energy (mass) of mH contribute to group graviton.
Since this was the first property of the Einstein graviton, the second property, by which the graviton is not of one dimension but two dimensional, is the precession. Further one has to realise that the graviton because of its acceleration is also a vector and therefore the components for the modulus are added accordingly. The component of the precession is always tangential and has a right angle with the component of linear acceleration as long as the velocities are smaller than c. The cross product of the vector components is perpendicular to the plane for the precession and the linear acceleration. This plane of these two components is called the plane of precession or inertia plane. In such a manner the graviton is a three dimensional object. Apparently this is the genius of the idea for the Einstein gravitons. By principle the 3D-world of Newton can be translated in another 3D- situation. How far it can go, is not clear yet, but considering the fact that dark matter and the dark energy in our cosmos making up 96% of the overall mass of the universe, are respectively composed of gravitons in different states of aggregation, it is an enormous leap forwards. The proof that 22% of the dark matter are polarized gravitons in equilibrium with 4% barons and 78% (74+4)% are gravitons for the space expansion in our cosmos, the dark energy. (Polarised gravitons, because they are three-dimensional)
Now why precession? A Hydrogen atom, neutral, under goes always an acceleration by the empty space, because empty space is accelerated everywhere. So for the acceleration at the atom an angular momentum exists, giving a precession. In quantum mechanic sense the angular momentum acts in such a manner that a precession is the consequence. Our original supposition that the precession is related to the Fermi spin of the electron or the H-atom was wrong (see following sections), but apparently the precession has to be associated with the inertia of matter, actually its rest mass, and it seems independent of the electromagnetic properties of these particles. It is the reason why one makes a distinction between the mediating mass mm = 250.81* me , connected to graviton energy of the empty space for the H-atom and it inertia mass mat = 1837.153 * me .
Generalise this idea of precession. Take an arbitrary mass, M = N * mat , then the precession is proportional with N-times the precession components of one atom. Always there should be equilibrium of force at a certain distance of M. Thus the generated gravity by M is somewhere equal with the outgoing acceleration of empty space. In that way one can make it plausible that the angle between the linear component and that of the precession is constant. From the relation of the equilibrium radius it turns out that the square power of this radius is proportional with the mass M. Thus to keep the precession angle constant, the requirement of the square power for the radius must be fulfilled everywhere. Even the gravitons generated by the H-atom with the Compton length for the graviton, comply to the rule of the constant angle, actually 60°.
In conclusion, the Einstein graviton is formed out of a precession and a linear accelerated component, of which the linear component follows the equivalence principle of Einstein.
The third property for the gravitons is the quantum mechanic equivalent for the event horizon of a mass, the Schwarzschild radius, which is discussed already extensively in Physics 2, chtr 6, Hindsight review . For the translation of Newton’s law of gravity this property is needed, of which here a quick derivation. For a gravity generation mass M the fundamental graviton λ is equal to the radius of the event horizon. The surface radius R of the mass is always longer than λ .Substituting λ = h/(mλ c) in the relation for the event horizon λ c² = 2G M gives 2Mmλ = mpl² It is thought that the Planck mass determines the quantum mechanic resonance condition to maintain the graviton properties of empty space. Also Hubble’s radius of the universe complies with an event horizon for the fundamental length of the graviton.
A thesis of the Einstein gravitons. An Einstein graviton is determined by its length, given by the energy or momentum
mg = h/(sg c) , in which sg² = sprec² + slin² with an angle of 60° between the modulus of s and the linear component. The angle of 60° is exact and follows from energy balance for equilibrium universe according to Newton. Even if the duration in time to accelerate the graviton is very long up to a value of ½ c , the angle is constant and does not change. Beyond the time of acceleration the graviton cannot exists, but must be converted in more or other gravitons. In a stationary process of gravity generation gravitons are continuously produced of a certain length and characteristic time of duration.
The mirror mass relation (2Mmλ = mpl²) with (λ = h/(mλ c)) in it self is astonishing. In terms of physics it means that the fundamental graviton of low energy λ belongs to an imaginary event horizon given by the macroscopic mass M. In other words the low energy gravitons in normal gravity conditions undergo interference, c.q. the interference of empty space. So empty space organizes it self accordingly along the well known principles of quantum mechanics.
It is amazing this was never realized in the theoretical discussion in the 20th century. Shafranov was the initiator in quantum gravity, placing the Planck mass on the map of physics. Probably the integration of GRT in super symmetry or quantum gravity put the blinkers on in theoretical physics.
There are many different possibilities, which could be the subject of the book Physics 3.
1.Use the Einstein gravitons to translate Newton’s law of gravity.
2.Use the gravitons to explain the pyramid model in the phase space of time.
3.Use the idea of the gravitons to explain the early universe, which could not be in equilibrium, for it is created from the Nothing.
4.Apply this concept of gravitons to explain the levitation mechanism between gravity and conjugated gravity.
5.It should be possible with graviton mechanism to explain the sun wheel drive effect for the Sun generating spin due to fusion burning.
6.The conservation of planetary angular momentum has to explain why the position of the rotation axis of Earth can be changed due to the sun wheel.
It looks like a lot of new physics that has never been explored. So have fun!
THE MANY DIMENSIONAL FORMS FOR NEWTON’S LAW OF GRAVITY
The first translation of Newton’s law into gravitons is given in, Physics 2, translation of Newton’s law of gravity into Einstein gravitons. For a first orientation it is useful to read back on this section.
In short.
The maximum energy of a gravity generating mass is given by the radius of its event-horizon. It is equal to the radius of a black hole, where a photon of zero-rest mass can not escape form this horizon.
½ M c² = G M² / λ ,
where λ c² = 2G M . If λ is the smallest Einstein graviton, for which is valid:
gλ λ = ½c²
. By substitution into Newton’s law : gλ λ² = GM one gets a similar expression
½λ c² = G M
However keep in mind that this smallest graviton can only reach a sub-light velocity of v = ½c due to precession.
There fore Newton’s law becomes: g(R) R² = 4 λ c² , but the scaling to c has preference, because the two vector components (precession) of the graviton, always have the modulus of v or c as a maximum velocity. So in the scaling relations we keep:
g(R) R² = 2λ c² (1a)
with a general definition of the Einstein graviton:
½ c² = Rlin (R) g(R) (1b)
formula (1) translates by elimination of g(R) into:
R² = Rlin (R) * λ (2)
This is a newly scaled relation for Newton’s law and note that R² has a parabolic behaviour with Rlin , where the minimum length for the graviton λ is the scaling constant.
For a mass point having a circular orbit around a gravity generating mass, the classic relation of Newton’s law is valid: v²(R)*R = G M or for the escape velocity: v²(R)*R = 2G M . By using (1a):
v² /c² = 2λ / R (3a)
Substitution of relation (2) gives:
(v(R) / c)(power4) = λ / Rlin (R)
or: v(R)(power 4) * Rlin (R) = λ c (power 4) = constant (3b)
Taking the graviton definition of (1b) and substituted in Newton’s law, one gets : 2c²* R² / Rlin (R) = GM
With Rlin (R) = h / (mg(R)* c) it translates into :
R² mg(R) = (hG/c³) M = λpl² M (4)
Only with the mirror mass relation M mλ = (hc/G) = mpl² (5)rel. (4) can be converted into
R² mg(R) = (h/c)² * (1/mλ)
and with λ = h /(mλ c) or (h/c)² = (λ* mλ)² one gets an important new relation for Newton’s law:
R² mg(R) = λ² * mλ = constant (6)
The importance of the mirror mass relation is not only that it gives a quantum mechanic resonance relation for the smallest graviton, but its energy is always very small while momentarily generating an enormous linear acceleration. The bigger the star mass, the smaller its energy, but the graviton length stays proportional to its mass.
The relation (6) is of importance, for it shows that the graviton mass at radius R is inverse proportional to the square power of the radius. If mg v is the momentum transfer at the radius, then if precession is involved, the component of the precession is proportional to mg (R) and there fore inverse proportional to the square power of the radius. It means that the angle between the two components of the vector of Rlin (R) is constant. The tangential component corresponds to the precession momentum. Of course classically, the force g(R) is always proportional with the momentum of the precession. So nothing is new. Only it was necessary to express the relation in the graviton mass mg .
Further is valid that the group velocity for the mg (R) -quanta is:
mg (R) *R = ½v²
So using rel (3a), R is also the radius of the mg –graviton to the gravity generating centre. It explains why the rel. (6) R² mg(R) = constant, while mg (R) *c is the transfer of available momentum belonging to the c-velocity at a point of the radius R.
One really should not overlook that the mirror mass relation comes from the equilibrium condition of force.
λ c² = 2G M and λ = h /(mλ c)
gives : M mλ = mpl²
In a real black hole the graviton λ could not escape, therefore a virtual quantum mechanic resonance relation.
THE PHYSICS OF THE EINSTEIN GRAVITONS FOR NEWTON’S LAW OF GRAVITY
The importance of Newton’s law is to realize that the transfer of flux as an energy of empty space through a spherical surface perpendicular to the flux is inverse proportional to the change of momentum.
The number of gravitons through a surface 4πR² is Ng (R) = M/mg (R) , where mg (R) is the energy of the graviton at radius R. M is the overall mass. Without weighing, the number of gravitons, Ng (R), with its energy, will always explode with increasing radius. Under certain conditions determined by Newton’s law, the product of
Ng (R)* mg (R) if summed over the interval, R to infinity, stays finite.
How to find such a relation? The Einstein gravitons are a simple idea, namely, a point mass for an accelerated straight trajectory. So suppose that a group of λ-quanta of gravitons, originating from a quantum resonator for the event horizon with an end velocity v at the radius R, represent the mg -quanta at that radius with the same kinetic energy (momentum).
At that radius R from the gravity generating source is valid:
Ng (R)* mg (R)*v(R) = Nλ(R)* mλ * v(R) (1)
If Nλ(R) = M/ mλ = Nλc = constant, meaning all quanta are contributing to the formation of the product Ng mg , then the differential :
mg * dNg/dR + Ng * dmg /dR = 0
where, if one substitutes mg (R)*R² = constant, as one of the derived rel. for Newton’s law, Ng (R) and also the weight product explode with increasing R after integration over the interval of R to ∞.
It is most likely that only a fraction due to conservation of momentum or conservation of energy for the number of Nλc of the λ-quanta contributes to formation of the momentum of mg (R) at radius R. So Nλ(R) is not a constant in rel. (1). Application of; mg (R)*R² = mλ * λ² = constant:
Ng (R) = Nλ(R)* (R/λ)²
Suppose that due to momentum conservation:
Nλ(R) = Nλc * (v/c) (2a)
And for energy conservation:
Nλ(R) = Nλc * (v/c)² = Nλc * (2λ/R) (2b)
due to applying Newton’s law.
The conservation of momentum in rel. (2a) leads always to exploding functions, sometimes even before integration over the interval of R to ∞.
So one is bound to except energy conservation of the λ-quanta. As one should expect, for the conservation of momentum includes energy loss. Note that the rel. (1) is also valid for the energy of the mg –quanta at radius R.
½Ng (R)* mg (R)*v²(R) = ½Nλ(R) mλ v²(R)
It is allowed to drop v or v² from both side of the equation, for the kinetic velocity is the same.
Or
Ng (R) * mg (R) = Nλ(R) * mλ (3)
It is now possible to calculate for the momentum, in (3) multiplied by v:
P(R) = 2*Nλc * mλ * c* (λ/R) (power 3/2) (4a)
For the energy at the radius:
W(R) = ½ * Nλc * mλ * c² * (λ/R)² (4b)
The relation (4b) converts back into the classic Newton by inserting λc² = G M and dividing by the graviton length of λ.
W(R) / λ = G M² / R² (5a)
W(R) / λ is dimensionally a force at the radius and after dividing by M one finds the force at a point R of the radius. But because of the division of
M = Nλc * mλ then w(R) / λ is also the force caused by one graviton mg .
Similarly the integral ∫w(R) dR is the potential energy, normalized again for λ.
(1/λ) ∫w(R)dR = λc²/ R = GM/Ro (interval R to ∞) (5b)
giving the potential energy for one quantum of Nλc
So far the idea of the mg (R)- quanta for the interaction of the energy of empty space seems to confirm Newton’s law of gravity.
However, rel.(1) for the momentum is much more revealing. Just take for c = ωλ , an angular velocity and divide in rel (4a) by Nλc then :
p(R) = mλ λ² ω √(λ/R) (1/R) (6a)
Instead of p(R), the momentum, one should use the angular momentum, b(R). Where mλ λ² is angular inertia for the graviton spin and
ωR = ω √(λ/R) , the angular velocity at radius R.
The total angular momentum is the ∫ p(R)d R over the interval Ro to ∞ is:
∫p(R) dR = -½ mλ λ² ω √(λ/Ro) (6b)
The rel. (6b) shows that the inertia moment, mλ λ², is a constant. With increasing radius the angular velocity, ω√(λ/R), reduces. In (6a) the angular momentum is inverse proportional with R.
The difference between the relations (5a&b) and (6a&b) is, that there is not divided by mλ * λ = h/c . Or apparently this product means the conversion from linear- to angular momentum. This conversion is allowed, because gravity is generated from a point source in the sense of quantum mechanics (only dimensional scaling). Contrary, due to action- reaction principle for matter, apparently the angular aspect disappears. The problem is, one can not prove this as yet and only a sophisticated in depth analysis may prove these deductions.
By taking half the c-velocity the supposition about the event horizon scaling, the virtual quantum mechanic interpretation seems allowed. It makes the Einstein gravitons discrete and determines the angle of 60° for the precession of the smallest graviton. The angle of 60° was assessed from the comparison of the overall energy for the universe between general relativity and Newtonian calculations, but it is not proven.
Second conclusion is that the graviton λ is not related to Fermi-particles, because this derivation does not need any atom characteristics. So the considerations on the fundamental symmetries for the Einstein gravitons and there fore the properties of the conjugated gravity and matter, are absolutely valid and firm. Although the precession is a hidden variable in Newton’s law of gravity and can only be revealed by supposition, the consequence of the angular moment is completely explained, including the constant-city of the angle between the components of the graviton and any graviton of other length.
The quantum mechanic switching of 60° for the fundamental graviton can be explained by the model for a double rotating dipole. Note, the graviton can be compared with a magnetic vector. It is a vector with an in and an out. So it is a dipole.
Firstly, it is unlikely that the graviton in our reality rotates continuously in the same plane, the plane of the precession and the linear acceleration. In this plane is the rotation axis of the spinning dipole, which displaces under de force of acceleration as a momentum of precession. Obviously the segment of 60° from this plane should be randomly distributed over the entire sphere it is contained in, similar to the quantum mechanic spin for Fermi particles. To be completely clear, our plane containing the rotation axis of the spinning dipole, is randomly switched around. It is the only way to explain, why the gravitons are evenly distributed around a sphere of matter.
Secondly, by interference of two equal gravitons in opposite phase relation in the game of the double rotating dipoles (down load the game from the website), the modulus of the momentary summed poles, follows spatially a line of symmetry. This line most likely the rotation axis of our spinning graviton, should project randomly in our reality. Has the double rotating dipole two different frequencies of rotation, ratio 2:3, then the above modulus has 6 intervals corresponding to a ratio of 1:3, showing the switching interval of 60°, apparently the 2-frequency is suppressed in the summation.
It is not allowed to have no integer ratio between the frequencies of the double rotation. In that way no other angles than 60° are allowed. So the angle of 58°.50, which was derived for the H-atom, shows the consequence of the electro magnetic weak interaction of the Fermi particles, making up the neutral H-atom.
Note, a distinction is made between a double rotation of a dipole in some other 3D-space and the reality of the spinning graviton around one axis subjected to precession and the uncertainty principle. See for a further discussion the section of the pyramid model.
A NON-LINEAR HARMONIC STRING THEORY FOR EINSTEIN GRAVITONS
Introduction and summary of the theory.
In the preceding section, the physics of the Einstein graviton for Newton’s law of gravity, it was outlined, how the gravitons behaved classically subjected to Newton’s law. Here it is tried to develop a quantum mechanic model for the properties of the gravitons in view of Newton’s law. Although Einstein gravitons are not absolute one dimensional objects, they certainly behave as such in the classic sense, there fore it does not seem too far fetched to compare them with the quantum mechanic analogue of the linear harmonic oscillator. The linear oscillator is a classic oscillator in one dimension, such as the Huygens pendulum or a mass connected to a spring and vibrating around an equilibrium point without any frictional loss.
The property of the harmonic oscillator is that it has one frequency of vibration independent of the magnitude of the deviation of the pendulum. The classic oscillator has a parabolic potential energy with respect to the deviation x, where m is the mass and ωo the angular or circle frequency. In ωo is included the constant of the spring or the constant gravity, g, at the surface of the Earth.
W(x) = ½ m ωo x²
W is the potential energy of the harmonic oscillator. The one dimensional equation of motion for x is converted in a 1D-Schrodinger equation. The mathematical treatment is rigorous and it reveals many interesting features of the quantified oscillator. For example, the quantum frequency of the 1D-oscilator is not constant.
ωn = ωo (n + ½)
But the change from one allowed state to the other is determined by the mathematical proven rule
Δ En = ± h ωo with Δn = ± 1 .
It is the only quantum step energy for the photon which is a constant irrespective of the value of ωn .
Has one instead of a parabolic potential the Coulomb potential then
Δ En , sequenced to the quantum number n, is not a constant any longer. This problem will be treated in the next section. Knowing the quantum properties of the 1D-harmonic oscillator and those of the Einstein gravitons, there are many similarities and it should be interesting to understand why the gravitons cannot behave as one single harmonic system or why they do not fit in the quantum mechanic sense in the Coulomb potential for gravity.
According to the title the non-harmonic oscillators are different. The graviton length, Rlin :
Rlin(R) = λo R² (1)
However the mass or energy, not rest mass, of the graviton string is different for each length. Meaning also that including the spring constant, which determines the frequency, it is a different harmonic oscillator. It turns out that the graviton string cannot reach higher excited quantum states, but each length in rel.(1) is in its zero-energy state. So every graviton belonging to the ensemble dictated by rel. (1) of Newton’s law of gravity, is always in the ground state of zero-energy, having each another oscillation frequency. It explains the title ‘non-linear’ for this section. Here the word ‘theory’ means that it is not possible to prove the non-linear behaviour of the strings consistently, but it is proved sufficiently that the strings cannot have the properties of one oscillator.
The end goal of all these exercises is to find out, if Newton’s law can be quantified and whatever consequences it has to a mass orbiting a star.
The quantum theory of the non-linear graviton strings.
The dimension scaling of Newton’s law for the graviton length is:
Rlin(R) = λo R² (1)
This relation is independent of the angle coordinates, φ and θ, in spherical symmetry. It gives only the radial dependence, R-coordinate, with respect to the gravity generating centre. However Rlin seems to be a one dimensional object, a radial line of a discrete length, operating in a spherical symmetric gravity field. To a surface with radius R all gravitons in any direction have the same length Rlin .
Now there are two ways to quantify rel.(1).
Either the radius: Rn = n λo (2a)
or the graviton Rlin(R) = λn = n λo (2b)
Where n = 1, 2, 3 , ……. ∞. Quantifying the radius is not interesting because Rlin behaves as a sequence of 0, 1, 4, 9, 16, …….. etc.. So the choice is simple and it is rel.(2b), the graviton length.
In other words:
Rn = √n * λo
And because:
vn² / c² = λo / Rn
the groups velocity of a bunch of gravitons is vn . Then:
vn = c / n (¼ -power)
the ¼ - power or the ¼ - root of n.
The sufficient proof.
For the energy state of a harmonic oscillator is valid:
En = h ωn = h ωo (½ + n )
Where: h = h/2π , the normalized uncertainty constant. With
n = 0, 1, 2, 3 , ……. ∞.
Δ En = En – En-1 = h ωo , Δn = ±1 .
Δ En are the allowed increments for the steps. The relation between the wavelength and the circle frequency ωn for the harmonic oscillator is:
m ωn = h λn (3)
Remember, ωn contains the spring constant. For the harmonic oscillator the mass m should always be a constant.
For the Einstein gravitons, the energy expressed in mass units in relation to its length, as per definition is:
Rlin(R) = λn = h / (mlin c) .
Thus it is impossible to reconcile this outcome with rel.(3).
In conclusion Rlin can be in first instance a harmonic oscillator, but for each length, the frequency of the ground state ωo should be different.
If all gravitons are in the ground state then mn of rel.(3) becomes:
mn ωn = h λn (4)
Here λn is related to Rlin in rel.(1) and it is also:
λn = (n + ½) λo
The factor a half has here a different meaning, because n begins at n = 0
and it is related to λo .
To satisfy rel.(4) mn and ωn should behave as:
mn = mo /√(n + ½) and ωn = ωo / √(n + ½) (5)
By using the ground state energy for each harmonic oscillator of λn :
En = ½ h ωn = ½ mn vn² (6)
With vn = ωn Rn and Rn = √(n + ½) * λo combined with rel.(5) one can prove the rel.(6) provided ωo λo = c .
In conclusion, all harmonic zero-energy states are related to the fundamental graviton λo of rel.(1).
Note, that vn is the group velocity of the gravitons at radius Rn and mn
is the effective energy expressed in mass units for the gravitons belonging to that discrete radius.
The ground state and the three orbital states of any graviton.
So far by simple means, it was sufficiently proven that any graviton at a distance of the centre should be in the ground state having a different effective mass at each radius. However the one dimensional assessment cannot be correct, if the graviton with wave properties operates in a potential well of three dimensions. A careful analysis of the quantum mechanics of a spherical symmetric harmonic oscillator is needed and further the angular momentum of a small mass (graviton energy) with quantum mechanic properties should be considered. Happily enough the whole frame work for the different situations of the Schrödinger equations is well documented and it is i.e. reviewed (German language): The rechenmethoden der Quantun theorie, S Flugge. Ed. 1965; Heidelberg taschenbuch, Springerverlag.
The analysis of the spherical oscillator has three degrees of freedom and the ground state has instead of the factor a half, a factor 3/2 for the zero- energy of a quantum particle.
Eo = 3/2 h ωo (7)
In fact the factor 3/2 means the graviton is simultaneous present in the x,y,z-direction or has a spherical probability wave. This cannot be valid because the graviton is mainly one dimensional. So the ground state should stay with a factor a half. On the other hand the orbital momentum for a quantum particle cannot exist in the ground state. Generally this can be formulated in a rule for all the eigenfunctions of the angular momentum in spherical symmetry. The number of states is: m= 2L + 1 , where L is the orbital integer and m the azimuthal integer or magnetic quantum number. So valid is:
m= L, -L+1, -L+2, ….., -1, 0, +1, ……, L-1, L .
It means that if m = 0 then the orbital quantum number is L = 1 , but if
m = 1 , then L = -1, 0, +1 , three states are available. These are the states closest to the ground state and the other states are not of interest here.
So if instead the quantum energy at every state can be written as :
Eo = (1 + ½) h ωo
It means that the graviton is simultaneous in a ground state and an orbital position. This energy state could contain a degenerated state for the orbital momentum.
The next step is perhaps too speculative but it shows why empty space maintains its three dimensionality around a gravity generating centre. Now take three gravitons at the radius Rn in the ground state, then if m = 1 , there are three orbital numbers for L . The three gravitons are placed in such a phase relation of orbital momentum that for m = -1, 0, +1 the three gravitons occupy these states, but the ground state represents also the factor half-state. Because there are many gravitons per radius, these gravitons may comply with the degenerated states of the orbital excitations.
Consider again relation rel.(7), written slightly different,
En = (1 + ½) h ωn
For the energy states should be valid for each radius Rn , as is shown already for the classic case then also is valid:
En = ½ mm vn²
The radial group velocity for the graviton scales in two manners classically:
vn² / c² = 2 λo /Rn , falling to the centre
vn² / c² = λo /Rn , circular orbit.
The reason for the distinction in the factor two is the angular momentum or the centre petal force. For a kinetic point following the graviton radially the kinetic energy is twice that of the orbital energy at the same radius.
In other words the energy gained cascading downwards is always a factor two different with respect to the circular kinetic energy at that radius. It means a radial cascading point represents a lower quantum state than the orbital state, which requires a higher excitation. Here one finds the quantum mechanic explanation for the factor two. It is a neat confirmation of Newton’s classic case for these two situations. Having 6 quantum states or three graviton groups spatially arranged per radius instead of one graviton for two quantum states, does not change the reasoning of the factor two.
An overview discussion is in the section: A quantum theory of Einstein gravitons in a conserved potential gravity field.
A QUANTUM STRING THEORY OF EINSTEIN GRAVITONS IN A CONSERVED GRAVITY FIELD
Since the harmonic nonlinear string theory is understood, it is not difficult to investigate the behaviour of the gravitons in the Coulomb potential for gravity.
The quantum mechanic model of an electron in the Coulomb potential of the proton is determined by the Schrödinger equation. Here one finds interesting similarities with the gravitons. Only one has to keep in mind that all levels of excitation are occupied by the gravitons. It is not the same where the electron can jump from one excited state to the next by releasing or absorbing a photon. The essential difference between gravitons quanta and the photon-electron mechanism of the atoms is that empty space of the gravitons completely fills up all possible states available in the potential field due to quantum graviton interference. In stationary condition an endless stream of quanta cascades downwards to the material surface, which is the boundary condition for the probability waves of the graviton oscillator.
A comparison with the harmonic string theory is given after explaining the Coulomb string model.
Description of the gravitons in a conserved energy field.
The conserved potential of the electron of the H-atom and that of the graviton is given by:
V(r) = - e² / r and V(R) = - λo c² / R
With λo = G M /c² and 2mo M = mpl²
For the gravity potential all parameters are known. For the electron potential, where e is the static charge of both electron and proton, the dielectric constant of vacuum is normalized to the e-charge. The derivation of the Schrödinger equation for the Coulomb potential can be found in previously given literature or by looking at the scientific section of the internet.
The different quantum states of the energy En in the potential well are dependent on the rest mass m of the electron and it is given by:
En = - m e(4-power) / (2h² n²) = - e² / (2ro n²) = - e² / (rn n)
With rn = 2 n ro and ro = h² / (m e) . ro is the Bohr-radius of the electron or the zero-energy state. The minus sign indicates the negative energy with respect to the continuum at V(r) = 0 for r = ∞ . h is again the normalized uncertainty constant.
With increasing n , the main- or principal quantum number (integer), the increments of the En-states gets smaller at higher levels of the of the potential till V(r) = 0 is reached. Every step (En+1 – En) = h ωn is the photon energy which is not constant but it is dependent on the main quantum number n. Only sequential steps for most of the situations of the H-atom between the energy levels are allowed. So (En+1 - En) or
(En – En-1).
The angular momentum having the quantum integers L, m , is related by strict rules to n . For every m there are (2L + 1) orbital positions and they are restricted to the maximum value of the n-integer. It gives some complicated rule of summation for the overall number of the discrete quantum states:
∑m = n² ( L = 0 & L = n -1) or ∑(2L + 1) = n² ( L = 0 & L = n -1)
With no external electric or magnetic fields acting at the electron quantum ensemble, all states of L, m are degenerated to the same energy level En (no splitting of levels at the main number n). All degenerated levels can be occupied by electrons if an atom with more electrons is considered. It is the ground state for the non-excited neutral atom.
Having described the H-atom as a photon quantum ensemble, some similarities with the gravitons are more or less self evident. The fundamental graviton mass (energy expressed in mass units) is mo corresponding to the rest mass of the electron. If the gravitons are just point like without a length or a precession component, then the energy of the graviton at that quantum level is distributed over an effective mass and a kinetic velocity (angular velocity at a radius Rn ). For the energy level En is valid:
En = - Eo λo / (n Rn)
With Eo = ½ mo c² = ½ h ωo if mo = h / (λo c)
In practical terms for the Sun with a radius of Rn = 7*10exp(+8) metres,
λo = 1.5*10exp(+3) m , then the number of states is of the order of 10exp(+10) , the square power of n . It is the quantum state at the surface of the Sun and En is operating in the continuum. The Earth orbit is 150*10exp(+6) km or 1.5*10exp(+11) m then the number of states is of the order of n² = 10exp(+16), still further away in the continuum. It gives an idea for the range for the quantum levels.
The step energy of Δ En = En+1 – En is the effective energy of h ωo ,related to the fundamental quantum λo and an extreme small fraction of
mo = h/( λo c) . Δ En = Eo / (n (n+1)) and it is the measure of graviton energy which increases cascading downwards to the matter surface. The graviton effective mass and the kinetic velocity increase while the radius Rn decreases. Every radius Rn represents again an great number of degenerated states belonging to En . All graviton lengths are defined by: Rlin(Rn) = λo Rn² . The length of the graviton, Rlin , is as usual determined by its energy at the c- velocity.
In the ground state Ro =λo or ½ λo , the probability wave of the graviton has no nodes. At Rn , the consequence of the point like nature of the quantum, there are node surfaces (node-points in the radial sense), where no gravitons can be present. So, the higher n, the more nodes exist, which are spaced closer together, but these nodes reach also deep in the inner radii. It is the analogue classically for the eccentric elliptic orbits of a particle. Here one finds a difference with the harmonic model. The existence of the nodes for gravitons is not understood.
Discussion
Compared with the harmonic string theory, the Coulomb model of the graviton strings gives some other results. It represents a free range for the occupation of allowed quantum states by not considering the specific properties of the gravitons. What is shown that quantum interference exist in a conserved energy field. The fundamental graviton of λo has already such a low energy that the uncertainty relation is valid, but the gravitons occupying all energies belonging to greater radii than the smallest of λo give still lower energies, consequently giving higher uncertainty interferences.
The problem, which makes the theory less valuable, is that the graviton energy can be broken down in fractions of any dimension, always standing for another mass energy. It contradicts the formalism which dictates that the mass is constant. There fore the distribution of graviton energy and the number of generated states at the discrete radius Rn cannot be correct, but it shows at least that the uncertainty condition can be applied. Also the existence of nodes is not explained.
A variation for this string model to overcome above objections, can be by assuming that each Rn is a ground state for the Coulomb potential.
So En = - Eo / 2(n Rn) with n = 1, 2, 3,…. ∞
Apparently it is nearly the same as the previous expression, but here one meets a main obstacle. How to relate the integer sequence of n to the step function? Either relate to the λo or assume that the circle frequency ωn = ωo / F(n ) or some other common sense assumption. In the harmonic theory this was resolved consistently. Here one has to make an ad hoc supposition. Using the assumption of the ground states, converts the Coulomb theory in the non-linear harmonic theory. To confirm this then the number of energy quantum states of n, the main quantum number, at the Rn should be the same for either model. Rn has the square root of n while the number of discrete gravitons goes by n for the harmonic model and the number of quantum states for the Coulomb model goes by the square power of n and the radius is proportional with n. But both n’s are different. If n(harmonic) = √n(coulomb) then the total quantum states should be the same.
Because of the number of degenerated states is different in the Coulomb model, there are two scenarios. As long as no formalism of the gravitons based on a special adapted model exist, the two scenarios are hopefully good enough to study the macroscopic behaviour of a mass operating in quantum gravity.
THE ALL-AROUND ROTATION OF EMPTY SPACE
A problem stretching our imagination is, how the Einstein gravitons disappearing into a material surface in every direction, equally distributed, act on a body. The graviton is two dimensional and acts according to a plane of preference. It is the plane for linear acceleration and that of the precession. In every point of the surface of the body a vector along the modulus has an angle of 60° to the tangent plane of the sphere while the linear component extrapolates to the centre of the sphere. The plane of preference, precession plane, of the graviton is perpendicular to the tangent plane of the sphere and the centre of the sphere is in that plane.
At first instance a thought experiment has two possibilities. Release of the gravitons equally distributed along the surface (or going into it), either simultaneously or at random, but statistically the mean of the number of gravitons covers the simultaneous release. So will the sphere start to rotate? The answer is, no; despite the fact that the cross-product of the two components of the graviton are always the same along the sphere.
In whatever symmetry plane through the centre one finds that along the circumference there are equally impinged vectors under an angle of 60°, which suggests strongly a rotation for the plane. Now take a polar axis through which this plane can rotate according to fig.1. Is it possible to imagine an overall rotation composed of all possible individual circles as of fig.1? The answer is there is no overall preference for the gravitons released or disappearing in a moment in time.
In other words imagine as many circles through the polar axis as possible, each representing an angular momentum for the gravitons belonging to the circle. But the summation of all these vectors around the polar axis does not give a resultant, the sum is zero. However if one does not make a full swing of 360° but only 180°, one finds the maximum net resultant, which is not zero any longer. It may be strange but a most important feature.
In conclusion one can say that every cross-section through the centre of the sphere has a circle representing the angular momentum of the gravitons, but all circles added up around an arbitrary polar axis, give a net result of zero. In other words, yes, empty space rotates, but it is impossible to pin point a finger and say how or in which direction space is turning. A brilliant concept, a concept overlooked for centuries of observing our own real space around a gravity generating object.
As soon as not one gravity body is present in vacuum, a direction of preference presents itself and all other possibilities for the gravitons are cancelled out, except the ones generating the orbit of the two bodies. With two bodies separated from each other they circle around, provided the linear component is equal along the circumference (representing the centripetal force)
Having analysed above problem, it should not be too difficult to trace this result back by considering Newton’s law of gravity. So let us find the angular momentum of the gravitons.
b(R) = mR R² ω(R) (1)
where mR = h/(Rlin c) and R² = λ Rlin Rlin is the length of the accelerated graviton for the modulus of the c-velocity of the two components. Then
b(R) = ( λ ω(R)/c) h (2)
this relation is similar to relation (6a) and (6b) in the previous section, The physics behind the Einstein gravitons in N-law of gravity. Here rel. (2) is an expression independent of R and with a circle frequency determined by a circular orbit around the gravity generating centre.
ω(R) = c √λ / √R³ giving b(R) = h{λ/R}power 3/2 (3)
b(R) expresses the angular momentum of empty space in polar direction with the origin of the sphere. Therefore the vector b(R) can not have an angular preference with respect to empty space. As soon as a real material point is put into reference, a circular or an elliptical path, the momentum is real and measurable, but it is one selected out of many quantum mechanical options.
It still is made more clear by the summation of b(R) over a complete radial interval.
∫ b(R) dR = h ∫ {λ/R}power 3/2 dR = - 2h{λpow 3/2 / √Ro}
provided the angular momentum is zero infinitely faraway of the gravity centre.
In other words with a material point around a gravity centre only a semi-sphere of our reality can be observed, but depending on the initial condition for the same kind of matter, a left handed or a right handed semi sphere is revealed. Always one part stays hidden in the quantum mechanics of empty space. Still it is not a trivial discovery. It is something new about the perception of Newtonian space of three dimensions, which had its onset more than three centuries ago.
Note: another treatment of above didactic explanation and leading to the same result is given in fig 2. Integrate the cross-product vector of the components of the gravitons for an angle φ over an interval of (-π to +π) rad. The result is not zero and this vector of the equatorial plane has to be summed for all vector directions for the angle θ . Over 180° the net result is maximum and over 360° degrees the net result is again zero.
THE STORAGE OF THE FLYWHEEL ENERGY FOR THE UNIVERSE
The proof in the previous section, The all around rotation of empty space, is valid for one kind of matter. For a follow up discussion on the sense of rotation of empty space one is referred to the section Conjugated gravity and conjugated matter. With the proof it is suddenly possible to understand the old question: Does the universe rotate as a flywheel or not?
Whatever the distribution of baron matter the statement is that because most of the energy of gravity is stored in empty space, our cosmos behaves as a flywheel, but it is not a flywheel because of the initial condition for the universe. The reason is that because matter is created at a moment in time, the initial condition reflects only a hemi-sphere of the angular momentum. The other hemi-sphere is not observable to us.
In other words each galaxy in the present-day universe could have represented its own hemi-sphere, either left or right handed, provides each galaxy was created under individual conditions. Since the generation of matter one should have expected this outcome for one hemi-sphere, for instance a right handed hemisphere. In fact it is well known that the opposite is observed, the angular momentum, or the spin of the galaxies is randomly distributed. But that is not because of the separate integration of angular momentum but because of the existence of conjugated matter in the early universe.
Turning to the atomic scale a similar consideration as above confirms the outcome. Baron matter evenly distributed without clustering, all atoms having a graviton pole have to release gravitons due to the balance of force, of small characteristic dimensions, Compton length. Then again the tiny angular momentum of the gravitons can be summed and equally, an overall hemi-sphere is the result of all possible orientations. The last can only be grasped by the condition that all angular momentum was generated simultaneously.
The symmetry of the coupled creation of matter and gravitons at the onset prevented a zero angular momentum of the universe. To get a zero angular momentum for 3D-space of our observable cosmos, our reality, the simultaneous creation of conjugated matter is a necessity.
A further discussion is given in the section The sense of rotation of empty space.
THE SENSE OF ROTATION OF EMPTY SPACE
The sense of the rotation of empty space can be detected by an orbiting test mass around a gravity generating centre of normal matter. The test particle can be either consisting of matter or anti-matter. If the orbit velocity of the test mass is the same for either matter or antimatter then the vector of the angular momentum is the opposite. If the orbital momentum is the same for the either the matter of the anti-matter test mass then one finds opposite orbit velocities.
Now if the gravity source consists of anti-matter then everything is reversed. So in total 4 options are available, which might be observed in an experiment using matter and anti-matter in a clever manner.
If one considers the conjugated matter in its two modes then one has 8 possibilities in total. The reason is that internally in the neutral baron structure the spin rotation is the opposite to that of normal matter (taking normal conjugated matter not the conjugated anti matter). As is reasoned before this internal spin, g-spin is not necessary the Fermi spin of the electron, proton or neutron. See the section, Notes about conjugated gravity and conjugated matter .
These 8 options arise the big old question: Are the matter and the anti-matter universe a 3D-hypersurface with two event cones for time and anti-time in 4D-time space or has one really two homogenous quantum 3D spaces of gravitons??
In case of the one hyper surface, it means that the nature of the 4 modes of angular momentum do not give rise to destructive quantum interference. The destructive quantum interference might mean two things:
1.The decomposition of graviton space in electro magnetic energy, mainly lossless.
2.The complete annihilation of quantum space.
In case of the complete annihilation two separated 3D-spaces linked by a phase velocity are needed. Obviously in the book Physics two, intuitively, due to the option of conjugated gravity, the case one was chosen as a hypothesis. But even then, complete annihilation of graviton space is not excluded, if anti-matter is considered.
The question about complete annihilation is difficult to resolve. In the section Notes about conjugated etc., the 4 options of the gravitons are explained and as shown here, are somehow linked to the angular momentum for a graviton group.
Tentatively, it appears that the graviton space generated by normal matter is not compatible to the graviton space generated by anti matter. Mainly because of the anti-commutation law for the vector cross product in the angular momentum and secondly, the linear acceleration for the gravitons generated by anti-matter propagate in the opposite direction to that of normal matter. So anti-matter might not stay in orbit very long with a gravity source of normal matter and vice versa.
If above reasoning is valid then the conclusion for the big question is that two separated universes have to exist, which are not allowed to annihilate each other. They are kept separated by the pull of matter and anti-matter along their common time axis.
Comment: The above reasoning is not wrong, but the old question might be the same only viewed in different ways. A hyper surface in a many dimensional geometry, having opposite event time cones, seems to prevent the destruction of this surface, (by way of speaking). The author is not expert enough to develop the mathematics to this question.