Quantum Gravitational Field Part 1: Fundamentals of

Quantum Gravitational Field Part 1: Fundamentals of the compatibility between fields José Alberto Pardi Honorary R&D Director – Advanced Knowledge Engineering (*) Member of the “Cruz Del Norte” Astronomical Association The replacement of 0, the vacuum dielectric constant, in the energy equation of electromagnetic fields is a key element in any attempt to quantize the gravitational fields. Usually, the choice falls on the inverse of the Universal Gravitational Constant G. This seems reasonable, since 0 has an inverse relationship with the Coulomb constant K, used to determine the force between two charges. This choice leads to the replacement of the unit charge by a mass not very common in the Universe, the Planck’s mass. However, some structures found in the Cosmos in the last century and the dimension of Time Crystals found in 2016, seem to differ from this choice. The analysis made in this paper, leads to determine a different constant for the replacement of 0, with a result more concordant with the cosmological data and Quantum Mechanics. This article is the first in a series, dedicated to explain these cosmological structures applying the new constant to the second quantization of Quantum Mechanics.

Table of Content A-Introduction ................................................................................................................................................................ 2 B-The Rosenfeld’s proposal ............................................................................................................................................ 4 C-Why G is not a good replacement of 0 ...................................................................................................................... 5 Empirics evidences .................................................................................................................................................... 5 Finding the unit mass without using G ...................................................................................................................... 6 Breaking-up the gravitational field ....................................................................................................................... 7 New unitary mass for the gravitational field ........................................................................................................ 7 D-Quantum connection between electromagnetism and gravitation ........................................................................... 8 Review of the unit of mass’ magnitude ................................................................................................................... 10 The physical meaning of the Sp constant ................................................................................................................ 10 The graviton-particle interaction as a measure of the graviton field ................................................................. 11 The physical meaning of the unit mass ................................................................................................................... 12 Uncertainty in the determination of the unit mass at lower speed ................................................................... 13 E- Equations of the quantum gravitational field when v ≪c ........................................................................................ 14 F- Checking the results with the Cosmos ..................................................................................................................... 15 The Treder’s New Uncertainty Principles ................................................................................................................ 15 Congruence of the Treder results with the graviton .......................................................................................... 15 Determination of the constant g ...................................................................................................................... 16 The Wilson's limit on the gravitational potential of cosmic bodies......................................................................... 17 The Chandrasekhar’s cosmic masses structure ....................................................................................................... 17 The Faizal's, Khalil's and Das' Time Crystals ............................................................................................................ 18 G-Conclusions .............................................................................................................................................................. 19 Addendum ................................................................................................................................................................... 20 A.1- Units of measures system used in this study ................................................................................................... 20 A.2 - List of the celestial bodies of the Solar System considered in this study ........................................................ 20 A.3- Bibliography ..................................................................................................................................................... 20

(*) https://www.researchgate.net/profile/Jose_Pardi2 - [email protected] DOI: 10.13140/RG.2.2.35700.65929/1

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Fundamentals of the compatibility between fields A-Introduction Those who try to develop a common model to unify gravitational fields with electromagnetic ones, face serious problems due to very different and complex mathematical formalisms. The physicist Hans-Jürgen Treder, tried to give a physical meaning to the quantization of the gravitational field, using a high-level vision that avoids involvement with these formalisms and mathematical instruments that make it difficult to visualize the natural process that is being described (Treder, H-J.-1979). Treder began explaining the physical foundations that justify the application of quantum theory to gravitational fields, defining with this process a simple method to find the bases of a new and less complex matching. This is the method that will be followed in this project, whose main objective is to explain the discretizations found in the Cosmos. This article is only the first phase of the project and its purpose is only to lay the foundations for a more complete later development. The whole project will be based only on existing and proven physical theories and any application of them in a different environment, should be validated with the reality of the Cosmos, when possible in the Solar System, where the necessary data are known with more precision and reliability. Since it is about explaining "discretizations in the Cosmos", the most appropriate physical theory seems to be Quantum Mechanics, so the first step would be the choice between wave mechanics (First Quantization) and the theory of the quantum fields (Second Quantization). If the First Quantization is used, the dimensions of the cosmic masses in motion will produce discretizations too small in their orbits, making necessary the formulation of new paradigms, which determine waves of lengths comparable to those of the orbits (Greenberger, D.M.-1983) to allow obtaining quantum numbers similar to those found in cosmic discretizations. This puts this alternative outside the methodology of this project, which only uses consolidated physical theories. The most suitable branch seems to be the Second Quantization, since the discretizations found in the orbital velocities do not depend on the masses of the celestial bodies involved (Agnese A., Festa R. - 1997, 198), as it should happen in a theory of quantum gravitational fields. Another advantage of the field’s quantization is that it allows overcoming the substantial difference between the nature of the electromagnetic and gravitational fields, using in its definition elements common to both, such as: energy, volume and time.

DOI Spanish version: 10.13140/RG.2.2.35700.65929

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Fundamentals of the compatibility between fields But not all are advantages, there are also some difficulties. One of them is that 0, the vacuum dielectric constant used to determine the field’s energy, uncovers the electromagnetic nature of the definition. This obstacle seems easy to overcome, since 0 is proportional to the inverse of Coulomb's constant K, the electric equivalent of G, the universal gravitation constant, so there is a general consensus to replace 0 with the inverse of G. But there are also more serious conceptual problems in the application of the logic of the Theory of Quantum Fields to the gravitational fields, among them that while the quantum fields are variable in time, the gravitational ones are static. Fortunately, there are different types of fields in electromagnetism: classical, semiclassical and quantum fields that, under certain conditions, may be related to each other. So it is possible to suppose that the same could happen with the gravitational fields, although if, of course, this must be proven. Specifically, the situation facing this analysis is not that of a pure field, but the interaction between a field and a mass. In this case, it is possible to use the theory of The Measurability of Electromagnetic Fields Quantities (Bohr N., Rosenfeld L. - 1933), a subset of the Quantum Fields Theory, which starts from classical fields such as the electric one, to reach fields defined by quantum theory, with the support of all its logical and mathematical formalism. Since this was also the choice of Treder in the publication already mentioned, using this formalism would give the advantage of being able to apply one of its new Uncertainty Principles to the discretizations observed in the Cosmos, to verify the conclusions that are going to be obtained in this study. Another serious problem from the conceptual point of view is that electromagnetic fields are defined as "the force exerted on a unit of charge". While in electromagnetism the elementary electric charge plays this role, in gravitation does not seem to be an equivalent mass. Although this mass does not appear explicitly in the equations of the field energy, making it evident becomes necessary to understand some processes. Replacing the charge's unit by its mass does not solve this problem since there are two elementary charges that meet this condition: that of the electron and that of the proton and both have different masses then, the issue is which one selects. Leon Rosenfeld also faced the quantification of the gravitational field, getting to establish the mass of Planck as the minimum acceptable limit (Rosenfeld L.-1965).


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Fundamentals of the compatibility between fields Treder, on the other hand, did not identify it as a limit, but used it in its new uncertainty principles as a consequence of them. Whether Rosenfeld or Treder, they came to the conclusion that it was impossible to prove any quantum theory of gravitation in regions with magnitudes similar to Planck's dimensions. But, as will be seen, this conclusion is a consequence of using the inverse of the Universal Gravitation Constant G as a replacement of 0. Rosenfeld, on the other hand, went further, concluding that "no logical compulsion exists for quantizing the gravitational field; rather the question of whether and how such a quantization is to be carried out can only be decided empirically "(sic). Although if all the aforementioned physicists converted the equation of field energy by simply replacing the constant 0 by 1/G, none of them considered the substantial difference between the two fields since the gravitational field is acceleration while the electromagnetic field is not. This omission has consequences in its conclusions and its equations. Until the end of the '80s the empirical data required by Rosenfeld to quantize the gravitational field did not exist, but after this date, they have begun to appear in the gigantic Cosmos laboratory, so it is time to review his claim, beginning for the analysis that led him to the mentioned conclusion. B-The Rosenfeld’s proposal Shortly after delivering the theory of electromagnetic field measurement developed with Niels Bohr, Rosenfeld gave a lecture at the NORDITA institute in Copenhagen, which he repeated in 1965 at the Einstein Symposium in Berlin, proposing to relate the Quantum Theory with gravity through the equation of the electromagnetic field energy, using the average value of the electrical field component. In fact, the average value of a component of the field F of a photon of electromagnetic radiation of dimension L is determined within the space-time volume defined by its own volume and the inverse of its oscillation frequency ω. Since in a photon L ≈ c/ (2π∙ω), this operation can be performed using the definition of the field energy: ≈

∙

≈

∙

∙

≈

∙

(B.1)

Rosenfeld proposed to begin the analysis of the possible quantization of the gravitational field by replacing in (B.1) the constant 0 by the inverse of the Universal Gravitation Constant G (Rosenfeld L.-1965), to estimate the photon’s field of a hypothetical weak gravitational radiation, taking advantage of the fact that the other variables do not refer to the electromagnetic environment: DOI Spanish version: 10.13140/RG.2.2.35700.65929

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Fundamentals of the compatibility between fields ∙ ∙

(B.2)

From this definition it can be determined the energy of the field: ∙

≈ ∙

∙

(B.3)

When trying to measure the field of this radiation photon, with a test body that uniformly fills the volume L3, following the reasoning of the theory of measurement of electromagnetic fields quantities, Rosenfeld finds that the measurement conditions impose the Planck mass as lower limit for the test body, without analyzing the impact that the constant G had in the determination of this limit. It is not difficult to show that the unit of mass implicit in (B.3) is the Planck mass ∙ , due to the constant used is just G. Using (B.3) and extracting the square root of the product from the product of the second term by the third: ∙

≈

∙

∙

∙

∙

∙

∙

∙

(B.3.1)

C-Why G is not a good replacement of 0 Whereas the elementary electric charge is the fundamental "brick" on which the electric bodies are "constructed", the same cannot be said for the Planck mass that is the product of a theoretical analysis, in addition, much greater than the mass of the fundamental component of the Universe: the Hydrogen atom. This restriction seems very limiting, since it leaves a large range of masses uncovered. It was from this and other results of his analysis, that Rosenfeld concluded by suggesting that the question of the quantization of gravitational fields should be decided empirically, that is, by consulting the "opinion of Nature", as already mentioned. Empirics evidences At the moment, the only empirical evidences of discretization of weak gravitational fields a cosmic scale that could be used for this purpose, are: 1. A discretization in the cosmic masses (Chandrasekhar S. – 1937), 2. The lowest limit of the gravitational potential in the cosmic bodies (Wilson A.G.-1968), 3. A discretization at cosmic scale (Agnese A., Festa R. – 1997 y 1998), where arises the universal constant αg that relates the tangential velocity of the lower energy orbit with the speed of light (αg =v/c ≅ 1/2086). Due to its characteristic, Agnese and Festa establish an analogy with the α constant of


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Fundamentals of the compatibility between fields the fine structure and hence its name. The astronomical or physical origin of this new constant was, before this article, unknown. Finding the unit mass without using G Of the few quantum principles proposed by Rosenfeld, it is possible to obtain a couple of conclusions, using data from the Solar System. Recalling the condition that the mass must uniformly fill the volume of the hypothetical radiation quantum that would be acting on it, it is possible to combine the equation of the impulse with the resulting velocity of the constant g (vg ≈ ag∙c = 1,437 105) and the average density of Solar System bodies (~ 3∙103), to find the size L of the hypothetical quantum, corresponding to the lowest orbit of a two body system. From (B.3.1), the impulse exerted by the field on a unit mass mu in the interval L/c is given by: ≈

∙

≈

≈

≈

∙

∙

(C.1)

From the second and fourth terms, it can be obtained the value of L replacing mu by the equation of the body density: ≈

∙ ∙ ∙

∙

∙

(C.2)

The result is quite similar to the neutral H atom’s size a0, in the lowest energy state (~10-11). The magnitude of mu can be determined from the average density of the celestial body, since it must uniformly fill the quantum volume L3:

∙

∙

≈

∙

∙

∙

≈

∙

≈

∙

(C.3)

From the results obtained from this empirical evidence applying to the gravitational field simple and basic quantum principles of the electromagnetic environment, it is possible to draw two conclusions: 1. The value obtained for the mass, leads unequivocally to an order of magnitude more comparable with the proton mass mp (~ 10-27), than with the Planck mass (~10-8), 2. The dimension L of the hypothetical gravitational radiation quantum has a magnitude more comparable to the dimension of the neutral H atom in its state of lower energy, than to the Planck length or to the dimension of a celestial body.


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Fundamentals of the compatibility between fields Breaking-up the gravitational field This last conclusion suggests that the action of the quantized gravitational field is not exerted on the totality of the mass of the celestial bodies, at least in their lowest orbital, but on the atoms that compose it or on fractions of it. This leads to suspect that, as in the electromagnetic case, the gravitational field, at least on a planetary scale, does not behave as a classical field, but as a semi-classical one and therefore could be composed of quantum oscillators that obey the Bose statistics, as approximately represented in Fig. 1 and 2. Those oscillators, together, would be who are impelling the mass of the celestial body.

New unitary mass for the gravitational field The order of magnitude of the mass (~10-29) of the "quantum of hypothetical weak gravitational radiation", which could be associated with that of a "graviton", seems to be much higher than that currently attributed (~10-48 - K Olive et al.- 2014) and also higher than the inferred from the gravitational waves of the source GW150914 (~10-39 - BP Abbott et al.- 2016), even though the latter were deduced from higher energies and, therefore, it should have also a higher magnitude. However, it is encouraging that the new value is 109 orders of magnitude higher than the former one, a jump similar to the one that separates it from the value deducted in this article and, in any case, very different from the Planck mass. The discrepancies in the obtained results with currently accepted values, justify a deeper analysis which allows determining the correct replacement of the constant 0. An analysis that simultaneously compares the energies involved: electric, gravitational and quantum, of a real particle (that is an existing one) and that has a mass close to that pointed out by the Cosmos (the proton), allowing to deduce the true relationship between them in a real situation.


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Fundamentals of the compatibility between fields D-Quantum connection between electromagnetism and gravitation To establish a connection between electromagnetic and gravitational fields, it is necessary to find a common element that serves as a comparison pattern. This element is provided by the Relativistic Uncertainty Principle (Berestetskij, V.B. et al., 1978), applied to a proton of mass mp, at rest: ∙

∙

≈

∙

(D.1)

This uncertainty in the position is equal to one quarter of the measured proton radius rpm (~8,412∙10-16- Randolf Pohl et al.-2010), a logical result given that no measurement can be less than the minimum uncertainty given by the Heisenberg principle. The fact that the measurement is four times greater than the minimum uncertainty is only an indication of the "inaccuracy" of the instrument. Since this value can be decreased with new more precise instruments, it seems more reasonable to take as proton radius for this study rp ≈ ∙ -16, the value obtained in (D.1), that can never be lower. Actually, since the proton is pure energy, its radius should be given by its electric selfenergy (r ~1.5∙10-18), but since this value is less than the minimum possible uncertainty, it seems will never be measured by any instrument, making sense to take rp as the proton’s radius. From (D.1) it is possible to relate the proton’s relativistic energy at rest with its position’s uncertainty and its radius rp: ∙

∙

∙

∙

(D.2)

From the constant of the fine structure α and the relationship between the electric and gravitational forces exerted between a proton and a charge equal to the electron but with a mass equal to the proton, situated at a distance rp, it is possible to establish a connection between the proton energy at rest with its gravitational self-energy, assuming that, as in the case of the electron, the ½ factor is ignored due to the lack of knowledge of its internal structure in the layered integration needed to determine the self-energy of the proton (Alonso M. and Finn E. – 1996): ∙

∙

∙

∙

∙

∙

∙

(D.3)

The value of Sp arises directly from the first and third terms of (D.3) and its value is around S/(4∙π): ∙ ∙

∙

∙

(D.4)

That allows completing the equivalence between the four energies:


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Fundamentals of the compatibility between fields ∙

∙

∙ ∙

≈

∙

∙

∙

(D.2.1)

This comparison unifies the conceptual difference between the definition of the electric field and the gravitational one, reducing the first to acceleration, demonstrating that the constant Sp only affects the field. For this it is enough to divide all the terms of (D.2.1) by mp. That is, Sp is not distributed among the masses. All the former reasoning shows how applying the quantum mechanics of electromagnetic fields to gravitational fields. From (D.2.1) it is also clear that, for the energy of the field to be related to the selfenergy of the mass and its relativistic energy at rest, the correct replacement of 0 is G∙Sp. From here it can be tried to build a solid quantum mechanics model for gravitational fields based on the Cosmos’ reality. Although all this reasoning can also be reproduced with a Planck’s mass at rest, obtaining compatibles equations for the relativistic and quantum’s energies and also for the gravitational self-energy (even without the need for the Sp constant), there are two important reasons to invalidate this possibility: 1. The uncertainty in the position of the Planck mass at rest is the Planck length, a theoretical dimension not yet observed in the Cosmos, also difficult to measure, while in the proton the uncertainty in its position has been measured and coincides approximately with its radio, 2. The mass density resulting for this uncertainty is of the order of 1096, much larger than the maximum density known in the Universe (~1059), whether cosmic (Quasar TON618) or artificial (produced with the Large Hadrons Collider), while that of the proton is comparable to the maximum density of neutron stars present in the Universe. These two reasons confirm the improbability of the Planck mass as the replacement of the unit of charge, in a real cosmological quantum gravity environment. Relativistic energy of a neutral H atom at rest The relativistic energy of a neutral H atom at rest and the uncertainty in its position are similar to that of the proton, due to the similarity between both masses, which allows using the same expression (D.2.1) of the latter, changing only the name of the corresponding variables: ∙

∙

≈

∙

∙

(D.5)

This passage is fundamental for the quantization of the gravitational field since, following Rosenfeld's proposal; it must be produced in the interaction with objects DOI Spanish version: 10.13140/RG.2.2.35700.65929

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Fundamentals of the compatibility between fields that have meaning in the Cosmos, where the more frequent one is the neutral H atom. The question arises of what will happen when a neutral H atom at rest and with its electron at the minimum energy level interacts with a quantum of gravitational radiation of its size, since the gravitational field will not interfere with the electric field but will act on the masses of the proton and the electron simultaneously, being able to dislodge the structure of the atom. Since the force that the graviton will exert on the electron (~6∙10-9) is an order of magnitude less than that which holds the electron clinging to the nucleus (~8∙10-8) and that the energy of the orbital is 3 orders of magnitude higher (~2∙10-18) than the quantum one (~1∙10-21), it does not seem that the structure of the atom can undergo any change. Review of the unit of mass’ magnitude The introduction of the new constant in the replacement of ε0 will produce changes in the field equations. To simplify the writing it is advisable to introduce a single constant that should be called g=1/(

∙

). However, within the scope of this project, it will be

useful to remember "its composition" and therefore the denomination will be used 1/GSp, in a way that ∙ . In addition, to simplify the writing of the equations, from now on, the following replacements will be used: ∙ ∙ and to simplify the references to the "quantum of gravitational radiation" it will be called directly "graviton", although in this article it is attributed a very different value to the currently accepted, as already mentioned. Replacing G by the new constant GSp in the gravitational field’s energy equation defined in (B.3.1), mp is confirmed as the unit mass. Using the same equation and procedure than before: ≈

∙

∙

∙

∙

∙

∙

∙

(D.6)

The physical meaning of the Sp constant From a purely mathematical point of view (D.2.1) shows that the constant Sp is a conversion factor (or scale) necessary to equal the energy of a graviton with the gravitational self-energy of the unit mass mP. From the quantum-physical point of view, this can be better understood if one imagines a charge that interacts with a photon. If it evenly fills the entire photon’s volume ~L3, it will absorb all its energy: ≈

∙

∙

≈

∙

∙

∙


(D.7)

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Fundamentals of the compatibility between fields The expression of the field in the last term is a consequence of the of AmpereMaxwell’s law. If the interaction transmits an ultra-relativistic velocity to the charge, the uncertainty of its position will coincide with the photon dimension ( X ~L) and, consequently, the self-energy of the charge will appear: ≈

∙

∙

≈

∙

∙

∙

≈

(D.7.1)

∙

An increase in density in the charge, without changing its dimension, will produce a decrease in X proportional to the increase in the number of elementary charges contained in it, keeping the final result of (D.7.1) equal to the self-energy of a charge and therefore the equality between the field of the photon and that of the interaction. Only an increase in the number of photons will change the intensity of the field, conserving its equality. If the formalism of the gravitational fields is compatible with that of the electromagnetic fields, these equalities should also be maintained in them, but as it results from (D.2.1), the gravitational self-energy of the proton mass is much lower than the of the unit charge, making necessary to add the constant Sp in (D.2.1) to compensate the difference, either to express the field of the interaction ( ∙ ∙ ), as for its conversion to the graviton field (

∙

∙

≈

∙

). From the third

and fifth term of (D.2.1) it is possible to obtain the energy EF,p of the body’s internal field, using its density p or the field quantum expression: ∙

≈

∙

∙

∙

∙

∙

∙

≈

∙

∙

∙

∙ ∙

∙

(D.2.3)

From all this reasoning arises that the physical meaning of the constant Sp is not to be a simple scale factor but a way to equate the intensity of the graviton field with that of the test body, in order to use the Ampere-Maxwell law in the environment of quantum gravity. The graviton-particle interaction as a measure of the graviton field A difference between fields, as in this case, is not new. In the measurement of an electric field using a test body, the energy of the interaction is greater than that of the field to be measured since it results from a combination with the field of the mass used for the measurement. The ratio between both fields is identified with the Greek letter , which in this case must be much less than unity (Bohr, N, Rosenfeld L. - 1933). If an interaction between a graviton and a unit mass is considered as a measurement of the graviton field, the ratio  would be much greater than unity, violating this condition. Remembering that in a graviton Xp,0 ≈L:


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Fundamentals of the compatibility between fields ∙

≈

∙ ∙

∙

(D.8)

∙

This situation is not compatible with that of the electromagnetic fields, where, at these speeds,  = 1, since both fields are equal, therefore, this is another physical reason why the presence of the new constant Sp in the denominator of the quotients of (D.8) is necessary for both fields to be compatible. This equivalence shows that a graviton-particle interaction could also be considered a measurement of the graviton field using the particle as a test body. Although Bohr and Rosenfeld excluded the elementary charge as a test body, this was because at that time it was considered a point charge, with almost no spatial extent. But this situation has changed and now it is known that within the electron there are other particles and therefore it is no longer "punctual". In addition, its spatial extent (~10-15) is much larger than the Planck length (~10-35). The physical meaning of the unit mass The procedure to obtain the unit mass mp in (D.6) is purely mathematical and therefore does not indicate the physical reason for the existence of a mass that, in reality, does not appear in the field energy equation and, therefore, could be considered a "ghost mass". The determination of the same as "mass limit" using Rosenfeld's reasoning gives it a physical sense, but it does not connect it directly with the energy of the field but with the impulse that it transmits, therefore, it is a physical sense external to the field. To understand the correct meaning as an implicit component of the energy of the field during the interaction, it must be started from the uncertainty of the internal potential of the mass (Treder H-J. – 1979): ≈

∙

≈

∙

∙ ∙

∙ ∙

≈

∙

(D.9)

According to Bohr demonstrated to Einstein (Sixth Congress of Solvay, 1930 and Rosenfeld L.- 1965), the second term defines the uncertainty in the determination of a mass subjected to the action of a field ≈ ∙ ∙ along the time (L/c), with an uncertainty in the magnitude L≈X (Treder H-J.-1979): ≈

∙ ∙

≈

∙ ∙

∙ ∙

(D.10)

From the coincidence between the magnitude obtained in (D.6) with that obtained in (D.10) it can be concluded that the mass implicit in the equation of the energy of the field, is in fact an uncertainty in the determination of the mass. A true "ghost mass", as already said, that, at ultra-relativistic speeds, coincides with the mass of the proton. But, what happens at lower speeds?


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Fundamentals of the compatibility between fields Uncertainty in the determination of the unit mass at lower speed Given that all the cosmic objects that are objectives of this project have non ultrarelativistic speeds, it is convenient to analyze what other consequences can have this uncertainty in the definition of the field. As we have seen before, this uncertainty in the determination of the mass that interacts with the field is independent of the magnitude of the mass. However, it is not independent of the speed that the field transmits to the mass, since when it is not ultra-relativistic (v ≪ c), an additional uncertainty appears in the determination of the interaction time, which is transmitted to the determination of the energy of the graviton in the procedure defined by Bohr and Rosenfeld to measure the field (the subscript v is added to the variables, to emphasize the dependence of the low speed): ≈

≈

∙

∙

≈

∙

∙

∙

∙

≈

∙

∙

∙

(D.11)

For the same reason that Bohr gave Einstein, this change in field strength impacts the accuracy in the determination of the mass: ≈

∙

∙

∙

∙

(D.10.1)

This explains why the value obtained in (C.3), using the dimension of the "hypothetical weak gravitational radiation quantum", was not equal to the mass of the proton; it was the uncertainty in the determination of the same. The difference did not have relevance at that time, given that the objective was only to find the order of magnitude of it. This uncertainty in the determination of the mass, will introduce an additional uncertainty in the determination of the body position, such that X≈ћ/(m∙v) and also in the graviton dimension product of the interaction since of (D.11) it will continue to be X≈Lv. Therefore, if the density of the uncertainty is equal to that of the body:

∙

∙

(E.1)

The dimension of the graviton product of the interaction is also slightly lower than that of the original graviton, due to the conservation of the density. If the correction (E.1) is applied to the Bohr radius of the H atom (a0=5.3∙10-11) using g, it is obtained Lv≈1,5∙10-11, a value very close to (C.2) (Lv≈ ∙10-11) pointing out that equations (C.1) to (C.3) are describing the interaction with a "piece" of a celestial body of similar size to the neutral H atom, but with a slightly different density since the


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Fundamentals of the compatibility between fields one used in these calculations was the average of the Solar System (~3∙103) while that of the non-excited neutral H atom is ~2,7∙103. This verification on the Solar System would demonstrate the existence of two gravitons of different energies and dimensions at the moment of interaction at low speeds. If it is accepted that the original graviton energy is a "relativistic variation" (E=p∙c), the result of its transmission at lower speed must necessarily be lower (E=p∙v/2). This would "align" the theory of relativity with quantum mechanics through the Treder’s New Uncertainty Principles. Since the only variables that the uncertainty in the energy of the graviton shares without variation with its energy are the density and the speed transmitted to the body, they could be considered “quantum invariants” of the energy. Since the unit of mass changes its magnitude at low speeds, in quantized gravitational fields, it should be considered only a "reference mass". E- Equations of the quantum gravitational field when v ≪c The expression of the energy uncertainty of the graviton product of the interaction making the reference mass explicit is: ≈

≈

∙

∙

∙

∙

≈

∙

∙

∙

(D.11.1)

And the graviton’s impulse on the reference mass m: ≈

≈

∙

∙

∙

∙

∙

∙

∙

(E.2)

The final velocity transmitted to the body by the graviton and the interaction time uncertainties are now given by: ∙

≈

∙

∙

≈

≈

(E.3)

∙

The interaction of the new field with a neutral H atom of mass mH will produce an uncertainty X lower than the dimension Lv of the graviton containing it, fulfilling the Bohr and Rosenfeld condition for a valid measurement (X L): ≈ ≈

∙

≈

∙

∙

∙

≈

∙

≈

∙

≈

(E.4)

is the average speed between the start and end of the interaction.

Due to the independence between mass and speed, the (E.4) shows that when a graviton interacts with a mass m of greater density but equal volume, X will be


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Fundamentals of the compatibility between fields reduced by m/m, such as it happens with the charge of the test body, in the electromagnetic environment. This variation does not only happen at low speeds but also at ultra-relative speeds. This is the foundation of the invariant characteristic of the "speed" and "density" before mentioned. Therefore X ~L only for the graviton. For masses of higher densities, X is always less than L, fulfilling the condition for an optimal field measurement. These are the basic equations of the uncertainties involved in an interaction between a graviton and a neutral H atom. F- Checking the results with the Cosmos Other research obtained similar results based on totally different considerations and analyzes, but using data from the Cosmos. The Treder’s New Uncertainty Principles In his 1979 article for the centenary of Einstein's birth, Treder defines two New Uncertainty Principles, based on the Bohr and Rosenfeld field measurement error theory, applied to gravitational fields: ∙ ∙

≈

≈

∙ ∙ ∙

≈

∙

(F.0)

For non-ultra-relativistic speeds, the terms on the right of equality, in both equations, must be multiplied by (c/v) as indicated by the same Treder. Both principles are congruent with the study developed by the Russian physicist Matvey Petrovich Bronstein 43 years earlier, using the same conclusions of the Bohr and Rosenfeld study (Gorelik G. - 1992), but connecting them directly with the equations of the General Theory of Relativity. His conclusions reinforce the connection between the General Theory of Relativity and Quantum Mechanics, through the uncertainty in the field measurement and in the space-time metric, proposed by Treder. Since the conclusions of Treder and Bronstein are obtained from the incorrect replacement of ε0, it is necessary to apply the correction of the Sp constant to both if it is wanted to apply them to the Cosmos. Congruence of the Treder results with the graviton To test the congruence of the graviton defined here, with the results published by Treder in the aforementioned document, it is sufficient to use the first of the (F.0) to determine the speed transmitted by the field, replacing G with GSp and L with Lv, since Treder equals this last variable with the uncertainty in the position of the test body.


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Fundamentals of the compatibility between fields Since the test is performed in a low velocity environment, such as in the Solar System, the correction c /v must also be applied: ∙

∙

∙ ≈

∙ ∙

∙

∙

∙

≈

∙

∙

(F.1)

The momentum by unit of mass gives the same speed than in (E.2), confirming with a fully different process, that the graviton is the product of uncertainty. Clearing v in (F.1), it is obtained a result congruent with (E.2): ≈

(F.2)

∙

Determination of the constant g It is also possible to apply the same uncertainty principle to determine the constant g assuming that the graviton of the field produced by the central celestial body transmits the impulse to a portion of the mass of the orbiting celestial body of a dimension similar to the Bohr radius. Replacing v/c with g,c in (F.1) and applying (D.12): ∙

∙

≈

∙

∙

∙

(F.3)

Clearing g,c: ≈

∙

≈

∙

(F.3.1)

This value is very similar to that published by Agnese and Festa in the paper already mentioned of 1998 (g ≈4,8∙10-4). To explain this difference it is necessary to remember that the value of g does not depend on the density of the body in orbit, so the tangential velocity of the lower orbit does not either, but from the point of view of the quantization of the gravitational field here described this dependence exists. Consequently, the g magnitude must represent an average value of the densities of the celestial bodies used by Agnese and Festa to determine the its value, more in line with the average densities of the Cosmos than with those of the Solar System. Taking the published value of g and using the impulse equation (E.2), it can be find the dimension L of the graviton that produces it and from there calculate the average density of the celestial body to which the impulse has been transmitted, which turns out to be of ~1,1∙103, the same order of magnitude as the density of Enceladus, one of Saturn's moons (~1,1∙103). Indeed, this density is lower than the average value of the


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Fundamentals of the compatibility between fields Solar System (~3∙103), something more in agreement with the Universe that includes bodies of lower densities, such as the galaxies. This is probably the first time that the value of the constant g is obtained from a reasoning based, purely and exclusively, on existent physical principles, since its discoverers obtained it from the average speed of the planet Mercury in its orbit, inspired by the correlation of the velocities of the planets of the Solar System. All subsequent works of other researchers on this discretization were based on this value, not explaining its origin without the need of introducing new physical models not yet tested. Besides this, the result has the double value of confirming, based on the data of the Cosmos, the validity of the new constant Sp and the New Principles of Uncertainty of Treder corrected by using this new constant. The Wilson's limit on the gravitational potential of cosmic bodies Although the existence of a limit in the potential of the orbits of the Cosmos may seem surprising, the astronomer Albert G. Wilson (Wilson, A.G., Edelen D.G.B.-1968) also found a limit for the gravitational potential of the cosmic bodies (stars, galaxies and galaxy clusters) given by: ∙

≈

≈

∙

(F.4)

This expression of the mass minimum potential makes clear that the GSp constant is already present in the Cosmos what can be considered a further proof of the validity of the GSp constant, since it has gone from theory to reality. The Chandrasekhar’s cosmic masses structure But if this coincidence between Wilson and Field Quantum Mechanics is not enough, there is more. In 1937 Chandrasekhar (Chandrasekhar S. - 1937), found the following relation between the maximum values of the completely degenerated cosmic masses, according to the Fermi-Dirac statistics (stars, galaxies, Universe) and fundamental constants: ≈

∙

∙

≈

∙

(F.5)

The results are reproduced in the following table: Level Stars Galaxies Universe

v

Mass

3/2 7/4 2

~1031 ~1041 ~1051


Sample (Updated) Mass Object 31 ~10 VV Cepheid A ~1041 Milky Way 53 ~10 Observable

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Fundamentals of the compatibility between fields Chandrasekhar found an explanation only for the level v = 3/2 for the largest masses of his model of the stellar evolution (Chandrasekhar S. – 1983). This structure of the maximum limits of the masses of the Cosmos’ celestial objects, although approximate, shows that the masses of the neutral H and the proton are not only the most frequent, but also play a preponderant role in its structure, reinforcing the choice as reference masses in the quantum mechanics of gravitational fields and confirming the presence of the Sp constant in the Universe. The Faizal's, Khalil's and Das' Time Crystals A study published in 2016, based on a modification to Heisenberg's algebra and applied to atomic systems, suggests that time behaves like a crystal, that is, time can be seen as a discrete emission that arises in space and not as a wave keep going (Mir Faizal et al.-2016). According to these authors, the size of the crystals resulting from the rhythm of spontaneous emission of hydrogen atoms, would be of the order of 10 -17s, a magnitude much greater than the Planck time but similar to the uncertainty in the measurement of the interaction time between a graviton of dimension equal to that of the H atom with which come into collision: ≈

∙

∙

≈

(F.6)

According to the authors, in the same year, these crystals were observed in Floquet systems that are not in equilibrium and in disordered dipolar systems. This result contradicts Treder conclusions based on the Planck units and, instead, reinforces the result obtained here. As we have seen before, introducing the new constant Sp in its New Uncertainty Principles reaches these same magnitudes. Because of the characteristics thereof, the time uncertainty of the graviton-H atom interaction can be considered a time crystal since it is the fourth dimension of the interaction introduced by Bohr and Rosenfeld to mediate the field components and it is discontinuous, since it transmits the impulse to the body in orbit in a discrete way. These studies (that of Treder and that of the time crystals), have a common antecedent in physicist Bruce DeWitt. In fact, the uncertainty in the relativistic metric gik was also found by him (Treder HJ.- 1979) and in the case of the time crystals, they were predicted by a deformed version of the Wheeler-DeWitt equation (Mir Faizal et al.-2016).


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Fundamentals of the compatibility between fields G-Conclusions The proton and the neutral H atom at its fundamental level have turned out to be the basic components from which to develop a more complete Quantum Field Theory of Gravity that encompasses masses of larger magnitudes and dimensions in later investigations. The mass of the proton has turned out to be a valid "reference mass" to replace the electric charge unit, although unlike this, which is a relativistic invariant, the reference mass depends on the speed when it is below 1/3 of the speed of light. In spite of this, it has been possible to demonstrate with reasonable certainty through the verification with the reality, that the combination of the constants G and Sp, necessary to compute the graviton energy, is present in the Cosmos structure, although if G continues to be valid to determine the global intensity of the Newtonian field induced by a mass and its interaction with other masses. Regardless of the constant of the energy equation, the conditions of a graviton-mass interaction seem to reveal that quantized gravitational fields do not act on the entire mass of cosmic bodies, but on fractions of it of the same dimension of the neutral, non excited, H atom, so it is possible that also follow the rules of quantization of electromagnetic fields, what can explain the discretizations observed in the Cosmos. As we have seen, the new constant also allows distinguishing the quantum gravity theory from the classical theory, contrary to the assumption by Rosenfeld and Treder, as a consequence of using a wrong constant in their equations. The discretization found in the Cosmos by Agnese, Festa and Chandrasekhar would be the demonstration. It let to explain the quantization of the geometry of the Universe, also known as the space-time "structure of quantum-geometric foam" (Treder, H-J.-1979). The quantum energy of the gravitational oscillators that make up the fields, could, perhaps, explain the dark matter detected in the Universe. The replacement of the constant G in the Treder’s New Uncertainty Principles, has allowed proving the validity of the constant αg found by Agnese and Festa and, for the first time, to found a physical foundation for it based in well known physical theories for it. However, the gravitational quantum energy equation shows that it depends on the density of the body mass in orbit and therefore αg does not seem to be a true constant, but rather a universal average value. If a complete theory based on these fundamentals, confirms that the discretizations found in the Cosmos are due to the uncertainty in the relativistic metric gik, as proposed by Treder, it will have confirmed his proposal to unify the General Theory of Relativity with the Fields Quantum Mechanics, avoiding the need to reconcile the logical and mathematical formalisms of both theories, which are so dissimilar (Treder


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Fundamentals of the compatibility between fields HJ.-1979). A good beginning for the development of this theory seems to be the work of Bronstein before mentioned. The next stage of this project is to analyze the quantization of the orbits in the Solar System, taking advantage of the precision with which the necessary data for the application of the equations here found are known.

Addendum A.1- Units of measures system used in this study International System of Units, derived from the MKS System.

A.2 - List of the celestial bodies of the Solar System considered in this study Planets (9): Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto. Natural satellites (36): Moon, Phobos, Deimos, Métis, Adrastea, Amalthea, Thebe, Io, Europe, Ganymede, Callisto, Leda, Himalia, Lysithea, Elara, Ananke, Carme, Pasiphae, Sinope, Mimas, Enceladus, Tethys, Dione, Rhea, Titán, Hyperión, Iapetus, Phoebe, Miranda, Ariel, Umbriel, Titania, Oberon, Triton, Nereid, Charon.

A.3- Bibliography Abbott B.P. et al. (2016) – “Tests of general relativity with GW150914” - arXiv:1602.03841 – 9 June 2016 Agnese A.G., Festa R. (1997) - “Clues to discretization on the cosmic scale – Physics Letters A 227, 165171 – Elsevier. Agnese A.G., Festa R. (1998) – “Discretization on the Cosmic Scale Inspired from the Old Quantum Mechanics” - arXiv:astro-ph/9807186v1, Cornell University Library. Alonso M., Finn E.J. (1996) – “Physics” – Addison Wesley – Ed. 1996 Berestetskij, V.B., Lifsits, E.M., Pitaevskij,L.P. (1978) – “Teoria quantistica relativistica” – Física Teorica 4 di Landau L.D. e Lifsits E.M. – Editori Reuniti, Roma (Italia) Edizioni Mir. Bohr N., Rosenfeld L. (1933) - “On the question of the measurability of electromagnetic field quantities” - Copenhagen -Selected papers of Leon Rosenfeld, eds. R.S.Cohen and J.Stachel - Translated by Prof. Aage Petersen. Chandrasekhar, S. (1937) – “The Cosmological Constants” – NATURE, May 1 1937 th

Chandrasekhar, S. (1983) – “On stars, their evolution and their stability” – Nobel lecture, December 8 , 1983, The University of Chicago, Chicago, Illinois, USA Gorelik Gennady (1992) – “First steps of Quantum Gravity and the Planck Values” – Studies in the history of General Relativity (Einstein Studies Vol 3) – Eds. Jean Eisenstaedt, A.J. Kox. P. 364-379. Greenberger D. M. (1983) - “Quantization in the large”, Foundations of Physics, Vol. 13, nº 9. Mir Faizal, Mohammed M. Khalil, Saurya Das (2016) - “Time crystals from minimum time uncertainty”The European Physical Journal C. Olive K. et al. (2014) – “Summary Table of Particles Properties” - Extracted from the Particle Listings of the Review of Particle Physics, Chin. Phys. C38, 090001 - Available at http://pdg.lbl.gov


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Fundamentals of the compatibility between fields Randolf Pohl et al. (2010) - “The size of the proton”, Nature (466) - 8 July 2010. Rosenfeld L. (1965) - ”Quantum theory and gravitation” – Conference in NORDITA, Copenhagen and at the Einstein Symposium of Nov.2-5, 1965, Berlin, Germany. Sixth Solvay’s Congress (1930)– Einstein’s second criticism - Bohr-Einstein debates, Wikipedia https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates Treder H-J (1979) – “On the problem of physical meaning of quantization of gravitational fields”, Albert Einstein 1879-1979. Relativity, quanta and cosmology in the Development of the scientific thought of Albert Einstein, Authors: Pantaleo, Mario y de Finis, Francesco, Johnson Reprint Corporation, New York. Wilson, Albert G., Edelen Dominic G.B. (1967) – “Homogeneous cosmological models with bounded potential” – The Astronomical Journal, Vo,l. 151, page 1.171, March 1968


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