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Quantum gravity and the transformation of a neutron star or the merger of two neutron stars into a black hole are important topics in cosmology. According to the Schwarzschild radius relationship, a black hole arises when two times
of the
gravitational binding energy of the gravitational system, GBE, equal the annihilation energy of its total mass. From a quantum perspective, the integer number of neutrons defines the GBE and mass in the merger of binary pure neutron stars transforming to a black hole. Therefore, one can scale all gravitational binding energy relationships by using neutron mass, energy, distance, time, or frequency equivalents. We define
of the neutron as the binding energy, 1.4188 × 10^{󔼹} J, of a virtual system of two neutrons separated by the neutron Compton wavelength. The
divided by a neutron’s rest mass energy represents a fundamental, dimensionless proportionality constant, 9.4252 × 10^{󔼰},
. The square root of
, α_{G}_{}
, which we introduce here as a coupling constant
,
is identical in concept to the fine structure constant found in electromagnetic physics, but for gravity. Both α_{G} and
inter-relate the neutron, proton, electron, Bohr radius, speed of light, Planck’s constant, GBE of the electron in hydrogen, and Planck time. This paper demonstrates a direct conceptual and computational rationale of why the neutron and its negative beta decay quantum products accurately can represent a quantum gravitational natural unit system.

Developing a quantum model of gravity has been an important conceptual goal in physics [^{0}, neutron stars, NS, and black holes, BH, are ideal conceptual, computational, and observational settings to interrogate gravitational properties at the extremes of short and long distances, and high gravitational binding energies, GBEs [

A NS is related to the transformation of atomic matter, in the simplest case hydrogen, H, to a degenerate n^{0} state driven by gravity. NSs can be characterized by a few physical factors since they are composed predominately of neutrons, and these are referred to as equations of state [^{0} matter [

A BH is a mathematically defined region of spacetime exhibiting such a powerful gravitational force that no particle or electromagnetic radiation can escape. The classic Schwarzschild radius equation states that a BH arises when the annihilation energy of the total mass with a Schwarzschild radius of r_{s}, (mass_{BH}_{_rs})c^{2} in joules, equals two times GBE in joules for any BH horizon measuring r_{s} (see Equation (1) and Equation (2)). From a quantum perspective of a pure binary NS system transformation to a BH, Equation (2) is a mathematical imperative expressing the equivalence between the integer number of neutrons defining the annihilation mass and the paired masses defining the GBE [

or

This is a specialized physical/mathematical setting where two entities, annihilation energy of a mass and a different form of energy, two times of the binding energy in this case, must be identically scaled in a paired relationship. It leads to a dimensionless quantum number of neutrons relationship defining both simultaneously. Another similar physical single and paired example would be related to matter/anti-matter pair production which is also important in the properties of BHs [^{2}. In the transformation of a binary system of NSs to a BH, the GBE scaling can be defined equivalently and solely in integer multiples of the energy of n^{0}, equaling 1.5053 × 10^{−10} J, which is a natural energy unit. This energy unit could be also expressed in term of its equivalent in frequency, ^{0} Compton wavelength,^{0} units.

The goal of this paper is to evaluate the specific units from a quantum integer perspective of a natural scaling based solely on the neutron, which define a system of gravity that is conceptually and computationally consistent with standard gravitation models and methods for NSs and BHs.

We define the simplest virtual n^{0} based natural unit gravitational system, GBE_{n}_{0_unit}, as the GBE of virtual paired n^{0} masses separated by_{ n}_{0_unit }J, 1.4188 × 10^{−49} J, is described in Equation (3).

Here, ^{−27} kg, G, the universal Newtonian gravitational constant, and c the speed of light. v_{n}_{0} represents the frequency equivalent of the neutron, 2.2719 × 10^{23} Hz,^{0} Compton wavelength 1.3196 × 10^{−15} m. ^{−16} Hz (^{24} m, and mass 1.5787 × 10^{−66} kg.

GBE_{n}_{0_unit }J is related to a proposed dimensionless coupling proportionality constant ratio with the n^{0}_unit in J, with ^{−40} from Equation (4) in multiple physical units. Here n^{0} J represents the Joule equivalent of n^{0}, 1.5053 × 10^{−10} J. This proportionality constant is referred to as

Our proportionality constant, k_{n}_{0_unit} is also defined by several equivalent relationships involving GBE_{n}_{0_unit}; Planck’s constant, h; Planck time, t_{P}; and the frequency equivalents of the neutron, proton (p+), electron (e^{−}), Bohr radius (a_{0}), and GBE of the electron in hydrogen; respectively v_{n}_{0}, v_{p}_{+}, v_{e−}, _{GBE}_{_e} in Equation (5). The v_{GBE}_{_e} is derived and valid identical to the Bohr radius derivation. These are also natural unit constants. One is defined by the degenerate neutron state and the other equivalently by the hydrogen state that is non-degenerate. Additionally, the negative n^{0} beta decay products represent the closest natural gravitational system to the virtual pure n^{0} system. Thus we find that the n^{0} serves as a fundamental unit for the origin of all of the hydrogen products as well. The product of_{n}_{0_unit} is related to the product of _{n}_{0}, therefore the product of

From Equation (5) the ratio of v_{GBE_n}_{0_unit} divided by v_{GBE_e}_{−} is related to the ratio of quantum entities of _{p}_{+}, v_{e}_{−}, and^{7}, Equation (6).

We also find interest in computing the square root of^{−20}, a dimensionless constant similar to the fine structure constant, α, in both concept and computation, but for gravitational, non-electromagnetic physical systems. It is described herein as α_{G}, Equation (7). The fine structure constant α is related to the electromagnetic binding energy of the electron in H, and involves p+, e^{−}, and a_{0}. Additionally α_{G} represents the ratio of a velocity divided by c that is also a β value from Special Relativity. This is described as β_{G} for this specific setting, just as α represents a specific β for hydrogen. The mass of the electron times the product of

The kinetic energy of this virtual n^{0}-system is the mass of n^{0} times the product of (α_{G} × c) squared divided by 2. This equals^{0}-based unit gravitational system is a virtual system because it does not have a significantly long lifetime. Our derived constant α_{G} equals the square root of the product of 2 and _{GBE_e}_{−} divided by the product of v_{p}_{+}, v_{e}_{−}, and

The mathematical connection with the closest natural atomic system, which represents hydrogen, shows that v_{GBE}_{_e} is a fundamental constant with similar importance to quantum gravity, and is analogous to the significance of the electromagnetic binding energy of the electron in H to photon and electromagnetic systems. This constant cannot be measured, but it does not diminish its physical significant. In fact, the Bohr radius, rest mass of the electron, and the Rydberg constant that define the hydrogen quanta cannot be experimentally measured either. This fundamental frequency v_{GBE}_{_e−} equals, 2.9000 × 10^{−24} Hz; equivalently 2.1381 × 10^{−74} kg; 1.0337 × 10^{32} m; or 1.9216 × 10^{−57} J. The product of _{GBE_e}_{−} equals 0.65885 Hz^{2} so v_{GBE_e}_{−} is closely scaled by the mathematical scalar reciprocal of_{GBE_e}_{−} divided by the product of v_{p}_{+}, v_{e}_{−}, and ^{−43} s. All frequency equivalent gravitational systems fulfill the relationship of the _{GBE_n}_{0_unit}, Equation (7).

The value of α_{G} equals the product of square root of 2, _{P} as derived in Equation (7). Thus α_{G} is inter-related with the quantum gravitational property of H in an analogous fashion as α is inter-related to both the quantum and electromagnetic properties of H. Now GBE_{n}_{0_unit }can be calculated from ratio/product relationships of

As an added and quite surprising result the Newtonian gravitational constant G can be completely defined in the defined units of the n^{0} mass, its Compton wavelength, its unit GBE; or with c and neutron frequency equivalent as in Equation (9).

Equivalently, the Newtonian gravitational constant G can also be defined in units of n^{0}, c, h, t_{P}, v_{p}_{+}, v_{e}_{−}, v_{a0}, v_{GBE_e}_{−},and neutron beta decay products in H as shown in Equation (10). This is analogous to the inter-relationship of multiple fundamental quantum constants that can define a_{0} or α.

The GBE can be calculated for any system using the integer unit n^{0} perspective by Equation (11). ^{0}. Also ^{0}. We define #λ_{n}_{0} as the integer number of n^{0} Compton wavelengths of the masses separation distance, λ. All of the # values are dimensionless integers.

Consequently we have just established the transformation of binary symmetric NSs of identical mass, merging into a BH as in _{s} is arbitrarily chosen to be c × s for mathematical simplicity, and it is a fundamental unit distance linked to all of the other constants. A Schwarzschild radius, r_{s}, of c × s equals 299,792,458 m is utilized to demonstrate a relevant cosmological dimension in the Schwarzschild ratios shown in Equation (12). The mass of such a BH equals the product of c^{2} times c × s divided by the product of 2 times the gravitational

constant G, which computes to a total mass of 2.0186 × 10^{35} kg from Equation (12). This is described as^{52} J; frequency of 2.7379 × 10^{85} Hz; Compton wavelength of 1.0950 × 10^{−77} m; and integer number of neutrons 1.2051 × 10^{62}n^{0}. We evaluate the GBE of a paired neutron system starting with 1 n^{0} to 1.2051 × 10^{62} n^{0}, or 2.0186 × 10^{35} kg for the mass. The product of 2, the mass squared and G divided by c × s equals twice the GBE of each individual incremental pair neutron systems, Equation (13).

The factor 2 is taken into account in these calculations to define the point of transition of the binary n^{0} masses to a BH. Since the GBE increases with the square of the number of n^{0} there is a point where the two lines and energy values converge, ^{35} kg 2 times the GBE equals the annihilation energy of the mass fulfilling the mathematical definition of a BH, Equations ((2), (13)). This demonstrates that binary gravitational systems from any pairing of two neutrons, two stars, and two NS, and one BH can be described by a quantum n^{0}-based system.

These results show how gravitation systems, indeed the Newtonian Gravitational constant itself, can be completely scaled by a quantum or n^{0}-unit system (see Equations ((9)-(11)).) This is an imperative of the mathematical definitions of pure NSs transforming to simplest non-rotating BHs. The GBE of any system represents dimensionless integer fractional values of the neutron as the base unit, which provides dimensionless ratio coupling constants defined by k_{n}_{0_unit} and α_{G}. Surprisingly, these are related to quantum subatomic constants derived from the frequency equivalents of the neutron, hydrogen, and t_{P}, (Equation (10)). This demonstrates a system that bridges from quantum to gravitational. Similarly both α and α_{G} represent conceptually and computational parallel β values, but are relegated to widely divergent physical settings. The fine structure constant α relates to electromagnetic relationships of hydrogen, whereas α_{G} is for neutron gravitational relationships. The electromagnetic binding energy of the electron in hydrogen, related to the Rydberg constant, is a fundamental scaling factor for all electromagnetic relationships including α. The GBE of the electron in hydrogen plays an identical role for neutron quantum gravity. Though our derived GBE_e-value is not directly observable it still represents an important fundamental inter-relational constant identical in concept to the other hydrogen quanta.

The time and distance units of gravity may also be defined by n^{0}-equivalents. The time unit is one second divided by^{−24} s. The distance unit is λ_{n}_{0}, ^{−15} m. Any gravitational system can be interrogated where time, distance, matter, and energy can all be evaluated as one of these quantum states. Likewise in quantum physics, where there are only specific quantum states possible, the only possible gravitational states are related to integral numbers of neutrons for the masses and ^{0}-unit gravitational system, Equation (11). Clearly, all possible GBEs are also only integer or integer fraction values of

In conclusion, mathematical definitions of pure Neutron Stars and Black Holes define an imperative that the scaling of the gravitational binding energy must be defined in quantum unit equivalents of the neutron. The dimensionless ratios, k_{n}_{0_unit} and α_{G}, of a pure neutron gravitational system, are defined by four quantum entities, and the n^{0}, the speed of light, the products of negative n^{0} beta decay, p^{+}, e^{−}, a_{0}, v_{GBE_e}_{−}, as in Equations ((5), (7)). Newton’s Gravitational constant G, can be defined solely in n^{0}-units and/or the speed of light, t_{P}, and hydrogen quantum values as in Equations ((9), (10)). Therefore gravity can be viewed as being scaled by the neutron which is an imperative in the NS BH transition. Time (chronon), matter (second quantization), distance (Planck scales), and energy (Rydberg spacing) also must be quantum integer based on n^{0}. Since all of the dimensions of a gravitational system are scaled by a single natural unit, the neutron, the system can be computationally analyzed as dimensionless inter-re- lationships just as electromagnetic systems are with the fine structure constant as in Equation (11).

We would like to thank Richard Vento, Columbus State Community College― retired for his help in the preparation of the paper.

Chakeres, D. and Andrianarijaona, V. (2017) Quantum Neutron Unit Gravity. Journal of High Energy Physics, Gravitation and Cosmology, 3, 267-276. https://doi.org/10.4236/jhepgc.2017.32022