Developing a quantum model of gravity has been an important conceptual goal in physics [1] - [6] . Uniting cosmologic physics and quantum physics is a continuing incomplete process. Many different models have been proposed, but none have seamlessly united quantum, classical and cosmologic phenomena [7] . Neutron, n⁰, 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 [8] - [14] . Many new observations are being acquired that lend themselves to interpretations of the basic properties and scaling of gravitational systems. Recent gravity wave measurements are dependent upon a robust gravity model [15] [16] .

A NS is related to the transformation of atomic matter, in the simplest case hydrogen, H, to a degenerate n⁰ 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 [17] [18] . Some theoretical studies consider NSs as pure n⁰ matter [19] . Some models describe the core as superfluid neutron-degenerate matter (mostly neutrons, with some protons and electrons). About 5% of all known NSs are in binary systems [20] . In some theories of binary evolution it is expected that NSs also exist in binary systems with a BH. It has been proposed that coalescence of binaries consisting of two NSs may be responsible for producing short gamma-ray bursts [21] .

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})c² 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 [22] [23] .

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 [24] . In pair product there are photon and two anti-particles in which the total energies are scaled equally. In anti-matter matter annihilation there are two matter entities and two photons. The number 2 reflects the paired physical state relationship. Both are associated with physical transformation settings where the velocity of matter is related to the speed of light, c, and the annihilation energy of the rest mass, mc². 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 n⁰, 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, , (by dividing the energy by Planck’s constant h) or in terms of n⁰ Compton wavelength,. Therefore gravity can be scaled as a single natural unit system that is inter-related by integer multiples or integer fractions of n⁰ 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 n⁰ based natural unit gravitational system, GBE_{n0_unit}, as the GBE of virtual paired n⁰ masses separated by, a distance that is equal to the speed of light divided by the annihilation frequency of the neutron,. The GBE in Joules of this system, GBE _{n0_unit} J, 1.4188 × 10⁻⁴⁹ J, is described in Equation (3).

Here, denotes the mass of the neutron, 1.6749 × 10⁻²⁷ kg, G, the universal Newtonian gravitational constant, and c the speed of light. v_n0 represents the frequency equivalent of the neutron, 2.2719 × 10²³ Hz,. is the n⁰ Compton wavelength 1.3196 × 10⁻¹⁵ m. values in other physical unit equivalents include frequency 2.1413 × 10⁻¹⁶ Hz (), Compton wavelength 1.4001 × 10²⁴ m, and mass 1.5787 × 10⁻⁶⁶ kg.

GBE_{n0_unit} J is related to a proposed dimensionless coupling proportionality constant ratio with the n⁰_unit in J, with equaling 9.4252 × 10⁻⁴⁰ from Equation (4) in multiple physical units. Here n⁰ J represents the Joule equivalent of n⁰, 1.5053 × 10⁻¹⁰ J. This proportionality constant is referred to as, where times any physical unit value of the neutron equals in the corresponding unit.

Our proportionality constant, k_{n0_unit} is also defined by several equivalent relationships involving GBE_{n0_unit}; Planck’s constant, h; Planck time, t_P; and the frequency equivalents of the neutron, proton (p+), electron (e⁻), Bohr radius (a₀), and GBE of the electron in hydrogen; respectively v_n0, v_p+, v_e−, , v_{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 n⁰ beta decay products represent the closest natural gravitational system to the virtual pure n⁰ system. Thus we find that the n⁰ serves as a fundamental unit for the origin of all of the hydrogen products as well. The product of, the frequency of one mass, the frequency of the other mass, the frequency of the distance equals the GBE in Hz. In the neutron unit setting the k_{n0_unit} is related to the product of and, divided by v_n0, therefore the product of and, Equation (5).

From Equation (5) the ratio of v_{GBE_n0_unit} divided by v_{GBE_e−} is related to the ratio of quantum entities of cubed divided by the product of v_p+, v_e−, and, 7.3836 × 10⁷, Equation (6).

We also find interest in computing the square root of, 4.3417 × 10⁻²⁰, 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 a₀. 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 squared divided by two is equal to the ionization binding energy of hydrogen, which is the Rydberg constant equivalent. The kinetic energy and the binding energy are similar in concept to the transformation point to a BH in Equation (2). An identical gravitational mathematical property will be shown later in Equation (8).

The kinetic energy of this virtual n⁰-system is the mass of n⁰ times the product of (α_G × c) squared divided by 2. This equals, Equation (8). Thus we find the Gravitational Binding energy neutron unit exactly equals the kinetic energy of the binary neutron pair identical to the pattern seen with one definition of α. This pure n⁰-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 squared times the frequency equivalent of the GBE of the electron in hydrogen, v_{GBE_e−} divided by the product of v_p+, v_e−, and from Equation (7). This unites gravitational scaling and multiple other fundamental quantum entities.

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 × 10³² m; or 1.9216 × 10⁻⁵⁷ J. The product of and v_{GBE_e−} equals 0.65885 Hz² so v_{GBE_e−} is closely scaled by the mathematical scalar reciprocal of. Whence v_{GBE_e−} divided by the product of v_p+, v_e−, and also equals the fundamental gravitational, quantum, and relativistic constant Planck time squared,. Planck time (h not the reduced Planck’s constant, ħ) equals 1.35141(08) × 10⁻⁴³ s. All frequency equivalent gravitational systems fulfill the relationship of the ratio including that of hydrogen and v_{GBE_n0_unit}, Equation (7).

The value of α_G equals the product of square root of 2, , and t_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_{n0_unit} can be calculated from ratio/product relationships of as seen in Equation (8).

As an added and quite surprising result the Newtonian gravitational constant G can be completely defined in the defined units of the n⁰ 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 n⁰, 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 a₀ or α.

The GBE can be calculated for any system using the integer unit n⁰ perspective by Equation (11). is the integer number of equivalent neutrons of one mass, #1n⁰. Also equals the integer number of neutron equivalences of the second mass, #2n⁰. We define #λ_n0 as the integer number of n⁰ 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 Figure 1. r_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 c² times c × s divided by the product of 2 times the gravitational

constant G, which computes to a total mass of 2.0186 × 10³⁵ kg from Equation (12). This is described as. Other unit equivalents of are: 1.8141 × 10⁵² J; frequency of 2.7379 × 10⁸⁵ Hz; Compton wavelength of 1.0950 × 10⁻⁷⁷ m; and integer number of neutrons 1.2051 × 10⁶²n⁰. We evaluate the GBE of a paired neutron system starting with 1 n⁰ to 1.2051 × 10⁶² n⁰, or 2.0186 × 10³⁵ 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 n⁰ masses to a BH. Since the GBE increases with the square of the number of n⁰ there is a point where the two lines and energy values converge, Figure 1. This is the transformation point to a BH. At neutron star pairing with a mass of 2.0185 × 10³⁵ 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 n⁰-based system.

These results show how gravitation systems, indeed the Newtonian Gravitational constant itself, can be completely scaled by a quantum or n⁰-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_{n0_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 n⁰-equivalents. The time unit is one second divided by, 4.4017 × 10⁻²⁴ s. The distance unit is λ_n0, equals 1.3196 × 10⁻¹⁵ 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 for the distances in a purely n⁰-unit gravitational system, Equation (11). Clearly, all possible GBEs are also only integer or integer fraction values of. Perhaps the quantization of the gravitational field may be discovered in our approach.

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_{n0_unit} and α_G, of a pure neutron gravitational system, are defined by four quantum entities, and the n⁰, the speed of light, the products of negative n⁰ beta decay, p⁺, e⁻, a₀, v_{GBE_e−}, as in Equations ((5), (7)). Newton’s Gravitational constant G, can be defined solely in n⁰-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 n⁰. 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