Quantum Neutron Unit Gravity

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 0 _ unit n GBE 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 0 _ unit n GBE divided by a neutron’s rest mass energy represents a fundamental, dimensionless proportionality constant, 9.4252 × 10, 0 _ unit n k . The square root of 0 _ unit 2 n k , G α , 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 αG and 0 _ unit n GBE 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.

uing incomplete process. Many different models have been proposed, but none have seamlessly united quantum, classical and cosmologic phenomena [7]. Neutron, n 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 [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 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 [17] [18]. Some theoretical studies consider NSs as pure n 0 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 )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 [ 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, mc 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 n ν , (by dividing the energy by Planck's constant h) or in terms of n 0 Compton wavelength, 0 n c ν . Therefore gravity can be scaled as a single natural unit system that is inter-related by integer multiples or integer fractions of n 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.

Virtual Neutron Gravity Energy Unit, GBEn0_unit
We define the simplest virtual n 0 based natural unit gravitational system, GBE n0_unit , as the GBE of virtual paired n 0 masses separated by

Virtual Neutron Gravity Dimensionless Coupling
Proportionality Constant Unit, kn 0 _unit GBE n0_unit J is related to a proposed dimensionless coupling proportionality constant ratio with the n 0 _unit in J, with

kn 0 _unit, Planck Time, and Hydrogen
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 (a 0 ), and GBE of the electron in hydrogen; respectively v n0 , v p+ , v e− , 0 a v , 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 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 2 P t , 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 2 From Equation (5)

Proposal of a Gravitational Constant Analogous of the Fine Structure Constant, αG
We also find interest in computing the square root of 0 _ unit 2 n k , 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 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 c α × 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 n 0 -system is the mass of n 0 times the product of (α G × c) squared divided by 2. This equals  The value of α G equals the product of square root of 2, 0 n v , 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 elec-

Definition of the Newtonian Gravitation Constant in Neutron Units
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.

Gravitational Binding Energy of Any System from an Integer Neutron Perspective
The GBE can be calculated for any system using the integer unit n 0 perspective by Equation (11).

An Example of the Transition of a Symmetric Binary Pure Neutron NS System to a BH
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 c 2 times c × s divided by the product of 2 times the gravitational Figure 1. Transformation of a symmetric binary neutron star system to a black hole. Figure 1 is a plot showing the transformation of the binary symmetric neutron system to a black hole with a Schwarzschild radius of c × s. The X-axis is log 10 # of n 0 . The Y-axis is in log 10 Joules of the annihilation energy of the # of n 0 , the upper blue line; and 2 times the gravitational binding energy, GBE, of a system with # of n 0 separated at a distance of c × s, the lower green line, Equation ( 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.

Discussion
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 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 . Perhaps the quantization of the gravitational field may be discovered in our approach.

Conclusions
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 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-relationships just as electromagnetic systems are with the fine structure constant as in Equation (11).