Quantum Theory of Gravity - "QTG"

                One important result of the postulates of the Theory of Relativity is the relativistic nature of simultaneity. It has been shown experimentally that time progresses at different rates at reference points in relative motion. The relativity of time means that a given pair of events will occupy different time intervals for different observers. If we adopt the Bohr atomic model, the velocity of an electron in orbit around a nucleus will mean that it experiences temporal dilation in relation to that nucleus. In order to determine this dilation, we must establish the velocity of the electron, thus entering in conflict with the Uncertainty Principle, which establishes that a particle does not have a well-defined simultaneous position and momentum. It will be shown that the imprecision of the present time of the particle – the Temporal Uncertainty Principle – can take the place of the Uncertainty Principle.

                Where: ħ is Planck’s constant divided   by  2π  =  1.054572669125101932 x 10^{-34 kg.m2}/s,

                            Δx   is the imprecision of position [m], and

                            Δp   is the imprecision of momentum [kg.m/s].

                Where:  r_Bohr  is the Bohr radius of the atom [m].

                As every natural physical system tends towards its most stable state, an atom’s size is optimized so as to minimize its total energy. We can therefore simplify the calculation by using the mass of the electron instead of the reduced mass, calculating the potential energy “U”, as follows:

                Where:  q is the elementary electrical charge   = 1.60217733 x 10^-19   A.s, and

                        ε_o: is the electrical permissivity of a vacuum = 8.85418781762 x10^-12  Coulomb²/N.m²  or   (A².s⁴)/(kg.m³)

                Where: m_electron is the relativistic mass of the electron [kg].

                To find the atom in its most stable state, we must find the Bohr radius that minimizes total energy. We must therefore take the derivative of the energy in relation to the Bohr radius.

               

 

                Returning to the formulation of the Uncertainty Principle and separating the imprecision of momentum into its two parts, we have:

               

                Where: υ_electron is the velocity of the electron.

                Where:  c  is the velocity of light = 2.99792458 x 10⁸ m/s,

                    m_{o electron}  is the rest mass of the electron = 9.1093897 x 10^-31  kg.

               
                This is the velocity of the electron for the most stable state of the hydrogen atom. We can call it “the velocity of the electron with no gravity-time influence beyond its own atom”. As will be shown, it is the velocity of the electron in a universe that consists of a single hydrogen atom.

                In order to determine the Bohr radius, we must again ignore the impositions of the Uncertainty Principle, as a result of the impossibility of a particle having well defined position and momentum at a given instant. Solving equation 13, we find:

                r_Bohr  =  5.29163167614727303014376 x 10^{-11 m}

                We can verify that this υ_electron and this r_Bohr are in accordance with the condition for stability of the electron established by Quantum Mechanics, where the nuclear forces (Coulomb force and centripetal force) must be equal. Solving partially, we have the following expressions for the Coulomb force and for the centripetal force:

                Substituting the relativistic expressions for the mass and radius of the electron (equations 12 and 13), we obtain the following for the Coulomb force:

               
 

                For the centripetal force, we have:

               

 
                Regrouping, we find:

                Simplifying and solving, we find exactly the same relation for the velocity of the electron as in equation 14, demonstrating that this relation also yields equilibrium of the nuclear forces using the relativistic values.

                Electrons in orbit around an atomic nucleus undergo temporal dilation as a result of their velocity (according to the Special Theory of Relativity) or as a result of their acceleration (according to the General Theory of Relativity). We can calculate the value as follows:

                Considering an arbitrary time variation of 1 second in the nucleus, we have:

                Time variation at nucleus   =   Δ _{Time Nucleus}    = 1 second.

                From the General Theory of Relativity, we have the following relation for time change in the electron cloud (Δ _{Time Electroncloud}):

                Under both the General and Special Theories of Relativity, we arrive at the same result, because both lead to exactly the same formulation in a rotating system. We thus have a time difference of:

                Δ _Time  =   Δ _{Time Electron} – Δ _{Time Nucleus}     =   26.63....µs.

                For the sake of curiosity, and to illustrate the work with more practical examples, let us examine the distance represented by this time dilation of the electron in relation to the nucleus:

                Distance traveled  =  Δ _Time x υ_electron  =  58.25...m.

                In terms of orbits around the nucleus, this distance is equivalent to:

                With each second that passes, this electron is going 26.63..µs out of phase with the local time of the nucleus, or with the external time reference, if we take the nucleus to be the only time reference external to the electron.

                Working from the limitation of the speed of light, we can verify that all the information received at our point of reference in the present is derived from the past. In other words, everything around us, irrespective of the distance, is in the past, which is to say that from any other point, in relation to everything else, our point of reference is in the future. One of the properties of atoms, therefore, is that time passes more slowly in the electron cloud, polarizing the time of the nucleus towards the future and thus establishing a direction for time. This could be described as being one of the pillars of the Quantum Theory of Gravity.

                In order to assimilate the ideas set out here, it is essential to patiently follow them in their logical sequence. Understanding the different parts in isolation will not lead to a full grasp of the theory: only as a whole will it be comprehensible.

                A certain degree of repetition is inevitable in the interests of clarity, although perhaps at the cost of greater elegance. In order to better illustrate the quantities involved and their small differences, equations have been used wherever possible and/or necessary.

                As the work proceeds, it will become clear that the subtle connection that brings together the Theory of Relativity and Quantum Mechanics depends on an understanding of how atoms generate gravity and time. Many issues not previously understood, or at best poorly explained, find a satisfactory solution in this theory.