_{1}

Did any physics experts expect SUPERRELATIVITY paper, a physics revolution producing the EINSTEIN-RODGERS RELATIVITY EQUATION, producing the HAWKING-RODGERS BLACK HOLE RADIUS, and producing the STEFAN-BOLTZMANN-SCHWARZSCHILD-HAWKING-RODGERS BLACK HOLE RADIATION POWER LAW, as the author gave a solution to The Clay Mathematics Institute’s very difficult problem about the Navier-Stokes Equations? The Clay Mathematics Institute in May 2000 offered that great $million prize to the first person providing a solution for a specific statement of the problem: “Prove or give a counter-example of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes Equations.” Did I, the creator of this paper, expect SUPERRELATIVITY to become a sophisticated conversion of my unified field theory ideas and mathematics into a precious fluid dynamics paper to help mathematicians, engineers and astro-physicists? [1]. Yes, but I did not expect such superb equations that can be used in medicine or in outer space! In this paper, complicated equations for multi-massed systems become simpler equations for fluid dynamic systems. That simplicity is what is great about the Navier-Stokes Equations. Can I delve deeply into adding novel formulae into the famous Schwarzschild’s equation? Surprisingly, yes I do! Questioning the concept of reversibility of events with time, I suggest possible 3-dimensional and 4-dimensional co-ordinate systems that seem better than what Albert Einstein used, and I suggest possible modifications to Maxwell’s Equations. In SUPERRELATIVITY, I propose that an error exists in Albert Einstein’s Special Relativity equations, and that error is significant because it leads to turbulence in the universe’s fluids including those in our human bodies. Further, in SUPERRELATIVITY, after I create Schwarzschild-based equations that enable easy derivation of the Navier-Stokes Equations, I suddenly create very interesting exponential energy equations that simplify physics equations, give a mathematical reason for turbulence in fluids, give a mathematical reason for irreversibility of events with time, and enable easy derivation of the Navier-Stokes Equations. Importantly, my new exponential Navier-Stokes Equations are actually wave equations as should be used in Fluid Dynamics. Thrilled by my success, I challenge famous equations by Albert Einstein and Stephen Hawking [2] [3].

SUPERRELATIVITY is my unified field theory of physics entitled ZERO RELATIVITY, that had possibly taken me eight thousand hours to create, but I improved it as I adapted it for the Navier-Stokes Equations of Fluid Dynamics [

Obstructive problems, for mathematicians attempting to understand or improve the Navier-Stokes Equations, are many. Without realizing the fact, mathematicians, for nearly two hundred years, have been confused about what the dimensions are that should be used in Navier-Stokes Equations. Schwarzschild’s metric equation can be used to give dramatic changes that should be made to the Navier-Stokes Equations [

In the 17^{th} century, Rene Descartes created the very useful Cartesian coordinates. Points on the Cartesian plane can be identified with n real numbers with the Cartesian product R^{n}. Mathematics students usually understand Cartesian coordinates. Some understand complex numbers in the form a + bi in which i is the imaginary number so i^{2} = −1. Mathematicians can create infinite coordinate systems. Co-ordinates matter for reversibility or irreversibility of events with time. Should we use a 3-D, 4-D, or 12-D system? [

A major problem for mathematical physicists has been that they do not have an adequate theoretical gene- ralization from which the current NAVIER-STOKES EQUATION, for fluid dynamics, can be derived.

For more than a hundred years, most physicists have followed Albert Einstein who chose to use dimensions in coordinate systems for reversibility of events with time. Do you believe in reversibility of events with time? Or do you believe in irreversibility of events with time? Because I believe in irreversibility of events with time, in this paper, I give dimensions in coordinate systems that give the popular reversibility, then dimensions in coordinate systems that give the less popular irreversibility, and expand upon the mathematics to give equations similar to the Navier-Stokes Equations [

What are also very interesting are my changes to Maxwell’s Equations in my unified field theory of physics entitled ZERO RELATIVITY [

Maxwell’s Equations are not fully correct for many reasons. Appropriate mathematics would explain why attractive interactions become repulsive, and why repulsive interactions become attractive, at short radii if (δt_{a}) and (δr_{a}) are modified. Light-rays are bent by gravity, but light-rays are also bent by a charge. Above, I unify Coulomb and gravitational interactions by introducing variable permittivity ε_{ab}. In Zero Relativity, I improved Schwarzschild’s equations for (δt_{a}) and (δr_{a}) so they produced reversals in effects of gravity and charge inter- actions at very short distances. By modifying Maxwell’s Equations and adding four equations, in my unified physics paper Zero Relativity,

REFINEMENTS TO MAXWELL’S EQUATIONS apply to a multi-particled system of charged particles, use the centre-of-mass of the entire system as the reference point for distances involved in the equations. Include variables μ and ε rather than the constants μ_{o} and ε_{o}, because it is known that permeability and per- mittivity are variable. Predict a variable velocity-of-light that accords with the photon moving at different velocities, and the velocity-of-light changing during history. Include separate equations for each of particle a & particle b. Use _{ab} between the particles. Mathematically explain why mass, like energy, is variable with charge. When I created the eight Maxwell-Rodgers Equations, I spent a lot of time and effort on their modification or creation. That does not mean that they are perfect. But they are very useful, thought-provoking equations. Some of them are not fully necessary for this Navier-Stokes paper. Intellectually delving into those basic equations has helped me understand topics to greater depth in this paper.

The Clay Mathematics Institute in May 2000 offered that great $million prize to the first person providing a solution for a specific statement of the problem: “Prove or give a counter-example of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes Equations.” This is an extremely advanced mathematical topic to be solved, because it has not been solved during the last 180 years, but I comprehend what the topic is about because I have created a unified field theory of physics entitled Zero Relativity. In fact, my unified field theory of physics includes different attractive and repulsive forces between atoms or molecules. Lucky for me, I created advanced multi-massed equations, long ago. To solve this, I require many complicated equations beyond mainstream comprehension of relativity and other formulae that cannot be expressed easily in words. While creating this SUPERRELATIVITY paper, I have very often, maybe for about twenty hours, looked at the Navier-Stokes Equations in an internet website’s NASA Navier-Stokes Equations for 3-dimensional―unsteady. This Nasa display of equations has helped me immensely to comprehend the difficult Navier-Stokes Equations.

In SUPERRELATIVITY, I give several different examples of possible co-ordinate systems that all include three space dimensions, and all include time. My equation constructed from the Schwarzschild equation is very impressive, but my exponential energy equation excites more. The vector velocity in the equation similar to the Schwarzschild equation or the vector velocity in the exponential energy equation are both smooth and globally defined. The scalar pressure field for both comes from the kinetic molecular theory of gases where it is smooth and globally defined. They fit into a more complicated equation that can be simplified into the Navier-Stokes Equations.

When I was young, isolated, and mathematically semi-qualified, from 1980 to 1988, I sent six handwritten mathematical, theoretical physics papers about variable velocity of light, changes to Maxwell’s Equations, cos- mology, and lasers and masers, from Australia to overseas universities where my ideas were stolen or mocked as not being the ideas of Stephen Hawking and Albert Einstein [^{th} June 2014, in Mind Magazine, “Do Stephen Hawking’s Black Holes Exist?”, 13^{th} July 2014, in Mind Magazine, “No Hawking’s Black Hole”, 19^{th} July 2014, in Mind Magazine, ‘Relativity Black Hole Truth’, 29^{th} July 2014, in Mind Magazine. “Navier-Stokes Physics”, 2^{nd} August 2014, in Mind Magazine. “NAVIER-STOKES EQUATION”, 4^{th} August 2014, in Mind Magazine. “WHY NAVIER-STOKES EQUATION”, 7^{th} August 2014, in Mind Magazine. “RELATIVITY TO NAVIER-STOKES EQUATION”, 26^{th} September 2014, in Mind Magazine. “RELATIVITY TO IRREVERSIBILITY”. 28^{th} October 2014, in Mind Magazine. “ENTANGLEMENT EXISTS”, 12^{th} November 2014, in Mind Magazine. “ZERO RELATIVITY”, 14^{th} December 2014, in Mind Magazine. “NAVIER-STOKES EXISTENCE AND SMOOTHNESS”, 31^{st} January 2016. “EINSTEIN’S RELATIVITY SHOCK IN NAVIER-STOKES EQUATIONS”, 14^{th} February 2016. “SUPER-RELATIVITY”, 1^{st} March 2016. My mathematical, theoretical SUPERRELATIVITY paper comes after my progression of independent mathematical, theoretical papers. In an elite genius society, a member suggested to all members that, instead of boasting about our IQs, we could each attempt mathematical problems of Clay Mathematics Institute for major prizes. I read the Navier-Stokes problem and believed that I had already developed unified field theory mathematics leading to the correct solution. Note that my 2014 papers about Navier-Stokes equations really only mentioned that my UFT equations should give the Navier- Stokes Equations. During the last two months, I have been concentrating on this topic by creating in-depth, relevant equations, and papers becoming this SUPERRELATIVITY paper that is about relativity, cosmology, and Navier-Stokes Equations because they are all fluid dynamics.

SUPERRELATIVITY gives a major reason why Albert Einstein’s special relativity mathematics might be incorrect. SUPERRELATIVITY’s new three-dimensional and four-dimensional coordinate equations that can replace the four-dimensional coordinate equations of special relativity and general relativity are very exciting for theoretical physics. Not only do the equations lead to interesting mathematics of turbulence, but they lead to an inherent existence of kinetic energy from basics and significant implications about rarity of anti-matter. In SUPERRELATIVITY, I derived a very significant equation, involving a two-massed system, then involving a multi-massed system, by expanding the mathematics of Schwarzchild’s metric for a light-ray from a star bending as it passes by the sun during an eclipse. SUPERRELATIVITY gives the Total Energy E of a multi- massed system, the Total Kinetic Energy KE of a multi-massed system, the Total Momentum of a multi-massed system, and the Total Mass of a multi-massed system. Possible irreversibility equations and possible exponential equations leading to the Navier-Stokes Equations are given and discussed. The use of the exponential equations would lead to new explanations for sub-atomic particles and planetary positions. I am extremely proud of my new RODGERS’S EXPONENTIAL ENERGY EQUATION. Also, the equation for a force is very com- prehensive and necessary in mathematics, physics and engineering. In this paper, I explain where the (

NEW FOUR-DIMENSIONAL CO-ORDINATES―As I delved deeply into Navier-Stokes Equations, a surpris- ing intellectual event happened. Despite famous Albert Einstein’s great intuition and his meticulous approach, I discovered one reason why Einstein’s special relativity mathematics might be incorrect. Einstein related the

moving mass to the resting mass as

events with time. On 06/02/2016, I discovered a much better equation, the Einstein-Rodgers Relativity Equation:

… or squared as …..

This equation leads to a section about Turbulence. During my life, I lived by rivers and the ocean. During thousands of hours, I watched flowing currents of salt water and could not understand the remarkable variety in turbulences creating whirlpools. Nature seemed to be far beyond any rational mathematician’s predictions. Now, I believe I have intellectually grasped the mathematical magic causing the universe’s turbulence. MATHEMATICS OF TURBULENCE: Where k is a specific unknown number between 2 and infinity,

My mathematics of turbulence is very interesting, but ^{4}/4c^{4}. Does this component ensure this universe’s irreversibility of events with time? At present, I believe so! Note that this equation is for constant velocity of light, but, soon, I introduce equations suitable for constant velocity of light or variable velocity of light. When a mass moves at a very slow velocity, it seems that Albert Einstein’s equation is correct. But, as the mass moves at a very fast velocity close to the velocity of light, Einstein’s equation becomes incorrect. My modification to Einstein’s equation is very important because it mathematically explains the per- ceived inherent existence of kinetic energy from basics. Further, this equation seems to have implications about rarity of anti-matter and the square-root of matter [

Fluids Are Like Gases; The interactions between atoms or molecules in fluids are similar to those in gases. From my ZERO RELATIVITY paper, I now copy and paste the following relevant section about my past conjectures: RODGERS’S KINETIC MOLECULAR THEORY OF GASES … Rodgers’s Theory of Gases is a development from the Kinetic Molecular Theory for Gases, and requires most of the latter assumptions. In Rodgers’s Theory of Gases: ●The interactions between molecules are NOT negligible. ●Relativistic effects are NOT negligible. ●Quantum-mechanical effects are NOT negligible. ●The total volume of the individual gas molecules added up is NOT negligible compared to the volume of the container. Where pressure equals P, volume equals V, and v

equals velocity, the Kinetic Molecular Theory of Gases gives that_{Ti} for total volume, and V_{Mi} for the volume of a molecule of mass M_{i}. I introduce Kinetic Energy as K. For my system with variable velocity-of-light,

follows:

EXISTENCE AND SMOOTHNESS OF THE NAVIER-STOKES EQUATIONS; USING

For a multi-massed system:

Note that I have suddenly introduced Navier-Stokes terms to the multi-massed equation.

To convert to Navier-Stokes terms, substitute

Total Energy E of a multi-massed system

Total Kinetic Energy KE of a multi-massed system

Navier-Stokes Energy Equation:

EQUATION FOR FORCE―Force has always been very significant to physicists:

NAVIER-STOKES MOMENTUM EQUATIONS

If Volume is incompressible, it is constant so:

NAVIER-STOKES CONTINUITY EQUATION

Reduce the viscosity and pressure components to zero:

that becomes_{ }

If ν_{isc} and P are considered to be zero, the continuity equation is very simple. It is merely a mathematical state- ment that the density of the system does not change over time. That means that it is incompressible…

IRREVERSIBILITY OF EVENTS WITH TIME: As a relevant addition to this paper, I want to state that I, unlike Albert Einstein, believe in both variable velocity of light and irreversibility of events with time. Although my method built upon mathematics of the Schwarzschild metric is very good physics, I believe that use of the exponential leads to a better method. What is very significant is that an exponential equation is a wave equation. I believe that wave equations are most appropriate for fluid dynamics because fluids consist of waves.

NATURAL BASE e CONJECTURE: For

USING

An exponential equation gives much turbulence:

Useful energy equations related to Equation (12) that includes kinetic energy:

(25)

RODGERS’S EXPONENTIAL ENERGY EQUATION:

The exponential of time squared makes events irreversible with time.

Substitute

For a multi-massed system:

NAVIER-STOKES-RODGERS EXPONENTIAL EQUATION

for an incompressible volume.

ENERGY DENSITY

RODGERS’S EXPONENTIAL ENERGY EQUATION:

The exponential of time squared makes events irreversible with time.

RODGERS’S ENERGY DENSITY EQUATION FOR SPHERE

KINETIC ENERGY DENSITY EQUATION (limited to no gravity, no charge interaction, no pressure)

After considering my equation above and the existing Stefan-Boltzmann Law for many days, I decided to use

That result is equivalent to the Stefan-Boltzmann Law that gives

The energy density of the radiation in a container with reflecting walls is proportional to T^{4}. This leads to the Stefan-Boltzmann Law stating that the amount of energy radiated R is proportional to T^{4}.

But I suggest my Stefan-Boltzmann-Rodgers Law that I believe is slightly better.

Stefan-Boltzmann-Rodgers Law:

(Note that I used to agree with the law, but it is possibly that should have been used.)

In my SUPERRELATIVITY paper, I have proposed the two very significant equations for Total Energy E of a multi-massed system:

and

Although my equations differ slightly from current theoretical physics equations, I want to add to work about black holes that I pedantically, believing in variable velocity of light for 45 years, have always stated do not exist [

STEFAN-BOLTZMANN-SCHWARZSCHILD-HAWKING BLACK HOLE RADIATION POWER LAW

An extremely impressive physics equation is the Stefan-Boltzmann-Schwarzschild-Hawking black hole radi- ation power law. With my unified field theory knowledge, and with great respect for Stephen Hawking’s effort, I now intend to contribute to that magnificent equation. Say black hole radiation power is P_{ower}:

is the radius Stephen Hawking used.

is the power radiation equation Stephen Hawking attained.

HAWKING-RODGERS RADIUS OF A BLACK HOLE

STEFAN-BOLTZMANN-SCHWARZSCHILD-HAWKING-RODGERS BLACK HOLE RADIATION P_{ower}

TWO POSSIBLE THREE-DIMENSIONAL CO-ORDINATES: My introductory four-dimensional co-ordinates explain kinetic energy. But two possible three-dimensional co-ordinates can also explain kinetic energy.

At present, these possible three-dimensional co-ordinates do not seem mathematically necessary. Therefore, I will continue to pursue my new four-dimensional co-ordinates. My mathematics, for reversibility or irrever- sibility of events with time, provide useful equations to justify or replace the famous Navier-Stokes Equations, and to help physicists and engineers who need fluid dynamics equations.

c = velocity of light, ρ = density, ?= energy density, e = exponential, E = energy, F = force, kT = kinetic energy, h = Planck’s constant, i = (−1)^{1/2}, k = constant, M = mass at velocity v, M_{a} = mass a, M_{o}_{ }= mass at zero velocity, P = pressure, t = time, ν_{isc}_{ }= viscosity, δx_{a} = change in x_{a}, δt = change in time, v_{a} = velocity of M_{a}, π = pi, V = volume, N_{m} = number of molecules, G = gravitational constant, r_{ab} = radius between a and b, Q_{a} = charge on a, Q_{b} = charge on b, λ_{a} = wavelength of a,

In Superrelativity, I have used hc/lambda in my equations, but this is only correct for a specific situation when the velocity is that of light. I should have used the general formula hv/lambda in most of my equations. Note that the orbital number can also be used as for electrons to give the formula nhv/lambda.

While creating my next exciting physics paper, I realised that there is a slight omission error in Rodgers’s Exponential Energy Equation of my Superrelativity paper. This same slight omission error occurs in Equations (26), (27), (28), (29), (30), (31), (36), (43), and (44). Now, I give you Rodgers’s Rectified Exponential Energy Equation:

where

Rodgers’s Exponential Energy Equation is Superrelativity’s most impressive equation. Now, it has been rectified to a better equation. This highlight equation will be modified more in my next radical physics paper because its formulation can be improved. Enormous amounts of effort, to re-analyse equational components and to re-arrange them, have produced other impressive equations that I include in my next interesting paper. This exponential, included above, will impact upon radius and radiation results of my black hole equations.

Although theta equals velocity squared over two by velocity of light squared, that is a simplification because the velocity squared over two by velocity of light squared is actually multiplied by exponential to minus two by theta. The multiplication is usually insignificant, but is significant in some situations. Note that velocity equals c in this calculation.

The Navier-Stokes Equations are very mysterious, but useful. As my progression of physics papers proves, for many thousands of hours, very creatively, I have attempted to solve difficult mathematical physics problems, especially my unified field theory of physics entitled ZERO RELATIVITY. The Clay Mathematics Institute requested, “Prove or give a counter-example of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes Equations.” If I had not created Zero Relativity, I would have attempted a much more simplistic, but incorrect, paper than this mathematically sophisticated physics paper that has required and produced some baffling mathematics. More and more, physicists are realizing that this universe consists of irreversibility of events with time. I needed to consider reversibility and irreversibility of events with time. Very significantly, I created wonderful irreversibility equations to replace Albert Einstein’s reversibility equations. To be inspired, see RODGERS’S EX- PONENTIAL ENERGY EQUATION at Equation (26), see HAWKING-RODGERS RADIUS OF A BLACK HOLE at Equation (43), and see STEFAN-BOLTZMANN- SCHWARZSCHILD-HAWKING-RODGERS BLACK HOLE RADIATION POWER LAW at Equation (44). As I have believed for the last 45 years, I believe in variable velocity of light, but my typed velocity of light, in this paper, can be variable or constant. Further, very impressed by the Schwarzschild’s Equation, I have improved that to incorporate more formulae, for advanced equations, and developed great exponential equations useful for Cosmology that is really Fluid Dynamics in Space. Fluid Dynamics should consist of wave equations, so I hope that you enjoy reading my revolutionary NAVIER-STOKES-RODGERS EXPONENTIAL EQUATION at Equation (28). Importantly, I have shown that, “in three dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes Equations” in my paper entitled SUPERRELATIVITY.

THANKS TO THE FOLLOWING:

・ Open Journal of Fluid Dynamics for publishing my SUPERRELATIVITY paper;

・ The Clay Mathematics Institute for offering prizes to inspire mathematicians;

・ The Royal Brisbane and Women’s Hospital that used chemotherapy to save me from serious cancer in 2013;

・ Schwarzschild, Einstein, Newton, Hawking, and other great creators of stimulating physics equations;

・ internet’s Mind Magazine that published a succession of my papers;

・ psychologists who assessed that I was extremely intelligent and kept it a secret as I requested;

・ creators of internet intelligence tests for geniuses, because you helped me to escape from boredom;

・ internet geniuses because communicating with you made me feel normal;

・ St Joseph’s College, Gregory Terrace, where I was a double-visioned, diabetic, dyslexic, dysphasic, epileptic non-reader;

・ my old charismatic, funny-fellow school-friends;

・ my sons for fun times and loyalty;

・ good relatives for positive relativity;

・ my stepmother for trying hard during weird, tumultuous times;

・ my real father, John Cyril Rodgers, for admitting that he was my biological father, less than one day before he died;

・ my real mother who died when I was 7 years old;

・ my identical twin brother I met at 61 years old, because it was surprising and nice to meet my twin;

・ my maternal grandfather who told me to prove Einstein’s physics is wrong when I was 6 years old;

・ sweet MC, whom I remember each time I write MC in my physics equations of M for Mass and C for light.

Peter Donald Rodgers, (2016) Superrelativity. Open Journal of Fluid Dynamics,06,130-143. doi: 10.4236/ojfd.2016.62011