INTRODUCTION
It has been the strict belief in science that any calculation for speeds faster than light involves the use of imaginary numbers because the relativity equations for mass and time break down at the speed of light.
Solutions to the relativity equations at faster than light speeds produce answers containing square roots of negative numbers which then become imaginary numbers.
Since Einsteins Theory of Relativity became absolute dogma in scientific circles, no one has yet concieved of a solution which would allow calculations for speeds faster than light to be expressed in real numbers. That psychological barrier has been knocked down with the discovery of a solution so simple, it's been hiding in plain sight for nearly 100 years.
RELATIVITY EQUATIONS
In order to properly calculate the size of mass and the length of time of an object approaching the speed of light, we first have to look at the relativity equations for mass and time. The equation for the mass of an object as it approaches the speed of light is expressed by the following equation;
The equation for the time experienced by an object as it approaches the speed of light is expressed by the following equation;
Since "c" is a constant, you could easily convert "c" into "1" and simplify the equations even further.
When you use the equations to calculate mass or time, you simply enter a number that's proportional to the speed of light.
For example if an object is traveling half the speed of light;
If an object is traveling one fourth the speed of light;
RELATIVITY COORDINATE SYSTEM
When you plot the relativity equations on a graph using the Cartesian Coordinate System, invented by French mathematician and philosopher Rene Descartes, you will create a graph that is geometrically out of perspective.
In order to have proper perspective for relativity equations, you have to completely redesign the graph system itself.
Normally the origin, or center point, of a Cartesian Coordinate Graph is "0" for the X and Y axis while the X and Y axis are numbered in increments of 1, 5, 10, etc. In the original Cartesian Graph that was used to plot the curves generated by the relativity equations, "0" was the centerpoint of the Y axis and "1" was the centerpoint of the X axis.
In the modified "Relativity Coordinate Graph", "1" is designated as the center point of the graph for both the X and Y axis where they intersect each other.
The X axis will represent velocity. The Y axis above the center point will represent matter. The Y axis below the center point will represent time.
The centerpoint or "1" on the X axis will represent the speed of light. The centerpoint or "1" on the Y axis will represent the normal state of matter and time.
The increments on both the X and Y axis will be marked in base-10 numbers.
Numbers above "1" will be represented by base-10 increments with positive exponents. Numbers below "1" will be represented by base 10 increments with negative exponents. The center point or "1" on both the X and Y axis will be represented by 10^0.
The "1" at the centerpoint and the base-10 increments on the X and Y axis form the foundation of the "Relativity Coordinate System".
MASS AND TIME EQUATIONS PLOTTED ON RELATIVITY GRAPH
When the equations for mass and time are plotted on this new graph, you get the following curves;
The curve for the size of matter curves upward from the X axis to the Y axis as matter approaches the speed of light. The curve for time curves downward from the X axis to the Y axis as the matter approaches the speed of light.
Of course you don't see any curves beyond the speed of light because the equations break down when "v = 1", the speed of light;
v = 1 = c
THE BREAKTHROUGH
Since the equations for mass and time break down at the speed of light, a very simple modification to the original equations allows us to calculate mass, time and even energy at speeds faster than light in real numbers.
In the original equation for mass;
if you were to reverse the locations of "v^2" and "c^2", you would invert the "v^2/c^2" fraction into its reciprocal "c^2/v^2" and you would get a new equation;
when you convert the constant "c" to "1" you get;
When this equation is plotted on the graph you get a curve that curves downward from the Y axis to the X axis on the other side of the speed of light.
When you make the same "c^2/v^2" modification for the time equation;
you get;
when you convert "c" to "1" you get;
When this equation is plotted on the graph you get a time curve that curves upward from the Y axis to the X axis on the other side of the speed of light.
The modification of the original equations for mass and time gives us two new equations for speeds faster than light. In order to prevent confusion between the original equations and the new equations, subscripts need to be added to the symbols "m" and "t" which respectively represent matter and time.
Since the original equations involve masses slower than light, the subscript "STL" should be added to "m" and "t" in the mass and time equations;
The new equations involving mass and time faster than light should have the subscript "FTL" added;
APPLICATIONS FOR NEW EQUATIONS
In Quantum Physics there are three classifications for sub-atomic particles;
BRADYONS - a term describing subatomic particles which travel slower than the speed of light.
LUXONS - a term describing subatomic particles which travel at the speed of light.
TACHYONS - a term describing subatomic particles which travel faster than the speed of light.
By using these classifications as a guide, we now have real number equations which can be applied to tachyons as well as the original equations for bradyons;
BRADYON(SLOWER THAN LIGHT) EQUATIONS
TACHYON(FASTER THAN LIGHT) EQUATIONS
The plotted curves on the graph become mass/energy & time curves for bradyons and tachyons;
The difference between the bradyon and tachyon equations can be applied to luxons in a new set of equations. The subscript "SOL" is added to "m" and "t" to represent "Speed of Light".
LUXON (SPEED OF LIGHT) EQUATIONS
At the speed of light where "v = 1" and "c = 1" within these luxon equations, mass becomes non-existent;
and time also becomes non-existent when v=1 and c=1;
Both of these luxon equations mathematically predict the properties of luxons.
E=MC^2 FOR TACHYONS
We all know the famous equation discovered by Einstein;
Where "E" is energy, "m" is mass and "c" is the speed of light. As a bradyon particle is accelerated toward the speed of light, the energy required to get the particle to light speed becomes infinity because "m" in "E = mc^2" can be replaced with;
therefore;
The equation "E = mc^2" can also be applied to faster than light tachyons when the follwing equation is subsituted for "m";
therefore;
According to quantum physics, tachyons lose speed as they gain energy and gain speed when they lose energy. The slowest speed they can travel is the speed of light. When you look at the graph for the equation;
You'll notice that the mass of the tachyon is at it's greatest point near the speed of light. The mass of the tachyon returns to normal the faster it accelerates past the speed of light. The energy required for the tachyon, as predicted by;
is at it's greatest point near the speed of light and at it's lowest point the farther it accelerates past the speed of light. This equation mathematically fulfills the prediction made by quantum physicists for the behavior of tachyons.
ABSOLUTE NUMERATOR FUNCTION
All of the previous equations of mass and time for bradyons, luxons and tachyons can be simplified into a single equation for mass and a single equation for time.
By looking at the mass and time equations for bradyons, luxons and tachyons;
they all have a common trait within their deominators; a fraction containing c^2 and v^2.
Below the speed of light the numerator v^2 is divided by denominator c^2. Beyond the speed of light, the numerator c^2 is divided by the deominator v^2. At the speed of light the numerator v^2 and deominator c^2 are virtually interchangable; they're one and the same. In order to simplify the bradyon, luxon and tachyon equations into a single equation, a new mathematical function called the "Absolute Numerator" must be created.
The "Absolute Numerator" is identified by the following set of brackets surrouding the fraction containing the numerator "n" and the denominator "d";
The "Absolute Numerator" follows three simple rules;
1. The numerator must always be smaller than the denominator. Below the speed of light, the numerator v^2 is smaller than the denominator c^2.
2. When the numerator and the deominator are equal, they're interchangeable and reduce to 1. When the numerator v^2 is equal to the denominator c^2 at the speed of light, they're interchangeable and reduce to "1".
3. When the numerator becomes greater than the denominator, the original numerator becomes the new denominator and the original denominator becomes the new numerator. When the numerator v^2 becomes greater than the denominator c^2 beyond the speed of light, the original numerator v^2 becomes the new denominator and the original denominator c^2 becomes the new numerator.
When the "Absolute Numerator" function is added to the speed of light equations for bradyons, luxons and tachyons, they simplify into one equation for mass;
and one equation for time;
The "Absolute Numerator" function can also be applied to the energy equations;
and simplify them into one equation;
When the mass, time and energy equations containing the "Absoulte Numerator" function are used for luxons, simply subtract the mass, time or energy equation from itself.
CONCLUSIONS
The following conclusions can be made from this document;
1. By modifying the original relativity equations, new equations can be created which will predict the behavior of mass and time at faster than light(FTL) speeds in real numbers.
2. The modification of the original Cartesian Coordinate System allows us to see relativity equations plotted in their proper perspective when the origin, or centerpoint, of the graph is designated as "1" instead of "0" on both the X and Y axis. The number "1" represents the normal state of matter and time on the Y axis and the speed of light on the X axis.. The increments on the X and Y axis are marked in base-10 numbers with positive exponents above "1" and negative exponents below "1". This new coordinate system is designated as the "Relativity Coordinate System".
3. The mass and time of tachyons can be predicted at faster than light(FTL) speeds.
4. The mass and time of luxons can be predicted at the speed of light.
5. The behavior of energy associated with tachyons at faster than light(FTL) speeds can be predicted.
6. The creation of the "Absolute Numerator" mathematical function simplifies the mass, time and energy equations for bradyons, luxons and tachyons into one equation for mass, one equation for time and one equation for energy. When mass, time or energy equations containing the "Absolute Numerator" function are used for luxons, simply subtract the mass, time or energy equation from itself.
These discoveries will open new doors in the world of science and lead to more breakthroughs in Quantum Physics, Astro-Physics and Superstring Theory.
COPYRIGHT 2002 - Dan Perez
fasterthanlightrelativity@yahoo.com