The Mathematical Origins of the Separation of Mass, Momentum, and Energy by Powers of c in a Photon

 J. Rogers, SE Ohio, 22 Jan 2025, 0132

Abstract: This paper explores the mathematical origins of how our current system of physics creates a separation between mass, momentum, and energy through the use of powers of ‘c’ (or more specifically, velocity, which includes ‘c’ as a distance/time measurement) in the formulas for these properties. It shows that the scaling of different properties by different powers of ‘c’ is a mathematical artifact that is a byproduct of the way that they were defined in the formulas of motion and energy.

1. Introduction:
The fundamental constants of physics, like Planck's constant, the speed of light, and the gravitational constant, have created a set of mathematical equations that can reliably model the behavior of objects at all levels. However, these equations and models have also created a set of values that often appear to be distinct and unrelated, such as energy, momentum, and mass. This paper will explore the reasons behind the mathematical separation of these properties, and how that has been created by our choices in mathematical models and our systems of measurement.

2. The Role of Velocity and the Speed of Light (c):

Our system of mathematics has a specific relationship between mass, momentum, and energy, and that is due to the speed of light, ‘c’, which is how we measure distances in time, and then use those measures to build our values for the various properties of motion.

• Distance as a Fraction of the Speed of Light: It is key to understand that our units of distance are based on a relationship with the speed of light, where a meter is a fraction of a light second. When we say 10 meters what we are really saying is 10 m * 1/c s/m = 10c seconds of light travel. This is important to remember, when we start to examine the equations for the various properties of motion.

• Velocity as Distance per Unit of Time: It is equally important to understand that velocity is measured as a distance divided by a time (v = dist/time). And therefore, when we include a velocity term in our equations, we are including an implicit dimension of distance in that equation, as the power of ‘c’.

Mass, Momentum, and the Power of Velocity

Examining how mass relates to momentum and energy through the speed of light (or velocity), we can understand how our mathematical system creates a separation between these properties, particularly in the context of a photon.

• Mass by itself represents base, as it has no separation from itself. It possesses the fundamental unit of mass and serves as the basis for deriving momentum and energy.

• Momentum as Mass, Scaled by a Single Power of Velocity: The equation for momentum, p = m * v, demonstrates that momentum is the product of mass (m) and velocity (v). In the case of a photon, since its velocity is c, the equation becomes p = mc. This shows that, for a photon, momentum is mass scaled by a single power of c.

• Kinetic Energy as Mass Scaled by Two Dimensions of Space-Time: The equation for kinetic energy, KE = 1/2 * m * v², reveals that kinetic energy is the product of mass (m) and velocity squared (v²). For a photon, since v = c, the equation becomes E = mc².  We lose the 1/2 and the K because of relativity concerns. This indicates that kinetic energy is mass scaled by two powers of c in the case of a photon.

• The Separation By Powers of c: Therefore, for a photon, momentum and kinetic energy—which initially appear to be separate, independent qualities—can be understood as mass scaled by different powers of c. Momentum is scaled by one power of c, while kinetic energy is scaled by two powers of c. This distinction between momentum and kinetic energy arises from the unique properties of photons and their constant velocity, c. The replacement of v with c in the standard momentum and energy formulas adjusted for relativity leads to the separation between mass, momentum, and energy in our mathematical models for photons.

• Energy and momentum are derived quantiles: These are not actual properties of the thing itself, if it is a car or a photon.  This is a human description of how mass is interacting with space time curvature. And since mass itself is space time curvature then it is how space time curvature is interacting with space time curvature.The map is not the territory. 

4. The Underlying Basis

This has shown how the mathematical system is what is creating the separation of these terms, through the use of powers of the speed of light as our base unit for measuring all properties related to motion.

• It has shown how the underlying framework is that mass is the fundamental property, and that the other properties are derivations of mass, that have been scaled using units of velocity, which is just an implicit dimension of space-time.

• That our units of measurement of mass, momentum, and energy are based on our system of measurement, and that they are not a property of the universe, but a property of our math.

• And that they are also based on relationships that were originally designed to model motion, rather than necessarily to reveal the underlying physics.

5. Conclusion:

This paper has shown that, by understanding the relationship between momentum and energy, and that the mathematics has encoded the properties with different powers of velocity, which is a relationship that is implicitly encoded in our math by powers of ‘c’, we can gain a new insight into why these properties appear to be different in our current models, when in fact, the system is showing us that they are all based on mass and that all properties are scaling from the underlying relationship between mass and the dimensions of space-time.