The Geometric Framework of Energy, Momentum, and Mass

Abstract

This paper explores the fundamental relationship between energy, momentum, and mass through a geometric interpretation rooted in a cube with edges of length $c$ (the speed of light). It positions $K$ as a unit-scaling factor that bridges intrinsic geometric properties and human-defined measurement systems. This perspective unifies these quantities under a timeless framework while revealing the deeper simplicity of physical laws.

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Intrinsic vs. Measured Values

The cube's dimensional relationships—edges, faces, and volume—represent the intrinsic properties of energy, momentum, and mass. These quantities exist naturally and are unaffected by human-defined units.

• Energy ($E$): Proportional to the cube's edge length ($c$).
• Momentum ($p$): Proportional to one of the cube face areas ($c^2$).
• Mass ($m$): Proportional to the cube's volume ($c^3$).

The role of $K$ is to scale these intrinsic, geometric relationships into practical units such as joules for energy, kilograms for mass, and meters per second for speed.

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Scaling Examples

The scaling behavior of $K$ is evident in the following expressions:

1. Energy:

   $$E=\frac{K}{\mathrm{c}}$$

   Energy corresponds to the cube's edge length ($c$), with $K$ ensuring the result is in units of joules.

2. Momentum:

   $$p=\frac{K}{c^{\mathrm{2}}}$$

   Momentum corresponds to the cube's face area, scaled by $K$ into units like $\text{kg}\text{ m/s.}$

3. Mass:

   $$m=\frac{K}{c^{\mathrm{3}}}$$

   Mass corresponds to the cube's volume, scaled by $K$ into kilograms.

These relationships illustrate the intrinsic geometric unity between energy, momentum, and mass, with $K$ acting as a simple scaling factor.

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Why $K$ is Not Fundamental

While $K$ is essential for converting geometric relationships into practical units, it is not a fundamental property of nature. Instead, $K$ arises from the arbitrary choice of units in human measurement systems. For example, if we redefine units such that $c = 1$, the value of $K$ would adjust accordingly, further emphasizing its role as a translator rather than a fundamental constant.

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Final Insight

This framework reveals that the intrinsic geometry of energy, momentum, and mass remains unaltered by our choice of units. $K$ merely translates these natural geometric quantities into our measurement system. This perspective demystifies constants like $h$, $c$, and $K$, showing them as reflections of universal geometric principles rather than standalone entities.

By framing these fundamental quantities as projections of a timeless, geometric structure, we unify their behaviors and simplify our understanding of the physical world. Energy, momentum, and mass emerge not as separate entities but as interconnected dimensions of the same underlying geometric reality.