A Unified Perspective on Mass, Energy, and Frequency: Insights from Scaling Factors

J. Rogers, SE Ohio, 14 Dec 2024 1117 

Abstract

This paper presents a novel framework that reveals a fundamental relationship between mass, energy, and frequency through the equation $\frac{s_{mass}}{mas{s}_{1Hz}}=\frac{c^3}{s_{length}}$. By examining this relationship, we demonstrate how it leads to a deeper understanding of mass-energy equivalence and gravitational interactions. This framework suggests that length, mass, energy, and frequency are interconnected through scaling factors that redefine our understanding of these fundamental concepts in physics.

Introduction

The relationship between mass, energy, and frequency has long been a cornerstone of modern physics. The famous equation $E=m{c}^2$ encapsulates the equivalence of mass and energy, while the Planck relation $E=hf$ connects energy with frequency. However, these relationships often appear distinct and isolated. This paper introduces scaling factors for mass and length—denoted as $s_{mass}$ and $s_{length}$—to illustrate how these quantities are fundamentally linked.At the heart of this exploration is the relationship:

$$\frac{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}}}{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}\mathrm{<span\; style="background-color:\; white;">H</span>}\mathrm{<span\; style="background-color:\; white;">z</span>}}}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">l</span>}\mathrm{<span\; style="background-color:\; white;">e</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}}$$

This equation not only provides a unified framework for understanding the interplay between mass, energy, and frequency but also offers insights into gravitational interactions.

The Core Relationship

1. Derivation and Explanation

The equation $\frac{s_{mass}}{mas{s}_{1Hz}}=\frac{c^3}{s_{length}}$ can be understood by defining each term:

• $s_{mass}$: A scaling factor representing mass in a generalized unit system.
• smass = sqrt(hc/G) This isolates the product that represents the contribution of mass shared between h and G.  
• $mas{s}_{1Hz}$: The effective mass at a frequency of 1 Hz.
• $c$: The speed of light in a vacuum.
• $s_{length}$: A scaling factor representing length in the same generalized unit system.
• slength =   sqrt(hcG) This isolates the product that represents the contribution of length and time shared between h and G
  
• hc =  slength *  smass  
• h   = (slength *  smass )/c
• G  = slength /  smass   

This relationship indicates that the ratio of the scaling factor for mass to the effective mass at 1 Hz is equal to the ratio of the cube of the speed of light to the scaling factor for length. This suggests a profound connection between these quantities that transcends their traditional definitions.

How this relates to the famous equation

Starting with the well-known equation from Einstein's theory of relativity:

1. Mass-Energy Equivalence

$$E=m{c}^2$$

We also know that energy can be expressed in terms of frequency using Planck's relation:

2. Energy-Frequency Relationship

$$E=hf$$

3. Combining the Equations

By equating the two expressions for energy, we have:

$$hf=m{c}^2$$

This means we can express frequency as:

4. Rearranging for Frequency

$$f=\frac{m{c}^2}{h}$$

5. Introducing Scaling Factors

In my framework, Planck's constant $h$ is expressed in terms of scaling factors for length and mass:

$$h=\frac{s_{length}\cdot s_{mass}}{c}$$

6. Substituting into the Frequency Equation

Substituting this expression for $h$ into our equation for frequency gives us:

$$f=\frac{m{c}^2}{\left(\frac{s_{length}\cdot s_{mass}}{c}\right)}$$

This simplifies to:

$$f=\frac{m{c}^3}{s_{length}\cdot s_{mass}}$$

7. Rearranging the Expression

This can be rearranged to highlight the relationships between the variables:

$$f=\frac{m}{s_{mass}}\cdot \frac{c^3}{s_{length}}$$

8. At a Frequency of 1 Hz

At a frequency of 1 Hz, the following relationship holds true:

$$\frac{s_{mass}}{massat1Hz}=\frac{c^3}{s_{length}}$$

Mass of a Photon and Its Relationship to Energy

Starting with the expression for the mass of a photon, we have:

1. Photon Mass Equation

$$\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">f</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

2. Replacing Planck's Constant

In my framework, Planck's constant $h$ can be expressed in terms of scaling factors for length and mass:

$$\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">l</span>}\mathrm{<span\; style="background-color:\; white;">e</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}$$

3. Substituting into the Photon Mass Equation

Substituting this expression for $h$ into the equation for photon mass gives us:

$$\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">=</span>\frac{<span\; style="background-color:\; white;">(</span>\frac{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">l</span>}\mathrm{<span\; style="background-color:\; white;">e</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">f</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

4. Simplifying the Expression

This simplifies to:

$$\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">l</span>}\mathrm{<span\; style="background-color:\; white;">e</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">f</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}$$

5. Rearranging the Expression

Rearranging this equation highlights the relationship between frequency, mass, and scaling factors:

$$\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">l</span>}\mathrm{<span\; style="background-color:\; white;">e</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}}}$$

6. Highlighting the Relationships

This can be expressed as:

$$\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">m</span>}}{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}}}<span\; style="background-color:\; white;">\cdot </span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">l</span>}\mathrm{<span\; style="background-color:\; white;">e</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}}$$

7. At a Frequency of 1 Hz

At a frequency of 1 Hz, the following relationship holds true:

$$\frac{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}}}{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\mathrm{<span\; style="background-color:\; white;">1</span>}\mathrm{<span\; style="background-color:\; white;">H</span>}\mathrm{<span\; style="background-color:\; white;">z</span>}}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">l</span>}\mathrm{<span\; style="background-color:\; white;">e</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}}$$

This structured analysis of the energy and mass formulas show that they are basically the same in relativity and quantum mechanics.

Significance of the Equation

The significance of this equation lies in its ability to unify concepts of mass, energy, and frequency. The term $c^3$ serves as a bridge between spatial dimensions (length) and temporal dimensions (time), indicating that changes in one can affect the other. This perspective allows us to view mass not merely as an isolated property but as inherently linked to both frequency and energy through these scaling factors.

Mathematical proof

$$\frac{c^3}{s_{\text{length}}} = \frac{2.694400241737399 \times 10^{25}}{3.641172281610566 \times 10^{-18}} = 7.399815315922403 \times 10^{42}$$
$$\frac{\text{mass at 1Hz}}{s_{\text{mass}}} = \frac{7.372497323812708 \times 10^{-51}}{5.455511861334621 \times 10^{-8}} = 1.351385078284684 \times 10^{-43}$$

7.399815315922403×10^42 * 1.351385078284684×10^−43 = 1

This confirms the inverse relationship between these terms.

Implications for Physical Laws

1. Mass-Energy Equivalence

The relationship $\frac{s_{mass}}{mas{s}_{1Hz}}=\frac{c^3}{s_{length}}$ provides insight into why h and G have the precise through interactions with with both our unit system and the geometry of space time itself.  This ratio between s_mass and mass at 1Hz is the same ratio as between c^3 and s_length, just inverted in the formula for mass and enegy and frequency. 

This explains why when you redefine the meter you also have to scale the kg to a specific value to keep this ratio true. Without knowing this relationship it would be a lot of trial and error to find the new ratios that work correctly.

When we redefine our units of measure to make hc = G = 1, then h = 1/new_c and new_c has a non zero value.  In such a system in a photon at 1 H, rotation is still2 pi,  wavelength = new_c,  E = 1/new_c, p=1/new_c^2 and m=1/new_c^3.  

I won't give an example of such a system in this document, I am working on more papers that demonstrait this. but what s_length and s_mass are doing is to normalize our measurements so we can work with useful measurements while working with energy and force in more natural units. 

2. Gravitational Interactions

By redefining the gravitational constant $G$ in terms of scaling factors:

$$\mathrm{<span\; style="background-color:\; white;">G</span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">l</span>}\mathrm{<span\; style="background-color:\; white;">e</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}}{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}}}$$

we can express gravitational interactions using our established relationships. The gravitational force formula becomes:

$$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">G</span>}\frac{{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\frac{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">l</span>}\mathrm{<span\; style="background-color:\; white;">e</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}}{{\mathrm{<span\; style="background-color:\; white;">s</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}\mathrm{<span\; style="background-color:\; white;">s</span>}}}<span\; style="background-color:\; white;">)</span>\frac{{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

This formulation highlights how gravitational strength is intimately connected to the definitions of length and mass, suggesting that gravitational interactions are inherently tied to the same scaling factors governing quantum mechanics. These scaling factors, which dictate the curvature of spacetime in photons, are the same ones responsible for linking energy, momentum, and mass. The consistency of these scaling factors in both gravitational and quantum contexts implies that the underlying principles governing mass, energy, and gravitational force are unified through spacetime curvature. In essence, gravity, energy, momentum, and mass are all emergent phenomena arising from the same geometric structure of spacetime.

3. Frequency and Energy

The relationship also highlights how frequency is connected to energy through:

$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">f</span>}$$

where frequency can be derived from our earlier equations. The implications are profound; they suggest that all forms of energy can be understood as manifestations of underlying relationships between mass, length, and time.

Unification of Scales

This framework suggests that the relationships hold true across different scales, from subatomic particles to cosmic structures. By viewing these quantities as interconnected through scaling factors, we can better understand phenomena such as quantum mechanics and cosmology.

Experimental Predictions and Verifications

This framework does not in itself require experimental verification. The ratios predict the values of the constants h and G and these ratios can be seen in the formulas by replacing h in the standard with this new definition showing these fundamental interactions with our unit system and space and time. 

Conclusion

The relationship $\frac{s_{mass}}{mas{s}_{1Hz}}=\frac{c^3}{s_{length}}$ provides crucial insights into our understanding of fundamental physics. By emphasizing the interconnectedness of mass, energy, and frequency through scaling factors, we gain a unified perspective that clarifies existing theories while opening avenues for further research. This framework encourages us to reconsider how we define physical quantities and their relationships in our quest to understand the universe's underlying principles.