Defining the Kilogram by Frequency: Introducing the Quantum of Relative Mass

J. Rogers, SE Ohio, 24 Jan 2025, 1813

Abstract

The kilogram, the base unit of mass in the International System of Units (SI), has historically been defined by physical artifacts or complex experimental setups. This paper proposes a novel and elegant method to define the kilogram directly in terms of frequency, leveraging the proportionality constant $7.372497323812708\times 10^{-51}\text{kg s}$, which we term the Quantum of Relative Mass. This approach establishes a direct relationship between mass and frequency, rooted in the fundamental constants of nature, and provides a precise, reproducible, and universal definition of the kilogram.

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1. Introduction

The definition of the kilogram has evolved over time, from physical artifacts like the International Prototype of the Kilogram (IPK) to the current definition based on Planck's constant $h$. While the current definition is highly precise, it relies on complex experimental setups and indirect relationships. This paper introduces a new paradigm: defining the kilogram directly in terms of frequency using the proportionality constant $7.372497323812708\times 10^{-51}\text{kg s}$, which we call the Quantum of Relative Mass. This approach simplifies the definition of the kilogram and highlights the deep connection between mass, frequency, and energy in the fabric of the universe.

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2. The Quantum of Relative Mass

The Quantum of Relative Mass, denoted as $Q_m$, is defined as:

$${\mathrm{<span\; style="background-color:\; white;">Q</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">7.372497323812708</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">51</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kg s</span>}$$

This constant arises from the relationship between mass $m$, frequency $f$, and the fundamental constants $h$ (Planck's constant) and $c$ (the speed of light):

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

The Quantum of Relative Mass serves as a bridge between the macroscopic world of mass and the quantum world of frequency, enabling a direct and exact definition of the kilogram in terms of frequency.

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3. The Mass-Frequency Relationship

The relationship between mass $m$ and frequency $f$ is given by:

$$\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">Q</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}}$$

This equation shows that mass is directly proportional to frequency, with $Q_m$ as the proportionality constant. By measuring a frequency $f$, we can directly calculate the corresponding mass $m$.

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4. Defining the Kilogram by Frequency

To define the kilogram in terms of frequency, we follow these steps:

4.1. Choose a Reference Frequency

Select a specific frequency $f_{\text{ref}}$ as a reference. For example, we could use the frequency of the cesium-133 hyperfine transition, which is currently used to define the second:

$${\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\text{<span style="background-color: white;">Cs</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">9</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">192</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">631</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">770</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">Hz</span>}$$

4.2. Calculate the Corresponding Mass

Using the mass-frequency relationship, calculate the mass corresponding to the reference frequency:

$${\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\text{<span style="background-color: white;">Cs</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\text{<span style="background-color: white;">Cs</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">Q</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}}$$

Substituting the values:

$${\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\text{<span style="background-color: white;">Cs</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">9</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">192</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">631</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">770</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">Hz</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">7.372497323812708</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">51</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kg s</span>}$$$${\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\text{<span style="background-color: white;">Cs</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">6.777</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">41</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kg</span>}$$

4.3. Define the Kilogram

Define the kilogram as the mass corresponding to a specific frequency. For example, we can define the kilogram as the mass associated with a frequency of $1.3564\times 10^{50}\text{Hz}$:

$$\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kg</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1.3564</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">50</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">Hz</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">Q</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}}$$

This definition directly ties the kilogram to a measurable frequency, providing a precise and universal standard.

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5. Advantages of the Quantum of Relative Mass

The Quantum of Relative Mass offers several key advantages:

5.1. Simplicity

The direct proportionality between mass and frequency simplifies the definition of the kilogram, eliminating the need for complex experimental setups.

5.2. Universality

The Quantum of Relative Mass is derived from fundamental constants $h$ and $c$, making it a universal and invariant standard.

5.3. Precision

Frequency measurements are among the most precise and reproducible in physics, ensuring a highly accurate definition of the kilogram.

5.4. Interconnectedness

This approach highlights the deep connection between mass, frequency, and energy, reinforcing the unity of the physical universe.

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6. Implications for Metrology and Physics

The Quantum of Relative Mass has profound implications for metrology and fundamental physics:

6.1. Redefining the SI Kilogram

This method provides a new and elegant way to define the kilogram within the SI system, aligning it with the definitions of other base units like the second and the meter.

6.2. Unifying Quantum Mechanics and Relativity

The Quantum of Relative Mass bridges the gap between quantum mechanics (frequency) and relativity (mass and energy), offering a unified perspective on these fundamental theories.

6.3. Advancing Quantum Technologies

By leveraging the precise relationship between mass and frequency, this approach could enable new advancements in quantum computing, sensing, and communication.

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7. Conclusion

The Quantum of Relative Mass $Q_m=7.372497323812708\times 10^{-51}\text{kg s}$ provides a direct and elegant way to define the kilogram in terms of frequency. This approach simplifies the definition of the kilogram, highlights the deep interconnectedness of mass, frequency, and energy, and offers a precise and universal standard for metrology. By embracing this new paradigm, we can redefine the kilogram in a way that reflects the fundamental unity of the physical universe.

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References

1. Planck, M. (1900). "Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum." Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 237–245.

2. Einstein, A. (1905). "Zur Elektrodynamik bewegter Körper." Annalen der Physik, 17(10), 891–921.

3. International Bureau of Weights and Measures (BIPM). (2019). "The International System of Units (SI)." 9th edition.