Rethinking Physical Constants: The Emergence of Scaling Relationships in Unit Conversions

James Rogers, SE Ohio, 25 Nov 2024 1147

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Abstract:

In this paper, we examine the physical constants that define key aspects of the universe, such as Planck's constant $h$, the gravitational constant $G$, the Boltzmann constant $k$, and the permittivity of free space $\epsilon_0$, from a new perspective. Instead of viewing these constants as intrinsic, fundamental values of nature, we propose that they are emergent from the relationships between scaling factors that convert between different units of measurement. These constants are the result of composite operations that transform one set of units into another, and their significance lies not just in their individual values but in how they are related. We argue that the mathematical operations encoded in the formulas that produce these constants are as crucial to understanding the constants themselves as the constants’ individual scaling factors.

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1. Introduction

In modern physics, constants such as $h$, $G$, and $k$ are often regarded as fundamental features of the universe. These constants are integral to our understanding of physical laws, from quantum mechanics to general relativity. However, it has long been recognized that these constants are defined within specific unit systems, such as the International System of Units (SI). In this paper, we aim to explore the relationships between the scaling factors that contribute to the definition of these constants, suggesting that the true significance of these constants lies not only in the numerical values of the scaling factors themselves but in the interrelationships between them.

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2. The Role of Constants in Unit Conversions

The concept of physical constants is traditionally understood as a means to relate physical quantities to measurable units. For example:

• Planck's constant $h$ relates energy and frequency.
• The gravitational constant $G$ relates mass and gravitational force.
• The Boltzmann constant $k$ relates temperature to energy.

These constants are often seen as the fundamental pillars of their respective fields. But they are not simply arbitrary numbers; they are intimately connected to the unit system that defines them. It is essential to recognize that these constants do not emerge from the universe's intrinsic properties, but from the way we define and measure physical quantities.

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3. The Mathematical Operations Encoded by Constants

Rather than seeing constants as individual, static quantities, we propose that their significance lies in the relationships between their associated scaling factors. Each constant encodes multiple mathematical operations in a single unit conversion formula. For instance:

$$h = \alpha^3 \beta / c$$ $$G = \alpha^3 / \beta$$ $$k = \alpha^3 \beta / \gamma$$ $$\epsilon_0 = \delta^2 / (\alpha^3 \beta)$$

In these expressions, we see that constants like $h$ and $G$ involve not only individual scaling factors, but also the interactions between these factors, which ultimately combine to define the constant. Each formula encodes a complex transformation, consisting of several distinct mathematical operations — multiplication, division, and exponentiation — that convert from one unit system to another.

3.1. The Role of Scaling Factors

The scaling factors ($\alpha, \beta, \gamma, \delta$) are fundamental to the conversion process, but it is the relationship between these factors that makes the transformation possible. For instance:

• $h$ and $G$ both involve $\alpha^3$, but they differ in the way they interact with the other factors (e.g., $\beta$ and $c$ for $h$, and $\beta$ alone for $G$).
• The inversion of $\beta$ in $G$ compared to $h$ is a crucial operation, reflecting how changing the relative positioning of the scaling factors changes the overall constant.

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4. The Emergent Nature of Constants

The key insight of this framework is that these constants are emergent from the relationships between the scaling factors. While each individual scaling factor may appear to be arbitrary, it is the way in which they are combined, inverted, and related that produces the meaningful values of the constants. These relationships represent the mathematical operations that allow us to convert between different units of measure, ultimately forming the constants we use in physical laws.

This approach challenges the traditional notion that constants like $h$, $G$, and $k$ are intrinsic features of the universe. Instead, they emerge from the unit system we choose and the relationships between the quantities we measure.

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5. Implications for Understanding the Structure of Physical Laws

If the constants are indeed emergent from the relationships between scaling factors, then their apparent fundamental nature is questioned. This insight could lead to a rethinking of how we approach the geometry of the universe. Rather than seeing these constants as fixed truths about reality, we might view them as tools for navigating our measurement system.

This framework suggests that there is a deeper structure underlying the constants that may be linked to the scaling of units rather than being inherent properties of physical phenomena. Understanding the precise relationships between the constants could reveal new perspectives on how we measure the universe and offer fresh insights into fundamental physics.

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6. Conclusion

In this paper, we have presented an alternative interpretation of physical constants. Rather than viewing them as fixed, fundamental quantities that describe the nature of the universe, we propose that they are emergent from the relationships between scaling factors that convert between units of measure. The constants are not just individual values, but composite operations that perform unit conversions, and it is the relationships encoded by these factors that reveal their true significance. Future work will explore the implications of this approach for our understanding of physical laws and the potential for further unification of the constants in a more fundamental framework.

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References:

1. Planck, M. (1900). Über das Gesetz der Energieverteilung im Normalen Spektrum. Annalen der Physik.
2. Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
3. Feynman, R. (1963). The Feynman Lectures on Physics.
4. Einstein, A. (1915). Die Feldgleichungen der Gravitation. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften.