Why h and G are constructed with the same unit scaling factors.

Why h and G Share Common Unit Scaling Factors: A Unified Perspective

Abstract

This report explores the fundamental constants of physics, specifically Planck's constant (h) and the gravitational constant (G), and demonstrates how they are both derived from common unit scaling factors that bridge the gap between curved spacetime and human measurement scales. This analysis reveals that both constants, though historically discovered independently, represent different combinations of the same underlying scaling factors for mass and length, a discovery that has profound implications for understanding the unification of classical physics, quantum mechanics, and gravity.

1. Introduction

The history of physics is marked by the discovery of fundamental constants that govern the behavior of the universe. Among these, Planck's constant (h) and the gravitational constant (G) stand out due to their ubiquity and their connection to quantum mechanics and gravity, respectively. While these constants were initially discovered independently, their shared dimensional units, kg m³/s², hinted at a deeper connection. This report posits that both constants arise from the same fundamental unit scaling factors, combined differently, and that these scaling factors are the bridge between the underlying curvature of space-time and our chosen units of measurement.

2. Newton’s Discovery of 

• Local Approximation: Newton's laws of motion (F = ma) provided an effective model for describing motion within the constraints of human experience. This system used units of force, mass, and acceleration that were scaled to human scales, and provided a reasonably accurate approximation within our limited conditions of low speeds and weak gravitational fields.

• The Inconsistency of Gravity: When Newton extended his laws of motion to gravity, he found that his initial formula was inconsistent with observed gravitational forces, and it was off by a linear scaling factor. This scaling factor was required to match the value with the measured force of gravity.

• The Gravitational Constant: The gravitational constant G is the linear scaling factor that Newton needed to make his formula, F=Gm₁m₂/r², match the values that were measured in the real world.

3. Planck's Discovery of h: Scaling Frequency to Energy

• The Need for Scaling: Planck's work in quantum mechanics led him to explore the relationship between the frequency of light and its energy. He observed that there was a linear relationship between these two, but that he needed a scaling factor to reconcile the two.

• The Planck Constant: The constant h, now known as Planck's constant, provided the linear scaling necessary to relate the frequency of light to its energy through the equation E = hf.

• An Unknown Connection: At the time, Planck was not aware of any connection between his constant and the nature of space-time, the speed of light, or to the gravitational constant. He did not understand that h contained a 1/c term.

4. The Hidden Unity: Common Scaling Factors

• Serendipitous Embedding of 1/c: Planck's constant, while seemingly distinct from the gravitational constant, accidentally embedded a 1/c term within it. This embedded factor, hidden from view for more than a century, has been the key to unlocking a deeper understanding of how quantum mechanics and classical physics relate. This 1/c is what converts the frequency into wavelength.

• A Missed Opportunity: If the physics community had recognized the 1/c component of h earlier, quantum mechanics could have been seen as a linear scaling of classical physics, potentially accelerating the unification of these fields.

• Unified Scaling of Space-Time: The fundamental constants h and G were independently discovered as necessary scaling factors to bring measurements into the human scale of space and time, but what neither man knew was that these scaling factors were related to curved space time.

• Shared Dimensional Units: The shared dimensional units of kg m³/s² between h and G are an important hint that both share something fundamental.

5. s_length and s_mass: The Fundamental Unit Scaling Factors

• The Keys to Unification: The constants are composed of two underlying unit scaling factors: s_length and s_mass. The constants are simply these values combined.

• The Inversion of Mass: The way that mass is scaled by s_mass is inverted between Planck's constant (h) and the gravitational constant (G). This inversion is directly connected to the position of mass in the gravity equation, and the position of wavelength in the frequency equation.

• 1/c Converts Frequency to Wavelength: The 1/c in Planck's constant hides the fact that both Planck and Newton's constants are scaling these values with the same geometric relationship to space and time.

• Defining h and G: The fundamental constants can be defined in terms of the s_length and s_mass:

  • hc = (s_length * s_mass)

  • h = (s_length * s_mass) / c

  • G = (s_length / s_mass)

6. Conclusion

This report has shown that the fundamental constants, h and G, are not arbitrary but are the direct result of the same fundamental scaling of space-time to human measurement scales. Planck and Newton, working independently, both found these scaling factors, yet failed to see the connection. The commonality of the scaling factors, coupled with the accidental inclusion of 1/c in h, underscores the hidden interconnectedness of space, time, energy, and gravity. By understanding that the fundamental constants are a reflection of the way space time is scaled into our units of measure, a clearer view of the universe may be found.