A Unit-Invariant Relationship Connecting the definition of the Speed of Light and the Scaling of length in Fundamental Constants

J. Rogers, 18 Jan 2025 1340

Abstract:

This paper explores a novel unit-invariant relationship discovered within a framework that proposes a unified scaling of mass, length, and spacetime. This framework, based on the interconnectedness of Planck's constant (h), the speed of light (c), the gravitational constant (G), and the Planck mass (m_P), reveals an invariant quantity derived from the scaling factor of the meter and the speed of light. This quantity, with units of m²/s, suggests a fundamental connection between the rate of spatial change and the fundamental constants governing the universe, potentially offering new insights into the nature of spacetime and the propagation of light.

1. Introduction:

The fundamental constants of physics, including Planck's constant (h), the speed of light (c), and the gravitational constant (G), are typically expressed with specific numerical values tied to our chosen system of units. However, the underlying physical reality these constants represent should be independent of any arbitrary unit system. This paper explores a recently discovered unit-invariant relationship that emerged from an analysis of the scaling behavior of these constants across different unit systems. This relationship suggests a deeper connection between the speed of light and the scaling of space itself, potentially offering new avenues for understanding the fundamental structure of the universe.

2. The Unified Scaling Framework:

Recent work has proposed a framework where h, c, and G are interconnected through a shared scaling factor, mediated by the Planck mass (m_P) and a characteristic volume per time squared factor (m³/s²). Within this framework, a specific numerical length scale (m ≈ 1.53843945e-06 meters)  and defining time as per 1 second emerges as a crucial structural element in expressing the relationships between these constants.

3. The Invariant Relationship:

A key finding within this framework is the invariance of the product of the linear meter scaling factor (m) and the speed of light (c) across different unit systems. While the numerical values of both 'm' and 'c' vary depending on the chosen unit of length, their product remains constant:

m * c ≈ 461.21254569389 m²/s

This relationship holds regardless of how the unit of length is defined, indicating a fundamental connection between the scaling of space and the speed of light that is independent of human-defined measurement systems.

4. Interpretation and Implications:

The units of this invariant quantity, length²/time, :

• Scaling the mass to the length: Removing the factor makes the mass too small by 461.  If we compensate by making the mass 461 times bigger we just end up with the same values for h, hc, and G.  It appears that there is an adjustment to the length to scale the mass to the correct value in the constant. Individual scaling for the specific definition of mass in a unit system, or adjustment to the definition of the meter is independent of this ratio between mass and length.  

5. Connection to Planck Units:

Further research should explore the relationship between the length scaling as represented by that value of m and Planck mass (m_P). This definition of the meter, and the Plank mass factor actual define how h, hc, and G work the way they do and how they are related.

6. Conclusion:

The discovery of a unit-invariant relationship between the scaling of length in fundamental constants and  the definition of the speed of light offers a novel perspective on the fundamental constants of the universe. This relationship suggests a deeper connection between the properties of spacetime and the propagation of light, potentially leading to new avenues for research in quantum gravity, cosmology, and our understanding of the fundamental laws of physics. Further investigation into the physical interpretation and implications of this invariant quantity is warranted. This observation challenges the conventional understanding of fundamental constants as fixed numerical values and opens up exciting possibilities for a more unified and fundamental understanding of the universe.