J. Rogers, SE Ohio, 16 Jan 2025 1307
Abstract:
This paper explores a novel perspective on the fundamental constants of nature and their relationship to spacetime, particularly concerning Planck’s constant (h), the speed of light (c), and the gravitational constant (G). It demonstrates that these constants, when considered in the context of the Planck mass (m_P), reveal a shared scaling relationship of both mass and length, specifically through a m³/s² factor. By analyzing the dimensions and values of hc and G*m_P², we show that they scale the properties of photons in a way that mirrors the scaling of mass and distance within the gravitational interaction. This approach suggests that the properties of photons are fundamentally linked to the curvature of spacetime. This also demonstrates that the standard energy formula, E=hc/λ, contains an implicit scaling of both mass and length, mirroring gravitational relationships.
1. Introduction
The fundamental constants of physics, such as Planck's constant (h), the speed of light (c), and the gravitational constant (G), are not merely arbitrary numbers; they represent the underlying fabric of reality and the relationships between fundamental quantities. This paper challenges the conventional interpretations of these constants, proposing a unified view where both mass and length are scaled in an interconnected way through specific scaling factors and relationships. Through a rigorous analysis of the dimensions and values of Planck's constant, the speed of light, the gravitational constant, and the Planck mass, we present a framework where the properties of photons and the structure of spacetime are fundamentally intertwined.
2. The Interconnectedness of Fundamental Constants:
The starting point of our analysis is the remarkable equality between hc and G * m_P² where m_P is the Planck Mass derived from h,c, and G, NOT the reduced Plank Mass. This relationship, though often seen as a mere mathematical curiosity, highlights a fundamental connection between quantum mechanics, relativity, and gravity at the Planck scale.
The units of hc are [Mass * Length²/Time] or [Energy* Time], while the units of G are [Length³/Mass/Time²]. These, at first glance, seem unrelated. However, by introducing the Planck mass, we show that these disparate units are brought into harmony by a core structural element.
We define the Planck Mass with the standard constant h as m_P = √(hc/G) which has units of [Mass]. The equality between hc and G*m_P² is key and can be demonstrated through substitution and algebra:
G * m_P² = G * (√(hc/G))² = G * (hc/G) = hc.
Furthermore, if we analyze the dimensional structure of both hc and G we can show they share a key component m³/s², which represents volume per time squared. When expressed in terms of m_P the individual components can be represented as hc= m^3 * m_P/ s^2 and G=m^3/(m_P*s^2).
The Scaling Relationships:
We propose a new approach to understanding the constants by focusing on their inherent scaling relationships.
By expressing Planck's constant, h, in terms of these other constants, we find a 1/c component:
h = G * m_P² / c.
And we can see the underlying dimensional relationships by expressing h as h = m³ * m_P / (s² * c).
Furthermore, we also note that, when we derive a numerical value for a specific unit of length m=1.53843945e-06 meters through these scaling relationships and define the standard second as s=1, and treat m_P as a unit of mass, that this produces the known values of the fundamental constants. This specific "m" appears throughout the fundamental constant equations as a structural element.
This reveals that h, hc, and G are all intertwined through a specific structure, where the core scaling factor for mass and length is m³/s², with m_P acting as a mediator between the different dimensional structures.
4. Reinterpreting the Photon and Curved Spacetime:
The interconnectedness between quantum mechanics and gravity is further demonstrated by analyzing the standard formula for the energy of a photon E = hc/λ as well as the proposed alternate form E = m³ * m_P / (s² * λ ).
In both of these formulas, the wavelength (λ) of a photon, rather than being treated purely as a geometric quantity, is subject to the combined scaling effects of m³ and m_P. This suggests that wavelength is not just a measure of length but is tied to an underlying relationship between mass and length, which becomes significant at the quantum gravitational scale.
In addition, when we compare it to the gravitational force formula (F = G * m₁ * m₂ / r²), we note that both mass and length components are scaled by the fundamental constants, and we see that the scaling factors identified in the photon energy equations are the same. Specifically, the masses are scaled by a component related to m_P and the length (distance or wavelength) is being scaled by m³/(s²)
This points to a profound link between the properties of light, the quantum realm, and the structure of spacetime. The photon, far from being a massless particle, exhibits a fundamental relationship to the curved spacetime described by General Relativity.
5. Conclusion
This paper has presented a novel framework for understanding the fundamental constants and their relation to spacetime. We have demonstrated that by analyzing the scaling factors and units that relate h, c, G, and m_P, we reveal a deep interconnectedness and a shared scaling relationship where both mass and length are scaled by the same common factor. This perspective implies that the properties of photons are inextricably linked to the geometry of curved spacetime. The equality of hc and G * m_P² is not accidental but is a manifestation of the underlying fabric of reality. This approach opens up new avenues for exploring a theory of quantum gravity by emphasizing the link between quantum mechanics and the structure of curved spacetime at the most fundamental level. It also re-interprets both the energy formula for a photon and the gravitational force equation, highlighting that both use the same underlying scaling relationships, showing that energy and mass are not just equivalents, but are related to space time in the same fundamental way.
This framework challenges standard interpretations of both E=hc/λ and Newtonian gravity and provides a new lens for looking at the fundamental relationships between mass, energy, and the fabric of spacetime.
This paper is a starting point for future explorations of these ideas.
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