The Frequency of Gravity: Deriving Gravitational Oscillations from Compton Frequency Interactions

 J. Rogers, SE Ohio, July 31, 2025

Abstract

We demonstrate that gravitational systems possess a natural oscillation frequency that emerges as the geometric mean of the Compton frequencies of interacting masses, scaled by the ratio of Planck length to separation distance. Using dimensional analysis and Planck unit normalization, we derive that the gravitational frequency between two masses is given by f = √(f₁ × f₂) × (l_P/r), where f₁ and f₂ are the respective Compton frequencies. This reveals that gravitational interactions are fundamentally harmonic phenomena arising from coupled temporal oscillations at the Planck scale.

1. Introduction

Recent work has shown that physical constants emerge as coordinate transformation factors encoding the Planck scale in various unit systems. This framework suggests that all physical phenomena reduce to simple relationships at the Planck scale that become complex only through coordinate projection into human measurement systems.

In this paper, we apply this insight to derive the natural frequency of gravitational interactions between masses. We show that gravity possesses an intrinsic oscillation frequency that can be understood as the geometric coupling of individual mass frequencies, providing a harmonic interpretation of gravitational phenomena.

2. The Dimensional Postulate

We begin with the dimensional postulate that gravitational frequency scales as:

f ~ √(M₁ × M₂)/r

This postulate suggests that the gravitational interaction frequency depends on the geometric mean of the interacting masses, inversely proportional to their separation distance.

3. Planck Unit Normalization

Following the standard dimensional analysis procedure, we normalize all quantities by their respective Planck units:

• f → f/f_P (dimensionless frequency ratio)
• M₁ → M₁/m_P (dimensionless mass ratio)
• M₂ → M₂/m_P (dimensionless mass ratio)
• r → r/l_P (dimensionless length ratio)

The dimensionless equation becomes:

f/f_P = √(M₁ × M₂) × l_P/(m_P × r)

Solving for f:

f = √(M₁ × M₂) × f_P × l_P/(m_P × r)

4. Planck Unit Substitution

Substituting the definitions of Planck units:

• f_P = c/(2π l_P) = √(c⁵/(ℏG))/(2π)
• l_P = √(ℏG/c³)
• m_P = √(ℏc/G)

After algebraic simplification, this yields:

f = √(G × M₁ × M₂ × c)/(√ℏ × r)

5. Connection to Compton Frequencies

The Compton frequency of a mass M is defined as:

f_Compton = Mc²/ℏ

For our two masses:

f₁ = M₁c²/ℏ
f₂ = M₂c²/ℏ

We can solve for the masses in terms of their Compton frequencies:

M₁ = f₁ℏ/c²
M₂ = f₂ℏ/c²

6. The Geometric Mean Derivation

Substituting the mass expressions into our gravitational frequency formula:

f = √(G × (f₁ℏ/c²) × (f₂ℏ/c²) × c)/(√ℏ × r)

Simplifying step by step:

f = √(G × f₁ × f₂ × ℏ² × c/c⁴)/(√ℏ × r)

f = √(G × f₁ × f₂ × ℏ²/c³)/(√ℏ × r)

f = √(f₁ × f₂) × √(Gℏ/c³)/r

7. The Planck Scale Reduction

The key insight comes from expressing the scaling factor √(Gℏ/c³) in terms of Planck units:

• G = l_P³/(m_P × t_P²)
• ℏ = m_P × l_P²/t_P
• c = l_P/t_P

Substituting these:

Gℏ/c³ = [l_P³/(m_P × t_P²)] × [m_P × l_P²/t_P] / [l_P/t_P]³

= (l_P³ × m_P × l_P²)/(m_P × t_P² × t_P) × (t_P³/l_P³)

= l_P⁵/t_P³ × t_P³/l_P³ = l_P²

Therefore: √(Gℏ/c³) = l_P

8. The Final Result

The gravitational frequency simplifies to:

f = √(f₁ × f₂) × l_P/r

This reveals the profound relationship: The gravitational frequency between two masses is the geometric mean of their Compton frequencies, scaled by the ratio of Planck length to their separation distance.

9. Physical Interpretation

This formula reveals several key insights:

9.1 Harmonic Nature of Gravity

Gravitational interactions are fundamentally oscillatory phenomena. Every gravitating system has a natural frequency determined by the constituent masses and their separation.

9.2 Frequency Coupling

The gravitational frequency emerges from the geometric coupling of individual mass frequencies (Compton frequencies). This suggests that what we call "mass" is actually a manifestation of underlying temporal oscillations.

9.3 Planck Scale Origin

The l_P/r scaling factor shows that gravitational coupling strength is determined by how close the system is to Planck scale separation. At r = l_P, the gravitational frequency equals the geometric mean of Compton frequencies.

9.4 Universal Scaling

The formula works for any gravitating system - from elementary particles to black holes - revealing the universal harmonic structure underlying all gravitational phenomena.

10. Testable Predictions

This framework makes several testable predictions:

10.1 Binary Systems

The natural orbital frequency of binary systems should approach:

f_orbital ≈ √(f₁ × f₂) × l_P/r

10.2 Gravitational Waves

The characteristic frequency of gravitational waves from merging objects should be related to this fundamental gravitational frequency.

10.3 Quantum Gravity Scale

At separations approaching the Planck length, gravitational effects should become strongly coupled to the individual Compton frequencies of the particles.

11. Implications for Fundamental Physics

11.1 Mass as Frequency

This derivation supports the view that mass is not a fundamental property but rather a manifestation of temporal oscillation patterns. The Compton frequency represents the "natural vibration rate" of what we perceive as mass.

11.2 Gravity as Harmonic Coupling

Rather than a force or spacetime curvature, gravity emerges as the harmonic coupling between temporal oscillation patterns, with coupling strength determined by spatial proximity on the Planck scale.

11.3 Unification with Quantum Mechanics

The appearance of ℏ and the reduction to Planck scale relationships suggests a natural bridge between gravitational and quantum phenomena through shared harmonic structure.

12. The Planck Scale Unity

The most profound aspect of this derivation is how all complexity reduces to simple Planck scale relationships. The elaborate formula:

f = √(G × M₁ × M₂ × c)/(√ℏ × r)

Simplifies to the elegant Planck scale expression:

f = √(f₁ × f₂) × l_P/r

This exemplifies how physical laws become transparent when expressed in natural units, revealing the underlying simplicity that coordinate systems obscure.

13. Conclusion

We have shown that gravitational systems possess a natural frequency given by the geometric mean of constituent Compton frequencies, scaled by Planck length to separation ratio. This reveals gravity as a fundamentally harmonic phenomenon arising from coupled temporal oscillations.

The derivation demonstrates the power of Planck unit analysis to reveal simple relationships underlying apparently complex physical laws. When expressed in natural units, gravitational frequency emerges as a straightforward geometric relationship between fundamental oscillation patterns.

This work suggests that all physical phenomena may ultimately reduce to harmonic interactions at the Planck scale, with apparent complexity arising only through coordinate projection into human measurement systems. The universe operates as a symphony of coupled temporal oscillations, with what we call "matter" and "gravity" representing different aspects of this fundamental harmonic structure.

References

1. Rogers, J. (2025). "The Structure of Physical Law as a Grothendieck Fibration." Independent Research.

2. Rogers, J. (2025). "From Temporal Experience to Spacetime Curvature: Deriving General Relativity from the m/r Axiom." Independent Research.

3. Planck, M. (1899). "Über irreversible Strahlungsvorgänge." Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.

4. Compton, A. H. (1923). "A Quantum Theory of the Scattering of X-rays by Light Elements." Physical Review 21 (5): 483–502.