A Frequency-Based Transformation of Gravity and the Role of the Quantum of Relative Mass

J. Rogers, SE Ohio, 29 Jan 2025, 1155

Abstract

This paper explores a transformation of Newton's Law of Gravitation by introducing a fundamental constant Q_m (the quantum of relative mass) and its relation to the Planck mass and the gravitational constant G. We show how a unit system where mass becomes numerically equivalent to frequency allows for a re-expression of gravity in terms of frequencies, distances, and the speed of light. We identify a fundamental, unit-invariant ratio related to the Planck time, and show that this ratio naturally emerges in this transformed system and acts as a scale-setting factor for the gravitational force.

1. Introduction

The conventional formulation of Newton's Law of Gravitation F = Gm₁m₂/r² relies on units of mass (kg), length (m), and time (s), often obscuring potentially deeper connections between gravity and other fundamental forces. This paper presents a transformation of this equation by introducing a fundamental constant we term the "quantum of relative mass," denoted by Q_m, and by considering a unit system in which mass and frequency are numerically equivalent. This transformation, along with the identification of a related unit-invariant quantity, reveals a re-expression of the gravitational law in terms of frequency and length.

2. Defining the Quantum of Relative Mass ( Q_m kg s )

We define the quantum of relative mass Q_m as:

    Q_m = h/c² = G*m_p²/c⁵

where:

• h is the Planck constant, and

• c is the speed of light in vacuum.

Q_m has units of kg⋅s (kilogram-seconds), and its numerical value is approximately 7.3725 × 10⁻⁵¹ kg⋅s.

What this constant does is convert the units  and values for frequencies to mass and mass to frequency.  
        
        m = f * Q_m
        f   = m / Q_m

3. Relationship of 

We define the Planck mass with the full Planck constant h:
    m_p = √(hc/G)

We then note that the following quantity, which we denote 1/x, is dimensionless and invariant under a unit change:
    1/x = (Q_m / m_p)² = hG/c⁵ 

We also calculate the inverse of this quantity as:
    x s^-2= c⁵ / hG = c**3/(G*Q_m)
The value of x is approximately: 5.476 × 10⁸⁵ s^-2 .

This can be seen as directly related to the Planck time squared if the Planck time is defined using the non-reduced Planck constant.
    t_p = √(hG/c⁵)
    t_p² = hG/c⁵
    1/x = t_p²  So x has units of s^-2

Using the new definitions:
    t_p = √((Q_m * c² * c³) / (Q_m * x * c⁵))
    t_p = √((Q_m * c⁵) / (Q_m * x * c⁵))
    t_p = √(1/x) with units of second.

4. A Unit System with Mass as Frequency

We consider a unit transformation where the numerical value of mass becomes equal to frequency (in Hz) by setting Q_m to be numerically equal to 1, although its units do not disappear. In this system, the new kilogram (kg') is defined such that the mass equivalent of a 1 Hz photon is equal to 1 kg' (where Q_m = 1 kg'⋅s).

In this system, the units are rescaled so the kilogram is now re-interpreted as a frequency. We also define force, length, and time in terms of fundamental constants.

In this unit system, we can express mass (m) in terms of frequency (f):
m = Q_m * f

5. Transforming the Gravitational Constant (G)

Starting with the definition of x:
    x = c³ / (G*Q_m) with units of s^-2
we can rearrange to solve for G:

    G = c³ / (Q_m * x)

6. Transforming Newton's Law of Gravitation

We start with the standard formulation of Newton's Law of Gravitation:
    F = G * m₁ * m₂ / r²

Using the relationship we have found for G, we substitute this into the equation:
    F = (c³ / (Q_m * x)) * m₁ * m₂ / r²
    F = c³ * (m₁ * m₂ / (Q_m * x * r²))

Using the fact that in our new system, m = Q_m * f (or f=m/Q_m), we substitute for m₁ and m₂
    m₁ = Q_m * f₁
    m₂ = Q_m * f₂
    F = c³ * ((Q_m * f₁) * (Q_m * f₂) / (Q_m * x * r²))
    F = c³ * (Q_m² * f₁ * f₂ / (Q_m * x * r²))
    F = c³ * (Q_m * f₁ * f₂ / (x * r²))


Where a cancellation has occurred, and the final equation is:
    F = c³ /x * (f₁ * f₂ * Q_m/ r²)

7. The Role of frequency in Gravity using Q_m and x.

The term c³ / xacts as a new gravitational scaling factor and has units of m/s² and is equivalent to G*Q_m, multiplied by a dimensionless numerical quantity related to h, G and c.
We now have a version of Newton's Law that has only c, length, time (which is implicitly in frequency), and a dimensionless number:
    F = (c³ / x) * (f₁ * f₂* Q_m/ r²)

We note that when the ratio f₁ * f₂ / r² =1, the force simplifies to just c³ * Q_m / x, the value that you have shown to be related to the Planck time squared (if we define it with the full Planck constant). This suggests that the factor x and thus 1/x is the scale at which gravity operates with respect to frequencies and lengths when our system of units transforms such that a unit of mass is a frequency.

What this means is that we have 

F = (Q_m c³) * (f₁ * f₂/ (x r²))

So this is a wavenumber times hc

F = hc * 1/x * (f₁ * f₂/ ( r²))

The (f₁ * f₂/ ( r²)) is a simple ratio that is applied to the default wavenumber 1/x.

F = hc *  (default wavenumber for force of gravity) * (ratio of frequency of mass/length squared)

As we can see this relates gravity to frequencies and shows that G contains Q_m which is the conversion of mass to frequency, and x, the default wavenumber squared.

8. Conclusion

This transformation reveals a new perspective on gravity, highlighting the deep connection between mass, frequency, and Planck time, and re-writing the law of gravity in terms of the speed of light, lengths, frequencies, and the constant that sets the scale for gravity in this system. The quantity 1/x= (Q_m/m_p)², which we have shown to be directly proportional to the square of the non-reduced Planck time, is a fundamental quantity that emerges from the interrelation between fundamental constants. This ratio appears to be invariant under unit rescaling and provides a framework for exploring gravity in a frequency based system. 

This exploration demonstrates the power of questioning standard unit conventions and seeking deeper relationships through a re-expression of physical laws in terms of more fundamental constants.