Re-examining the Gravitational Constant: A Simplified 'Default Force' Interpretation and Implications for Planck Mass Units

 J. Rogers, SE Ohio, 18 Feb 2025, 2230

Abstract:

The gravitational constant, G, in Newton's Law of Universal Gravitation is often perceived as a dimensionally complex and somewhat mysterious fundamental constant. This paper proposes a conceptually simpler interpretation of G as a "default force" – the baseline gravitational force magnitude when the mass-distance ratio m1*m2 / r² = 1. This perspective, aligned with Occam's Razor, demystifies G and highlights the scaling nature of Newton's Law. Furthermore, when this simplified understanding of G is applied to the Planck mass formula (using the non-reduced Planck constant, h), it reveals a potential unit inconsistency, suggesting a re-evaluation of the dimensional interpretation of Planck units and the underlying complexity of G itself.

1. Introduction: The Perceived Complexity of G

Newton's Law of Universal Gravitation, F = G * m1*m2 / r², is a cornerstone of classical physics. The gravitational constant, G, plays a crucial role in this law, determining the strength of the gravitational force. However, G is often presented as an empirically determined constant with dimensionally complex units (m³/(kg⋅s²)), which can obscure its fundamental physical meaning. This paper aims to present a simpler, more intuitive understanding of G and to explore the implications of this simplification for Planck mass units.

2. A Simplified Interpretation of G: The 'Default Force' Concept

We propose understanding G not as an abstract constant of "gravitational strength," but as a "default force". Consider the condition where the ratio m1*m2 / r² = 1. Under this condition, Newton's Law simplifies to:

F = G * 1 = G

This reveals that the numerical value of G directly represents the gravitational force itself when the ratio m1*m2 / r² equals unity. This condition, m1*m2 / r² = 1, is not a single point but encompasses an infinite set of mass and distance combinations. It defines a scenario where the gravitational force magnitude is precisely equal to G.

Thus, G can be conceptually understood as the baseline gravitational force that exists when the mass-distance ratio is unity. Newton's Law then becomes a simple scaling law: the gravitational force between any two masses at any distance is obtained by scaling this "default force" G by the dimensionless ratio m1*m2 / r².

This interpretation aligns with Occam's Razor by providing the simplest possible explanation for the role of G. It demystifies G, making it understandable as a physically meaningful force value rather than just an abstract constant. It highlights the fundamental scaling nature of Newton's Law, where the ratio m1*m2 / r² linearly scales this "default force."

3. Units of G and Dimensional Consistency

The standard units of G (m³/(kg⋅s²)) are necessary to ensure dimensional consistency in Newton's Law. The term m1*m2 / r² has units of kg²/m², which are not units of force (kg⋅m/s²). G provides the "missing" units to convert this mass-distance quantity into a force. However, these complex units can obscure the simpler interpretation of G as a "default force." For conceptual clarity, we can focus on the numerical value of G as representing this baseline force magnitude, with the ratio m1*m2 / r² acting as a dimensionless scaling factor.

4. Implications for Planck Mass Units: A Potential Inconsistency

Planck units are derived from fundamental constants, including the gravitational constant G and Planck's constant h (or reduced Planck constant ħ). Let's examine the Planck mass formula using the non-reduced Planck constant, h, and substitute our simplified understanding of G conceptually as a force, and consider its dimensional implications. The Planck mass formula is:

m_p = sqrt(hc/G)

Now, let's substitute h = Q_m * c² and conceptually treat G, in this dimensional context, as a force (while acknowledging its standard units are more complex for dimensional consistency within Newton's Law):

m_p = sqrt( (Q_m * c²) * c / G )

If we further conceptually substitute G with a form analogous to our quantum formulas, G ≈ Q_m * f_d * (c/second) (noting this is a conceptual substitution for unit analysis exploration, not a strict dimensional equality for G itself), we get:

m_p = sqrt( (Q_m * c²) * c / (Q_m * f_d * (c/second)) )
m_p = sqrt( (Q_m / Q_m) * (c³ / c) * (second / f_d) )
m_p = sqrt( c² * (second / f_d) )
m_p = sqrt( c² / f_d ) * sqrt(second)
m_p = c / sqrt(f_d) * sqrt(second)

Analyzing the units of m_p = sqrt(c²/f_d):

Units of m_p = sqrt( (m²/s²) / (1/s) ) = sqrt( m²/s ) = m / sqrt(s)

The internal unit analysis reveals that the Planck mass formula, when using simplified forms related to Q_m and a "default gravity frequency" within the square root, does not naturally yield units of mass (kilograms). The resulting units are meters per square root of seconds (m/√s). This suggests a potential unit inconsistency or an "artificial" imposition of kilogram units onto Planck mass through external definitions, as the internal structure of the formula, when viewed through this simplified lens, does not intrinsically produce mass units.

5. Discussion and Conclusion

This paper proposes a simplified interpretation of the gravitational constant G as a "default force," making Newton's Law of Gravitation conceptually clearer and highlighting its scaling nature. Applying this simplified view to the Planck mass formula (non-reduced, with h) reveals a potential unit inconsistency, where the internal units of the formula do not naturally resolve to kilograms.

This unit mismatch raises questions about the dimensional consistency and potential over-complication of Planck units, particularly Planck mass, when derived using the standard, dimensionally complex form of G. It suggests that a re-evaluation of the foundations of Planck units and a deeper exploration of the fundamental nature of G, potentially through a more unified framework that incorporates concepts from quantum mechanics and gravity, may be warranted.

Further research is needed to fully explore the implications of this unit inconsistency and to investigate whether a reformulation of Planck units or a revised understanding of fundamental constants can lead to a more dimensionally consistent and conceptually simpler picture of physics at the Planck scale and beyond.