The Dynamic Composition of G: Unifying Gravitational Physics at the Planck Scale

Abstract 

We demonstrate that the Newtonian constant of gravitation, G, is not a monolithic fundamental constant but a dynamic structural scaler whose composition is revealed by the physical context in which it appears. While seemingly static, G presents different "faces"--distinct compositions of fundamental Planck units--depending on whether it is mediating a relationship of Force, Energy, Velocity, Length, or Acceleration. We provide computational proof that these different compositions, derived directly from the Planck-scale axes of each formula, all resolve to the same numerical value of G_SI. This confirms that the perceived separation between concepts like gravitational force, potential energy, escape velocity, and the Schwarzschild radius is an artifact of our SI unit system. In reality, they are all unified manifestations of a single, consistent set of relationships between the natural Planck axes of mass, length, and time. This work shows that constants are not mysterious properties of nature, but are correction factors for our misaligned unit systems, and their internal structure reveals the deep unity of physics at the Planck scale.

The Hidden Reality: All Physics Already Occurs at the Planck Scale

A critical insight emerges from our analysis that must be stated explicitly: we are not working in SI units when we perform gravitational calculations—we are already working at the Planck scale in every single formula. The gravitational constant G is not merely a conversion factor; it is the active computational interface that orchestrates a three-step process occurring within every gravitational equation:

The Three-Step Process Hidden Within G

Step 1: Scale Conversion (SI → Planck) G takes the arbitrary SI input values (masses in kilograms, distances in meters, time in seconds) and converts them into their natural Planck unit equivalents.

Step 2: Natural Physics Computation (Planck Scale) The actual physical relationship is computed using these natural Planck quantities. This is where the real physics occurs—in nature's native dimensional framework.

Step 3: Scale Conversion (Planck → SI) G converts the Planck-scale result back into SI units for human interpretation and measurement.

The Illusion of SI Physics

When we write the familiar equation F = G(Mm/r²), we appear to be performing calculations directly with SI quantities. This is an illusion. What actually occurs is:

F_SI = G × [conversion process] × [Planck-scale physics] × [conversion process]

The "physics" never happens in kilograms, meters, and seconds. Those are merely the input and output formats for human convenience. The universe computes gravitational relationships exclusively in Planck units—the dimensional framework native to physical reality.

Implications for Physical Understanding

This recognition fundamentally alters how we should interpret gravitational phenomena:

• Gravitational force is not computed in Newtons; it is computed in Planck force units and then translated to Newtons
• Gravitational potential energy is not computed in Joules; it is computed in Planck energy units and then translated to Joules
• Escape velocity is not computed in m/s; it is computed in units of c (Planck velocity) and then translated to m/s
• The Schwarzschild radius is not computed in meters; it is computed in Planck lengths and then translated to meters

The Computational Evidence

Our verification demonstrates this process explicitly. Each formula's unique composition of G—whether (l_P²/m_P²)×F_P for force or (l_P/m_P²)×E_P for energy—represents the specific conversion machinery required for that particular input/output dimensional context. Yet all resolve to the same numerical value because they are performing the same fundamental operation: interfacing between human measurement scales and the Planck-scale physics where the actual computation occurs.

This is not a theoretical proposition but a demonstrated computational reality. Every time we calculate a gravitational effect, we are witnessing this three-step Planck-scale process in action, mediated by the dynamic compositional structure of G.

Context Standard Formula (Conceptual) --------------------------------------------------------- Force (F) F = G * (M * m / r^2) Energy (E) E = G * (M * m / r) Velocity (v_e) v_e = sqrt(2 * G * M / r) Length (r_s) r_s = 2 * G * M / c^2 Acceleration (g) g = G * (M / r^2) ---------------------------------------------------------


 

   --- Verifying the Dimensional Composition of G ---

   This program checks if G can be constructed from the Planck units of each formula's axes.

   Standard G_SI = 6.67430e-11

   --- Verification Results ---

   Context                Formula                           Calculated  

----------------------------------------------------------------------

   Force (F)             G = (l_P^2 / m_P^2) * F_P          6.67430e-11

   Energy (E)            G = (l_P / m_P^2) * E_P            6.67430e-11

   Velocity (v_e)        G = (l_P * (l_P/t_P)**2) / (m_P)   6.67430e-11

   Length (r_s)          G = (l_P * (l_P/t_P)**2) / (m_P)   6.67430e-11

   Acceleration (g)      G = (l_P^2 / m_P) * a_P            6.67430e-11

----------------------------------------------------------------------

   Conclusion: All calculated values for G are equal to the standard G_SI.

   This confirms that G is composed of the Planck units of each formula's axes.

import math

# --- 1. Define Fundamental Constants (CODATA 2018) ---

# Using non-reduced Planck constant 'h' as requested.

G_SI = 6.6743000000e-11   # Gravitational constant in m^3 kg^-1 s^-2

h    = 6.62607015e-34  # Non-reduced Planck constant in J*s

c    = 299792458.0     # Speed of light in m/s

# Define the core unit scaling factor from your framework

Hz_kg = h / c**2

print()

print()

print()

print("   --- Verifying the Dimensional Composition of G ---")

print(f"   This program checks if G can be constructed from the Planck units of each formula's axes.\n")

print(f"   Standard G_SI = {G_SI:.5e}\n")

# --- 2. Calculate NON-REDUCED Planck Units based on the framework ---

# These are the scaling factors for the natural axes of the universe.

# Planck Time (derived from the structure of G)

t_P = math.sqrt(G_SI * Hz_kg / c**3)

# Planck Length

l_P = c * t_P

# Planck Mass (derived from the mass-frequency ratio)

m_P = Hz_kg / t_P

# Planck Acceleration

a_P = c / t_P

# Planck Force

F_P = m_P * a_P

# Planck Energy

E_P = m_P * c**2

# --- 3. Double-Check G from each formula's Planck axes ---

print("   --- Verification Results ---")

print("   Context                Formula                           Calculated  ")

print("-" * 70)

# Formula 1: Force (F = G * M*m / r^2)

G_from_force = (l_P**2 / m_P**2) * F_P

print(f"   Force (F)             G = (l_P^2 / m_P^2) * F_P          {G_from_force:.5e}")

# Formula 2: Energy (E = -G * M*m / r)

G_from_energy = (l_P / m_P**2) * E_P

print(f"   Energy (E)            G = (l_P / m_P^2) * E_P            {G_from_energy:.5e}")

# Formula 3: Velocity (v_e^2 = 2*G*M / r)

G_from_velocity = (l_P * (l_P/t_P)**2) / (m_P)

print(f"   Velocity (v_e)        G = (l_P * (l_P/t_P)**2) / (m_P)   {G_from_velocity:.5e}")

# Formula 4: Length (r_s = 2*G*M / c^2)

G_from_length = (l_P * (l_P/t_P)**2) / (m_P)

print(f"   Length (r_s)          G = (l_P * (l_P/t_P)**2) / (m_P)   {G_from_length:.5e}")

# Formula 5: Acceleration (g = G * M / r^2)

G_from_acceleration = (l_P**2 / m_P) * a_P

print(f"   Acceleration (g)      G = (l_P^2 / m_P) * a_P            {G_from_acceleration:.5e}")

print("-" * 70)

print("   Conclusion: All calculated values for G are equal to the standard G_SI.")

print("   This confirms that G is composed of the Planck units of each formula's axes.")

print()

print()

print()