Redefinition of the kg shows that G and hc are deeply related.

 

The constants G and hc are the same value at one redefinition of the unit of measure for the kg.

The m and s are scaled the same, the kg is scaled as the inverse of the other.

Other than that one difference, both the units are identical.

Original values:

hc = 1.986446e-25 kg⋅m³/s²

G  = 6.674301e-11 m³/(kg⋅s²)

Ratio hc/G = 2.976260e-15

kg scaling factor = 5.455511e-08

After kg scaling:

h_new = 1.214564e-26 (scaled_kg)⋅m^2/s

hc_new = 3.641173e-18 (scaled_kg)⋅m³/s²

G_new  = 3.641173e-18 m³/(scaled_kg⋅s²)

Ratio hc_new/G_new = 1.000000000000000

Absolute difference = 0.000000e+00

Relative difference = 0.000000e+00

Implications for mass:

1 kg in original units = 5.455511e-08 kg in new units

1 kg in new units = 1.833009e+07 kg in original units

import numpy as np

def analyze_hc_G_scaling():

    """

    Analyze how hc and G relate when we scale kg to make their ratio 1

    """

    # Original constants

    h = 6.62607015e-34

    c = 299792458

    G = 6.6743015e-11

    # Calculate original hc

    hc_original = h * c

    print("\nOriginal values:")

    print(f"hc = {hc_original:.6e} kg⋅m³/s²")

    print(f"G  = {G:.6e} m³/(kg⋅s²)")

    print(f"Ratio hc/G = {hc_original/G:.6e}")

    # Calculate the kg scaling factor from sqrt(hc/G)

    kg_scale = np.sqrt(hc_original/G)

    print(f"\nkg scaling factor = {kg_scale:.6e}")

    # Calculate new values with scaled kg

    hc_new = hc_original / kg_scale  # Dividing by kg_scale because hc has kg in numerator

    G_new = G * kg_scale            # Multiplying by kg_scale because G has kg in denominator

    print("\nAfter kg scaling:")

    print(f"h_new = {h/kg_scale:.6e} (scaled_kg)⋅m^2/s")

    print(f"hc_new = {hc_new:.6e} (scaled_kg)⋅m³/s²")

    print(f"G_new  = {G_new:.6e} m³/(scaled_kg⋅s²)")

    print(f"Ratio hc_new/G_new = {hc_new/G_new:.15f}")

    print(f"Absolute difference = {abs(hc_new - G_new):.6e}")

    print(f"Relative difference = {abs(hc_new - G_new)/hc_new:.6e}")

    # Calculate what this means for mass scaling

    print("\nImplications for mass:")

    print(f"1 kg in original units = {kg_scale:.6e} kg in new units")

    print(f"1 kg in new units = {1/kg_scale:.6e} kg in original units")

    return hc_new, G_new, kg_scale

if __name__ == "__main__":

    hc_new, G_new, kg_scale = analyze_hc_G_scaling()

We could futher redefine the meter and make ch = G = 1

1. Current values:
   hc = 1.986446e-25 kg⋅m³/s²
   G = 6.674301e-11 m³/(kg⋅s²)
2. We've already scaled the kg to make hc = G = 3.641173e-18 (in their respective units)
3. To make both equal to 1, we need to scale the meter:
   Scaling factor for meter = (3.641173e-18)^(1/3) ≈ 1.533474e-6
4. New definitions:
   1 new_kg ≈ 1.833009e+07 old_kg
   1 new_meter ≈ 1.533474e-6 old_meter
   (second remains unchanged)
5. Result:
   hc_new = G_new = 1 (in the new unit system)