G gravitational_constant_G 6.6743e-11 1.82624163e-86 m^3 kg^-1 s^-2
k_e coulombs_constant_k_e 8987551787.0 0.001161409732 kg^1 m^3 s^-2 C^-2
--- Scaling Factors Used ---
m 2.99792458000000000000e+08 this is c
kg 7.37249732381270843547e-51 this is h/c
K 4.79924307336622100516e-11 this is h/k
s 1.00000000000000000000e+00 time stays the same
Hz 1.00000000000000000000e+00
mol 1.66053906717384659585e-24
C 1.60217663399999989376e-19
This results in natural units just by scaling the base units.
And yes, the 2.210219e-42 value looks familiar, that is h/c. The fact that this is unit scaling from natural units force to SI Newton unit gives you a good clue about what h is doing in SI units.
Constant Original Value Rescaled Value
--------------------------------------------------------------------------------
c speed_of_light_c 299792458 1.0 m^1 s^-1
Hz 1 Hz 1 1.0 Hz^1
h planck_constant_h 6.62607015e-34 1.0 kg^1 m^2 s^-1
k boltzmann_constant_k 1.380649e-23 1.0 kg^1 m^2 s^-2 K^-1
G gravitational_constant_G 6.6743e-11 1.82624163e-86 m^3 kg^-1 s^-2
k_e coulombs_constant_k_e 8987551787.0 0.001161409732 kg^1 m^3 s^-2 C^-2
? Wien wl d law 0.002897771955 0.2014052353 m^1 K^1
? neutron Compton wavelength 1.31959090581e-15 4.401681465e-24 m^1
e elementary_charge_e 1.602176634e-19 1.0 C^1
Na avogadro_constant_Na 6.02214076e+23 1.0 mol^-1
me electron_mass_me 9.1093837015e-31 1.235589964e+20 kg^1
mp proton_mass_mp 1.67262192369e-27 2.268731815e+23 kg^1
mn neutron_mass_mn 1.67492749804e-27 2.271859079e+23 kg^1
? Angstrom star 1.00001495e-10 3.33569082e-19 m^1
ε₀ vacuum_permittivity_epsilon0 8.8541878128e-12 68.51799954 C^2 kg^-1 m^-3 s^2
μ₀ vacuum_permeability_mu0 1.25663706212e-06 0.01459470514 kg^1 m^1 s^-2 C^-2
Rydberg rydberg_constant_Rydberg 10973731.56816 3.28984196e+15 m^-1
a₀ bohr_radius_a0 5.29177210903e-11 1.765145176e-19 m^1
E_h hartree_energy_Eh 4.359744722071e-18 6.57968392e+15 kg^1 m^2 s^-2
r_e electron_radius_re 2.8179403262e-15 9.399637152e-24 m^1
R_K klitzing_constant_Rk 25812.80745 0.9999999996 kg^1 m^2 s^-3 C^-2
Wb/K weber_per_kelvin 1.0 11604.51812 kg^1 m^2 s^-1 C^-1 K^-1
F faraday_constant_F 96485.33212 1.0 mol^-1 C^1
σ stefan_boltzmann_constant_sigma 5.670374419e-08 40.80262464 kg^1 s^-3 K^-4
Φ₀ magnetic_flux_quantum_Phi0 2.067833848e-15 0.4999999999 kg^1 m^2 s^-1 C^-1
K_J josephson_constant_KJ 483597848400000.0 2.0 kg^-1 m^-2 s^1 C^1
G₀ conductance_quantum_G0 7.748091729e-05 2.0 kg^-1 m^-2 s^1 C^2
R molar_gas_constant_R 8.314462618 1.0 kg^1 m^2 s^-2 mol^-1 K^-1
u atomic_mass_unit_u 1.6605390666e-27 2.252342719e+23 kg^1
μ_B bohr_magneton_muB 9.2740100783e-24 6.440443341e-22 m^2 s^-1 C^1
μ_N nuclear_magneton_muN 5.0507837461e-27 3.507575069e-25 m^2 s^-1 C^1
λ_e,C electron_compton_wavelength_lambda_eC 2.42631023867e-12 8.093299794e-21 m^1
m_P planck_mass_mP 2.176434e-08 2.952098732e+42 kg^1
l_P planck_length_lP 1.616255e-35 5.391246367e-44 m^1
t_P planck_time_tP 5.391247e-44 5.391247e-44 s^1
T_P planck_temperature_TP 1.416784e+32 2.952098859e+42 K^1
g₀ standard_gravity_g0 9.80665 3.271146334e-08 m^1 s^-2
--- Calculating Electrostatic Force Directly in SI Units ---
Coulomb's Constant (k_e_si): 8.987552e+09 N m^2 / C^2
Elementary Charge (e_si): 1.602177e-19 C
Speed of Light (c_si): 2.997925e+08 m/s
Planck Constant (h_si): 6.626070e-34 kg m^2 / s
------------------------------------------------------------
Scenario: Two charges equal to e_SI, separated by 1 light-second
Charge 1 & 2 (q_natural_in_si): 1.602177e-19 C
Distance (r_natural_in_si): 2.997925e+08 m (1 light-second)
------------------------------------------------------------
Force calculated directly in SI: 2.566970e-45 N
------------------------------------------------------------
--- Comparison with Natural Unit Calculation ---
Value of k_e in Natural Units (k_e'): 0.0011614097 (in Natural Force Units)
Value of 1 Natural Force Unit in Newtons:
kg_nat*m_nat/s_nat^2 = 2.210219e-42 N / (Natural Force Unit)
Expected SI Force = k_e' * (Value of 1 Natural Force Unit in N)
Calculated SI Force from k_e': 2.566970e-45 N
------------------------------------------------------------
Result: The direct SI calculation MATCHES the calculation using k_e'
and the force unit conversion factor.
$ cat ke_calc_nu01.py
import math
# --- SI Constants ---
k_e_si = 8.9875517873681764e9 # N m^2 / C^2 (Defined as 1/(4*pi*epsilon0) where epsilon0 is derived from c and mu0)
# Or more fundamentally: k_e = (mu0 * c^2) / (4 * pi)
# mu0_si = 4 * math.pi * 1e-7 # N/A^2 = kg*m/(s^2*A^2) = kg*m/C^2 (Exact by definition before 2019, now derived)
c_si = 299792458.0 # m/s (Exact by definition)
h_si = 6.62607015e-34 # J s = kg m^2 / s (Exact by definition)
e_si = 1.602176634e-19 # C (Exact by definition)
# --- Derived SI values based on your natural unit setup ---
# Charge corresponding to 1 natural charge unit (e_si)
q_natural_in_si = e_si
# Distance corresponding to 1 natural length unit (1 light-second)
r_natural_in_si = c_si * 1.0 # 1 light-second
# --- Calculate Force directly in SI using Coulomb's Law ---
q1_si = q_natural_in_si
q2_si = q_natural_in_si
r_si = r_natural_in_si
force_direct_si_coulomb = k_e_si * q1_si * q2_si / (r_si**2)
# --- Calculate Scaling Factor for Force (same as before) ---
# Value of 1 Natural Unit in SI Units
natural_m_in_si = c_si
natural_kg_in_si = h_si / (c_si**2)
natural_s_in_si = 1.0
# Value of 1 Natural Force Unit in Newtons
natural_force_in_newtons = natural_kg_in_si * natural_m_in_si / (natural_s_in_si**2)
# --- Value of k_e in the Natural Units (k_e') (from your previous script execution) ---
# k_e_natural = 0.001161409732 # kg^1 m^3 s^-2 C^-2 <-- Wait, these units are wrong for k_e'
# k_e' should represent the force (in natural units) between unit natural charges at unit natural distance.
# Let's use the value your script calculated for k_e:
k_e_natural = 0.001161409732 # This should be the dimensionless force factor in natural units.
# --- Calculate expected SI force from Natural k_e ---
# k_e_natural represents the force between unit natural charges at unit natural distance,
# expressed in natural force units.
# To get the force in SI units (Newtons), multiply this value by the
# value of 1 natural force unit in Newtons.
force_from_natural_ke = k_e_natural * natural_force_in_newtons
# --- Print Results ---
print("--- Calculating Electrostatic Force Directly in SI Units ---")
print(f"Coulomb's Constant (k_e_si): {k_e_si:.6e} N m^2 / C^2")
print(f"Elementary Charge (e_si): {e_si:.6e} C")
print(f"Speed of Light (c_si): {c_si:.6e} m/s")
print(f"Planck Constant (h_si): {h_si:.6e} kg m^2 / s")
print("-" * 60)
print("Scenario: Two charges equal to e_SI, separated by 1 light-second")
print(f"Charge 1 & 2 (q_natural_in_si): {q_natural_in_si:.6e} C")
print(f"Distance (r_natural_in_si): {r_natural_in_si:.6e} m (1 light-second)")
print("-" * 60)
print(f"Force calculated directly in SI: {force_direct_si_coulomb:.6e} N")
print("-" * 60)
print("--- Comparison with Natural Unit Calculation ---")
print(f"Value of k_e in Natural Units (k_e'): {k_e_natural:.10f} (in Natural Force Units)") # Used f-format for precision
print("Value of 1 Natural Force Unit in Newtons:")
print(f" kg_nat*m_nat/s_nat^2 = {natural_force_in_newtons:.6e} N / (Natural Force Unit)")
print("\nExpected SI Force = k_e' * (Value of 1 Natural Force Unit in N)")
print(f"Calculated SI Force from k_e': {force_from_natural_ke:.6e} N")
print("-" * 60)
if math.isclose(force_direct_si_coulomb, force_from_natural_ke, rel_tol=1e-9):
print("Result: The direct SI calculation MATCHES the calculation using k_e'")
print(" and the force unit conversion factor.")
else:
print("Result: There is a discrepancy between the two methods.")
print(f" Difference: {abs(force_direct_si_coulomb - force_from_natural_ke):.2e} N")
--- Calculating Gravitational Force Directly in SI Units ---
Gravitational Constant (G_si): 6.674300e-11 N m^2 / kg^2
Speed of Light (c_si): 2.997925e+08 m/s
Planck Constant (h_si): 6.626070e-34 kg m^2 / s
------------------------------------------------------------
Scenario: Two masses equal to (h/c^2)_SI, separated by 1 light-second
Mass 1 & 2 (m_natural_in_si): 7.372497e-51 kg
Distance (r_natural_in_si): 2.997925e+08 m (1 light-second)
------------------------------------------------------------
Force calculated directly in SI: 4.036394e-128 N
------------------------------------------------------------
--- Comparison with Natural Unit Calculation ---
Value of G in Natural Units (G'): 1.826242e-86 (in Natural Force Units)
Value of 1 Natural Force Unit in Newtons:
kg_nat*m_nat/s_nat^2 = 2.210219e-42 N / (Natural Force Unit)
Expected SI Force = G' * (Value of 1 Natural Force Unit in N)
Calculated SI Force from G': 4.036394e-128 N
------------------------------------------------------------
Result: The direct SI calculation MATCHES the calculation using G'
and the force unit conversion factor.
$ cat g_calc_nu01.py
import math
# --- SI Constants ---
G_si = 6.67430e-11 # N m^2 / kg^2 (CODATA 2018)
c_si = 299792458.0 # m/s (Exact by definition)
h_si = 6.62607015e-34 # J s = kg m^2 / s (Exact by definition)
# --- Derived SI values based on your natural unit setup ---
# Mass corresponding to 1 natural mass unit (h/c^2)
m_natural_in_si = h_si / (c_si**2)
# Distance corresponding to 1 natural length unit (1 light-second)
r_natural_in_si = c_si * 1.0 # 1 light-second
# --- Calculate Force directly in SI ---
m1_si = m_natural_in_si
m2_si = m_natural_in_si
r_si = r_natural_in_si
force_direct_si = G_si * m1_si * m2_si / (r_si**2)
# --- Calculate Scaling Factor for Force ---
# These are the factors that define 1 natural unit in terms of SI units
# (Value of 1 Natural Unit in SI Units)
# This is the inverse of the 'rescale_factors' dictionary values used before,
# which represented the value to divide SI by.
natural_m_in_si = c_si # 1 Natural length unit = c meters
natural_kg_in_si = h_si / (c_si**2) # 1 Natural mass unit = h/c^2 kg
natural_s_in_si = 1.0 # 1 Natural time unit = 1 s
# Force units: kg * m / s^2
# Calculate the value of 1 Natural Force Unit in Newtons
natural_force_in_newtons = natural_kg_in_si * natural_m_in_si / (natural_s_in_si**2)
# --- Value of G in the Natural Units (G') (from previous script execution) ---
G_natural = 1.82624162981050139182530428342e-86 # This is dimensionless IF using F=ma directly
# --- Calculate expected SI force from Natural G ---
# G_natural represents the force between unit natural masses at unit natural distance,
# expressed in natural force units.
# To get the force in SI units (Newtons), multiply this value by the
# value of 1 natural force unit in Newtons.
force_from_natural_G = G_natural * natural_force_in_newtons
# --- Print Results ---
print("--- Calculating Gravitational Force Directly in SI Units ---")
print(f"Gravitational Constant (G_si): {G_si:.6e} N m^2 / kg^2")
print(f"Speed of Light (c_si): {c_si:.6e} m/s")
print(f"Planck Constant (h_si): {h_si:.6e} kg m^2 / s")
print("-" * 60)
print("Scenario: Two masses equal to (h/c^2)_SI, separated by 1 light-second")
print(f"Mass 1 & 2 (m_natural_in_si): {m_natural_in_si:.6e} kg")
print(f"Distance (r_natural_in_si): {r_natural_in_si:.6e} m (1 light-second)")
print("-" * 60)
print(f"Force calculated directly in SI: {force_direct_si:.6e} N")
print("-" * 60)
print("--- Comparison with Natural Unit Calculation ---")
print(f"Value of G in Natural Units (G'): {G_natural:.6e} (in Natural Force Units)")
print("Value of 1 Natural Force Unit in Newtons:")
print(f" kg_nat*m_nat/s_nat^2 = {natural_force_in_newtons:.6e} N / (Natural Force Unit)")
print("\nExpected SI Force = G' * (Value of 1 Natural Force Unit in N)")
print(f"Calculated SI Force from G': {force_from_natural_G:.6e} N")
print("-" * 60)
if math.isclose(force_direct_si, force_from_natural_G, rel_tol=1e-9):
print("Result: The direct SI calculation MATCHES the calculation using G'")
print(" and the force unit conversion factor.")
else:
print("Result: There is a discrepancy between the two methods.")
print(f" Difference: {abs(force_direct_si - force_from_natural_G):.2e} N")
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