Exploring a black hole programatically.

So, I found something interesting, I was puttering around the inside of a blackhole and realized that the natural radius/ ( 2 * natural mass ) is exactly non reduced Planck time squared. Imagine my surprise!

t_P^2, non reduced: 1.826241629811e-86 s^2
r_n/(m_n*2): 1.826241629811e-86 s^2

What this means is that

t_P^2 = r_n/(2 * m_n)

we can just calculate the radius by

r_n = 2 * m_n * t_P^2

So, keep this in mind when doing renovations in your black hole.

r_n/ t_P = 2 * m_n * t_P
becomes
r_planck = 2 * m_planck dimensionless. This is the unit scaling that is always happening in this formula, from the scaling inside the units and what is encoded into the G_SI value and units. It has this definition. G_SI = non reduced t_P^2 * c^3 / Hz_kg The Hz_kg is the ratio between mass and frequency in SI units of measurement m/f. To the universe m=f and L=T, but we humans scale the kg, meter, and second wildly different than that. You can find Hz_kg = h/c^2 and so we can define h = Hz_kg c^2.

I calculated a black hole and am exploring the escape velocity formula under different unit scaling that is encoded into G and the constants.  

python escape.py 

--- Input SI Values ---

M_SI (Mass in kg):              2.019886382250e+31       

r_SI (Radius in m):               3.000000000000e+04       

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

--- Fundamental and Derived Constants (SI) ---

G_SI (Gravitational Constant):  6.674300000000e-11        m^3 kg^-1 s^-2

c_SI (Speed of Light):          2.997924580000e+08        m s^-1

h_SI (Planck Constant):         6.626070150000e-34        J s

Hz_kg (Mass/Frequency Ratio):   7.372497323813e-51        kg s (or kg/Hz)

G_n (Natural Grav. Constant):   1.826241629811e-86        s^2

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

--- Standard Escape Velocity (SI) ---

v_e_SI^2:                       8.987551787368e+16        m^2 s^-2

v_e_SI (Escape Velocity):       2.997924580000e+08        m s^-1

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

--- Naturalized Quantities ---

m_n (Natural Mass):              2.739758718835e+81        s^-1 (or Hz)

r_n (Natural Radius):            1.000692285594e-04        s

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

--- Naturalized Quantities  time scaled by t_P ---

m_n (Natural Mass * t_P):         3.702469050733e+38        dimensionless

r_n (t_P / Natural Radius):       1.350450181078e-39        dimensionless

    G_n * m_n / r_n               5.000000000000e-01        dimensionless

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

--- Beta Calculation (Dimensionless) ---

Numerator (2 * G_n * m_n):      1.000692285594e-04        s

β^2 (Dimensionless Ratio^2):    1.000000000000e+00       

β (Dimensionless Ratio):        1.000000000000e+00       

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

--- Verification ---

v_e_SI (from SI formula):       2.997924580000e+08        m s^-1

v_e_from_beta (β * c_SI):       2.997924580000e+08        m s^-1

Absolute difference:            5.960464477539e-08       

Verification successful: v_e_SI is very close to β * c_SI.

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

--- Detailed breakdown of β² calculation ---

G_n:                            1.826241629811e-86        s^2

m_n:                            2.739758718835e+81        s^-1

r_n:                            1.000692285594e-04        s

1/r_n:                          9.993081933333e+03        s

m_n/r_n:                        2.737863335488e+85        s^-2

r_n/m_n:                        3.652483259621e-86        s^-2

r_n/(m_n*2):                    1.826241629811e-86        s^2

m_planck (Natural Mass * t_P):  3.702469050733e+38  dimensionless planck mass

m_planck (2 * Natural Mass * t_P):7.404938101467e+38 dimensionless planck mass

r_planck (1/ t_P / Natural Radius)7.404938101467e+38 dimless planck length

r_planck (t_P / Natural Radius):  1.350450181078e-39 dimless planck length

(r_n/t_P)/(2 * m_n* t_P)        1.000000000000e+00   dimensionless

(G_n * m_n) / r_n:              5.000000000000e-01   dimensionless

2 * (G_n * m_n) / r_n (β²):     1.000000000000e+00       

sqrt(2 * (G_n * m_n) / r_n) (β):1.000000000000e+00   

    

(venv) $ cat escape.py 

import math

# CODATA 2018 values for fundamental constants (full precision)

G_SI = 6.67430e-11         # Gravitational constant (m^3 kg^-1 s^-2)

c_SI = 299792458.0         # Speed of light (m s^-1)

h_SI = 6.62607015e-34      # Planck constant (J s or kg m^2 s^-1)

# --- Your Derived Scaling Constants ---

# Hz_kg: Mass-to-frequency ratio (kg/Hz or kg*s)

Hz_kg = h_SI / (c_SI**2)

# G_n: Naturalized gravitational constant (s^2)

# G_n = G_SI * Hz_kg / (c_SI**3)

# Let's calculate it step by step to see units clearly

# G_SI * Hz_kg has units (m^3 kg^-1 s^-2) * (kg s) = m^3 s^-1

# c_SI^3 has units m^3 s^-3

# So (G_SI * Hz_kg) / c_SI^3 has units (m^3 s^-1) / (m^3 s^-3) = s^2

G_n = (G_SI * Hz_kg) / (c_SI**3)

t_P = G_n**(1/2)

# --- Input SI Values ---

M_SI = 1.0  # Mass in kilograms

M_SI = 1/1.356392489652e+50/t_P  # Mass in kilograms

M_SI = 2e31/9.901547025492e-01  # Mass in kilograms

r_SI = 1.0  # Radius in meters

r_SI = 3.335640951982e-09*t_P  # Radius in meters

r_SI = 3e4  # Radius in meters

print("--- Input SI Values ---")

print(f"M_SI (Mass in kg):              {M_SI:<25.12e}")

print(f"r_SI (Radius in m):               {r_SI:<25.12e}")

print("-" * 70)

print("--- Fundamental and Derived Constants (SI) ---")

print(f"G_SI (Gravitational Constant):  {G_SI:<25.12e} m^3 kg^-1 s^-2")

print(f"c_SI (Speed of Light):          {c_SI:<25.12e} m s^-1")

print(f"h_SI (Planck Constant):         {h_SI:<25.12e} J s")

print(f"Hz_kg (Mass/Frequency Ratio):   {Hz_kg:<25.12e} kg s (or kg/Hz)")

print(f"G_n (Natural Grav. Constant):   {G_n:<25.12e} s^2")

print("-" * 70)

# 1. Standard Escape Velocity Calculation (SI units)

# v_e = sqrt(2 * G_SI * M_SI / r_SI)

v_e_SI_squared = (2 * G_SI * M_SI) / r_SI

v_e_SI = math.sqrt(v_e_SI_squared)

print("--- Standard Escape Velocity (SI) ---")

print(f"v_e_SI^2:                       {v_e_SI_squared:<25.12e} m^2 s^-2")

print(f"v_e_SI (Escape Velocity):       {v_e_SI:<25.12e} m s^-1")

print("-" * 70)

# 2. Convert SI Mass and Radius to Natural Forms

# m_n: Natural mass (Hz or s^-1)

m_n = M_SI / Hz_kg

# r_n: Natural radius (s)

r_n = r_SI / c_SI

print("--- Naturalized Quantities ---")

print(f"m_n (Natural Mass):              {m_n:<25.12e} s^-1 (or Hz)")

print(f"r_n (Natural Radius):            {r_n:<25.12e} s")

print("-" * 70)

print("--- Naturalized Quantities  time scaled by t_P ---")

print(f"m_n (Natural Mass * t_P):         {m_n* t_P:<25.12e} dimensionless")

print(f"r_n (t_P / Natural Radius):       {t_P/r_n:<25.12e} dimensionless")

print(f"    G_n * m_n / r_n               {G_n * m_n / r_n:<25.12e} dimensionless")

print("-" * 70)

# 3. Calculate Beta (β) using Naturalized Quantities

# β^2 = 2 * G_n * m_n / r_n

beta_squared_numerator = 2 * G_n * m_n

beta_squared = beta_squared_numerator / r_n

beta = math.sqrt(beta_squared)

print("--- Beta Calculation (Dimensionless) ---")

print(f"Numerator (2 * G_n * m_n):      {beta_squared_numerator:<25.12e} s") # G_n (s^2) * m_n (s^-1) = s

print(f"β^2 (Dimensionless Ratio^2):    {beta_squared:<25.12e}")

print(f"β (Dimensionless Ratio):        {beta:<25.12e}")

print("-" * 70)

# 4. Verify: v_e = β * c

v_e_from_beta = beta * c_SI

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

print(f"v_e_SI (from SI formula):       {v_e_SI:<25.12e} m s^-1")

print(f"v_e_from_beta (β * c_SI):       {v_e_from_beta:<25.12e} m s^-1")

# Check for difference (due to potential floating point artifacts)

difference = abs(v_e_SI - v_e_from_beta)

print(f"Absolute difference:            {difference:<25.12e}")

if math.isclose(v_e_SI, v_e_from_beta, rel_tol=1e-12): # Relative tolerance for float comparison

    print("\nVerification successful: v_e_SI is very close to β * c_SI.")

else:

    print("\nVerification shows a notable difference. Check calculations or precision.")

print("-" * 70)

print("\n--- Detailed breakdown of β² calculation ---")

# (G_n * m_n) / r_n

# Term1 = G_n * m_n (units s)

term1_Gn_mn = G_n * m_n

# Term2 = (G_n * m_n) / r_n (dimensionless)

term2_Gn_mn_div_rn = term1_Gn_mn / r_n

print(f"G_n:                            {G_n:<25.12e} s^2")

print(f"m_n:                            {m_n:<25.12e} s^-1")

print(f"r_n:                            {r_n:<25.12e} s")

print(f"1/r_n:                          {1/r_n:<25.12e} s")

print(f"m_n/r_n:                        {m_n/r_n:<25.12e} s^-2")

print(f"r_n/m_n:                        {r_n/m_n:<25.12e} s^-2")

print(f"r_n/(m_n*2):                    {r_n/(m_n*2):<25.12e} s^2")

print(f"m_planck (Natural Mass * t_P):       {m_n* t_P:<25.12e} dimensionless planck mass")

print(f"m_planck (2 * Natural Mass * t_P):   {2 * m_n* t_P:<25.12e} dimensionless planck mass")

print(f"r_planck (1/ t_P / Natural Radius)   {1/(t_P/r_n):<25.12e} dimensionless planck length")

print(f"r_planck (t_P / Natural Radius):     {t_P/r_n:<25.12e} dimensionless planck length")

print(f"(r_n/t_P)/(2 * m_n* t_P)        {(r_n/t_P)/(2 * m_n* t_P) :<25.12e} dimensionless planck mass")

print(f"(G_n * m_n) / r_n:              {term2_Gn_mn_div_rn:<25.12e} (dimensionless)")

print(f"2 * (G_n * m_n) / r_n (β²):     {2 * term2_Gn_mn_div_rn:<25.12e}")

print(f"sqrt(2 * (G_n * m_n) / r_n) (β):{math.sqrt(2 * term2_Gn_mn_div_rn):<25.12e}")