In[]:=
(*TheHamilton-BekensteinIdentity*)​​​​HamiltonBekensteinIdentity=​​M==((Quantity["ReducedPlanckConstant"]*Log[3])/(2*Pi*Quantity["SpeedOfLight"]*R))*(A/(4*Quantity["PlanckLength"]^2*Log[3]))*eta
Out[]=
M
Aeta
1
8π
ℏ/(
2
l
P
c)
R
In[]:=
Solve[HamiltonBekensteinIdentity/.{​​MQuantity[5.972*10^24,"Kilograms"],​​RQuantity[6371,"Kilometers"],​​A4*Pi*(Quantity[6371,"Kilometers"])^2​​},eta]​​
Out[]=
{{eta1.39222×
-9
10
}}
In[]:=
Solve[HamiltonBekensteinIdentity/.{​​MQuantity[1.989*10^30,"Kilograms"],​​eta1,​​A4*Pi*R^2​​},R]
Out[]=
R
2954.13
m

In[]:=
(* The Micro Audit: Electron Informational Radius *)
​
Solve[HamiltonBekensteinIdentity /. {
M -> Quantity["ElectronMass"],
eta -> 1,
A -> 4 * Quantity["PlanckLength"]^2 * Log[3]
}, R]
Solve
:Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.
Out[]=
R
6.75197843×
-14
10
m

In[]:=
(*ThePlanckAudit:UniversalLedgerCalibration*)​​​​Solve[HamiltonBekensteinIdentity/.{​​MQuantity["PlanckMass"],​​eta1,​​A4*Quantity["PlanckLength"]^2*Log[3]​​},R]
Solve
:Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.
Out[]=
R
2.8260×
-36
10
m
