# Example Content

In $R\nu$ the [Planck Length](https://en.wikipedia.org/wiki/Planck_units#Planck_length) is the universal unit for measurement of distance, and is defined approximately to be:
$$\boxed{L_P=\sqrt{\hbar}=5.72928\times10^{-35}m=1 L}$$
Where $1\ L$, is 1 Planck Length of distance.  

### SI Conversion Factors

The following conversion factors can be used to convert observable quantities of measure from the _SI_ system of units to $R\nu$ to ~6 significant digits.

| Conversion Factor            | Symbol   | Value                                              |
| ---------------------------- | -------- | -------------------------------------------------- |
| meters to Planck Length      | $\chi_P$ | $1.74542\times10^{34} \frac{L}{m}$                 |
| seconds to Planck Length     | $\tau_p$ | $5.23264\times10^{42} \frac{L}{s}$                 |
| mass to Planck Length        | $G_P$    | $1.62871\times10^8 \frac{L}{kg}$                   |
| energy to Planck Length      | $E_P$    | $1.81219\times10^9 \frac{L}{J}$                    |
| momentum to Planck Length    | $P_P$    | $5.43280\times10^{-1} \frac{L\cdot s}{kg \cdot m}$ |
| temperature to Planck Length | $k_P$    | $2.501998\times10^{-14} \frac{L}{K}$               |
| charge to Planck Length      | $C_P$    | $1.89007\times10^{18} \frac{L}{C}$                 |

### Physical Constants
Applying conversion factors from the table above, we can convert SI values to Reduced Natural Units.  For example, performing this analysis on the the speed of light yields a unit-less number with a value of 1:

$c = 299792458 \frac{m}{s} = 299792458 \frac{m}{s} \cdot 1.74542\times10^{34} \frac{L}{m} \cdot \frac{1}{5.23264\times10^{42} \frac{L}{s}} = 1.00000$

| Quantity                       | Symbol       | SI                                                           | $\nu$                          |
| ------------------------------ | ------------ | ------------------------------------------------------------ | ------------------------------ |
| Speed of Light                 | $c$          | $299792458 \frac{m}{s}$                                      | 1                              |
| Reduced Gravitational Constant | $G_0$        | $8.38659\times10^{-10} \frac{m^3}{kg \cdot s^2}$             | 1                              |
| Boltzmann's Constant           | $k$          | $k=1.380649\times10^-23 \frac{J}{K}$                         | 1                              |
| Permittivity of Free Space     | $\epsilon_o$ | $8.854187817620\times10^{-12} \frac{C^{2}s^2}{kg \cdot m^3}$ | 1                              |
| Permeability of Free Space     | $\mu_o$      | $\huge{\frac{1}{\epsilon_{o} \cdot c^{2}}}$                  | 1                              |
| Reduced Planck's Constant      | $\hbar$      | $1.054571726\times10^-34 \frac{kg \cdot m^2}{s}$             | $1 L^2$                        |
| Mass of the Electron           | $m_e$        | $9.10938\times10^{-31} kg$                                   | $1.48366\times10^{-22} L$      |
| Charge of the Electron         | $e^-$        | $-1.60218\times10^{-19} C$                                   | $-3.02822\times10^{-1} L$      |
| Unit Cycle                     | $\Theta$     | $2\pi = 6.28318...\ Radians$                                 | $1 \tau = 6.28318...\ Radians$ |

## Fine Structure Constant

As a consistency check, we compute the _[Fine Structure Constant](https://en.wikipedia.org/wiki/Fine-structure_constant)_ using Reduced Natural Units which is a unit less ratio that should be independent of our system of units.

$\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar c}=\frac{e^2}{2\tau}=0.00729735≈\frac{1}{137}}$


#### Dimensional Analysis

The reader should be familiar with high school physics and chemistry [dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis).

- $1\ meter\ (m) = 100\ centimeters\ (cm)$
- $1\ kilometer\ (km) = 1000\ meters\ (m)$
- $1\ mile = 5280\ feet\ (ft\ or\ ')$
- $1\ foot\ (ft\ or\ ') = 12\ inches\ (in\ or\ ")$
- $1\ inch\ (") = 2.54\ centimeters\ (cm)$

How many kilometers are in 1 mile?
$1\ mile = 1\ mile \times \frac{5280\ ft}{mile} \times \frac{12\ in}{ft} \times \frac{2.54\ cm}{in}\times \frac{1\ m}{100 cm} \times \frac{1\ km}{1000 m} = \frac{5280 \times 12 \times 2.54}{100 \times 1000}\ km = \frac{160934.40}{100000}\ km = 1.6\ km$
Note that each unit in the denominator cancels with one if the numerator until we are left with only km.


## Newton's Law of Gravity

The force of gravity ($F_g$) between 2 masses, $m1$ and $m2$ separated by distance $r$ is given by [Newton's Law of Gravity](https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation):

$F_{g} = G \frac{m_{1} m_{2}}{r^{2}}$

Where $G$, is the [Gravitational Constant](http://en.wikipedia.org/wiki/Gravitational_Constant).

$G = 6.67430 \times 10^{-11}\ N\frac{m^2}{kg^2}$

The strength of gravitational force follow the inverse square law distributing gravitational flux over the surface area of a sphere ($4\pi r^2$).  

#### Inverse Square Law

Any source of a signal strength ($S_0$) that radiates isotropically in 3-dimensional space will distribute that signal strength ($S_0$) over the surface area of a sphere ($SA = 4 \pi r$) of radius ($r$). Such that the intensity ($I$) at distance ($r$) is:

$$I(r) = \frac{S_0}{4 \pi r^{2}}=\frac{S_0}{2 \tau r^{2}}$$
![inverse square law](lib/img/Inverse_square_law.svg.png)
#### $R\nu$ Reduced Gravitational Constant
In this version of Newton's Law of Gravity we introduce a new constant $G_0$, the reduced gravitational constant to accommodate for the factor of $4\pi = 2\tau$ which is has been integrated in the SI version of the gravitational constant.

$F_g =G \frac{m_{1} m_{2}}{r^{2}}= G_0 \frac{m_{1} m_{2}}{4 \pi r^{2}}=G_0 \frac{m_{1} m_{2}}{2 \tau r^{2}}$

Where: 

$G = \frac{G_{0}}{2\tau} = 6.67384 \times 10^{-11} \frac{N \cdot m^2}{kg^2}$

Analyzing the units:
$$\frac{N \cdot m^2}{kg^2} = \left( \frac{\left( kg \cdot \frac{m}{s^2} \right) \cdot m^2}{kg^2} \right)=\frac{m^3}{s^2 kg}$$
Converting seconds to meters with the SI speed of light as a conversion factor: 
$$\frac{m^3}{s^2 kg}\cdot\frac{1}{c^2}=\frac{m^3}{s^2 kg}\cdot\frac{s^2}{m^2}=\frac{m}{kg}$$

Thus where space and time are measured in units of meters, the reduced gravitational constant,  is:

$$\boxed{G_0=\frac{2\tau G}{c^2}=\frac{2\tau \cdot 6.67384 \times 10^{-11}}{299792458^2} \frac{m}{kg} = 9.33135 \times 10^-27 \frac{m}{kg}}$$

>  Observation
> This implies that not only can space an time be measure in units of meters, but so can mass.

### Relativistic Energy Momentum Relation

Einsteins [Relativistic Energy Momentum](https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation) relationship shows a Pythagorean relation between the total energy ($E$), rest mass ($m_0$) and momentum ($p$) of a system.

$E^{2} = (m_0 \cdot c^2)^2 + (p \cdot c)^2$

Where space and time are both measure in units of meters, c=1.

$E^2=(m_0)^2+(p)^2$

From this we can see that Energy, Momentum and Mass have equivalent units. 

>  
> _While we do not really know what energy, mass and momentum are we know that they are fundamentally “made” out of the same stuff because they have the same units._

##### Objects of mass at rest

For an object at rest with no momentum ($p = 0$) we see Einstein's famous equations:

$E = m_{0} \cdot c^2$

Or, with $c=1$, this is much simpler to understand.  Energy = Mass

$E = m_0$

##### Zero mass objects moving at the speed of light

And for objects with no mass, like photos, ($m_{0}= 0$):

$E=pc$

Or, with $c=1$, this is much simpler to understand.  Energy = Momentum

$E=p$

## Planck's Constant

The [Reduced Planck constant](https://en.wikipedia.org/wiki/Planck_constant) , ħ, represents a conversion factor for relating the frequency, $\omega$ (in $2\pi$ radians per second), of a photon to the energy of that photon. This can easily be seen from the simple but profound relationship:

$E=\hbar\omega$

Where:

$\hbar=1.054571726 \times 10^{−34} J \cdot s$

and

$J \cdot s = {kg}\cdot\frac{m^2}{s}$

>  Reduced Planck's Constant
> $\hbar = \frac{h}{2\pi} = \frac{h}{\tau}$


Simplifying our units by converting time and mass to units of meters:
$$\boxed{\hbar=1.054571726 \times 10^{−34} {kg}\cdot\frac{m^2}{s}\cdot\frac{G_0}{c}=3.282462\times10^{-69}m^2}$$

Which suggest that the Plank constant can be interpreted as an areas for which the square root of is suspiciously close to the Plank length:

$$\boxed{\sqrt{\hbar}=\sqrt{3.282462\times10^{-69}m^2}=5.72928\times10^{-35}m}$$

#### Planck Area

The [Planck Area](https://en.wikipedia.org/wiki/Planck_units#Derived_units) is the square of the [Planck Length](https://en.wikipedia.org/wiki/Planck_units#Planck_length).

$l_{P}= \sqrt{\frac{\hbar G}{c^3}}$     

and      $l_{P}^{2}= \frac{\hbar G}{c^3}$

In $R\nu$ units both $c$ and $G_o$ are 1.

$l_{P} = \sqrt{\hbar}$        

and        $l_P^{2}=\hbar$
## Bekenstein's Bound
After having recently read _Three Roads to Quantum Gravity_ by Lee Smolin, I now suspect the meaning of this areas is related to the [Bekensteins Law](https://en.wikipedia.org/wiki/Bekenstein_bound) as applied to a surface areas surrounding a mass. Where the [thermodynamic entropy](https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)), _S_, is proportional to the the enclosed surface area, $A$.

$S=\frac{1}{4}\cdot\frac{A}{G\hbar}$

$S=\frac{k c^{3} A}{4 G \hbar}$


$S \le \frac{2\pi k R E}{\hbar c} = \frac{\tau R k E}{\hbar c}$

From our new values for $G_0$and $\hbar$ we can likely rewrite this:

$S=\frac{\pi\cdot A}{\hbar G_0}$

With the limiting case being at the Plank scale.

$S=\frac{\pi\cdot \sqrt{\hbar}}{\hbar G_0}$

## Planck Length
https://en.wikipedia.org/wiki/Planck_length

The concept of the Planck Length comes from exploring the limits of Quantum Mechanics and General Relativity.  The limits of General Relativity can be seen a the event horizon of a black hole, described by the  Schwarzschild Radius.  And the limits of Quantum Mechanics can be found in the  Compton Wavelength for a given quanta. 

The [Schwarzschild Radius](https://simple.wikipedia.org/wiki/Schwarzschild_radius) is defined as the distance at which light cannot escape from the gravitational field of a mass (m):

Classic Derivation.

$r_S=\frac{2G m}{c^2}$

The reduced [Compton Wavelength](https://en.wikipedia.org/wiki/Compton_wavelength) represents a lower limit on the wavelength for quanta that can interact with a quantum particle with mass (m):

$\lambda_C=\frac{h}{m c}$

$\bar{\lambda_C}=\frac{2\pi\hbar}{m c}=\frac{\tau\hbar}{m c}$

And set the Schwarzschild Radius equal to the Compton Wavelength:
$r_S=\lambda_C$

$\frac{2Gm}{c^{2}}=\frac{h}{m c}$

$m^{2}= \frac{hc}{2G}$

$m = \sqrt{\frac{hc}{2G}}$

$l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^2}$
$l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^{2}}= \sqrt{\frac{2Gh}{c^2}}$

With reduced Compton Wavelength
$r_S=\bar{\lambda_C}$

$\frac{2Gm}{c^{2}}=\frac{\tau\hbar}{m c}$

$m^2=\frac{\tau\ \hbar\ c}{2G}$

$m = \sqrt{\frac{\tau\ \hbar\ c}{2G}}$

$l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\ c}{2G}}}{c^2}$

$l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\ c}{2G}}}{c^{2}}=\sqrt{\frac{4\ \tau\ G\ \hbar}{c^3}}$

If we reduce the units in these equation to those of mass and time measured in meters. 

$l_P=\sqrt{4\tau\hbar G_{o}}$

and 

$\lambda_C=\frac{\hbar}{m}$

$m=R_s=\lambda_C=\frac{\hbar}{m}$

This is known as the Planck Mass, $M_P$.
$M_P=m=\sqrt{\hbar}$

Solving the Compton Wavelength for distance we find the classic Plank Length:

$\lambda_C=\frac{\hbar}{\sqrt{\hbar}}=\frac{\sqrt{\hbar}}{\sqrt{\hbar}}\cdot\frac{\hbar}{\sqrt{\hbar}}=\sqrt{\hbar}=L_P$

Which is in precise agreement with the value we found in above. Thus the Plank Length is:

$L_P=\sqrt{\hbar}=5.72928\times10^{-35}m$

When we measure distance, time, and mass in units of distance, c=1, and the Plank Time, $T_P$, is equal to Plank Length, $L_P$, which is equal to the Plank Mass, $M_P$:

$$\boxed{L_P=T_P=M_P}$$

Conversion Factor | Symbol | Value
--- | --- | ----
meters to Planck Length | $\chi_P$ | $1.74542\times10^{34} \frac{L}{m}$
seconds to Planck Length | $\tau_p$ | $5.23264\times10^{42} \frac{L}{s}$
mass to Planck Length | $G_P$ | $1.62871\times10^8 \frac{L}{kg}$
energy to Planck Length | $E_P$ | $1.81219\times10^9 \frac{L}{J}$
momentum to Planck Length | $P_P$ | $5.43280\times10^{-1} \frac{L\cdot s}{kg \cdot m}$
temperature to Planck Length | $k_P$ | $2.501998\times10^{-14} \frac{L}{K}$
charge to Planck Length | $C_P$| $1.89007\times10^{18} \frac{L}{C}$

Applying conversion factors from the table above, we can convert SI values to Reduced Natural Units.
$c=\frac{1}{\sqrt{\epsilon_o \mu_o}}$

Quantity | Symbol | SI | $\nu$ 
--- | --- | --- | ----
Speed of Light | $c$ | $299792458 \frac{m}{s}$ | 1
Gravitational Constant | $G_0$ | $8.38659\times10^{-10} \frac{m^3}{kg \cdot s^2}$ | 1
Boltzmann's Constant | $k$ | $k=1.380649\times10^-23 \frac{J}{K}$ | 1
Permittivity of Free Space | $\epsilon_o$ | $8.854187817620\times10^{-12} \frac{C^{2}s^2}{kg \cdot m^3}$ | 1
Permeability of Free Space | $\mu_o$ | $\huge{\frac{1}{\epsilon_{o} \cdot c^{2}}}$ | 1
Planck's Constant | $\hbar$ | $1.054571726\times10^-34 \frac{kg \cdot m^2}{s}$ | $1 L^2$
Mass of the Electron | $m_e$ | $9.10938\times10^{-31} kg$ | $1.48366\times10^{-22} L$
Charge of the Electron | $e^-$ | $-1.60218\times10^{-19} C$ | $-3.02822\times10^{-1} L$

## Fine Structure Constant
https://en.wikipedia.org/wiki/Fine-structure_constant
As a consistency check, we compute the _Fine Structure Constant_ using Reduced Natural Units which is a unit less ratio that should be independent of our system of units.

$\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar c}=\frac{e^2}{4\pi}=0.00729735≈\frac{1}{137}}$

This check confirms that our system of Reduced Natural Units has internally consistent values for $c$, $\epsilon_o$, $\hbar$ and $e-$. And also $G_o$ which was used to computer prior values is also consistent.

## Sage Code
Unit Analysis computations have been performed with [Sage Math](https://www.sagemath.org/).

```bash
# Define constance
one = 1.n(digits=6)
pi = pi.n(digits=6)
tau = 2 * pi
t = tau

# Define the units
meters = var('m')
m = one*meters

seconds = var('s')
s = seconds

kilograms = var('kg')
kg = kilograms

newtons = kg * m / s^2
N = newtons

joules = N * m
J = joules

print("pi    =", pi)
print("tau   =", t)

# Speed of light in meters/second
speed_of_light = 299792458 * meters/seconds
sol = speed_of_light
c = sol
print("si c  =", c)

rnu_c = c / c
print("R\u03BD c  =", rnu_c)

# Gravitational Constant
gravitational_constant = 6.67384e-11 * N*(m^2/kg^2)
G = gravitational_constant
print("si G  =", G)

rnu_G = 4*pi*G/c^2
Go = rnu_G
print("R\u03BD Go =", Go)


# Planck's Constant
reduced_plancks_constant = 1.054571726e-34 * J*s
h_bar = reduced_plancks_constant
print("si \u210F  =", h_bar)

rnu_h_bar = h_bar * Go / c
print("R\u03BD "u"\u210F  =", rnu_h_bar)

# Planck Length
rnu_h_bar_str = str(rnu_h_bar) 
numerical_part_str = rnu_h_bar_str.split('*')[0]  
numerical_part_str = numerical_part_str.strip('()')
numerical_part = float(numerical_part_str)
rnu_sqrt_h_bar = numerical_part^(1/2)
# ^ Sage cannot process sqrt on units... Lame.
lP = rnu_sqrt_h_bar * m
print("R\u03BD \u221A\u210F =", lP)
```

#### Output
```
pi    = 3.14159
tau   = 6.28319
si c  = 299792458*m/s
Rν c  = 1
si G  = (6.67384e-11)*m^3/(kg*s^2)
Rν Go = (9.33135e-27)*m/kg
si ℏ  = (1.05457e-34)*kg*m^2/s
Rν ℏ  = (3.28246e-69)*m^2
Rν √ℏ = (5.72928e-35)*m
si lP = (1.61620e-35)*sqrt(m^2)
Rν lP = (2.77455e-47)*sqrt(m^3/kg)
```