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> [!established] Established Concept
> Einsteins Relativistic Dynamics Equations
> $$E^2 = (m_{0} \cdot c^2)^2 + (p \cdot c)^2 $$
> [!proposed] Proposed Concept
With the speed of light, $c = 1$:
$$E^2 = m_{0}^2 + p^2 $$
> [!speculative] Speculative Concept
With the speed of light, $c = 1$:
$$E^2 = m_{0}^2 + p^2 $$
> [!caution] Caution Note
> Beware of this section.
> [!warning] Warning Note
> Beware of this section.
> [!danger] Alerts
> Extreme Highlight
# 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:
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.
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:
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.
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:
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.
> 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:
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):
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 | $E_P$ | $1.81219\times10^9 \frac{L}{J}$
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.
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/).
John Haverlack