370 lines
15 KiB
Markdown
370 lines
15 KiB
Markdown
# Example Content
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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:
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$$\boxed{L_P=\sqrt{\hbar}=5.72928\times10^{-35}m=1 L}$$
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Where $1\ L$, is 1 Planck Length of distance.
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### SI Conversion Factors
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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.
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| Conversion Factor | Symbol | Value |
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| ---------------------------- | -------- | -------------------------------------------------- |
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| meters to Planck Length | $\chi_P$ | $1.74542\times10^{34} \frac{L}{m}$ |
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| seconds to Planck Length | $\tau_p$ | $5.23264\times10^{42} \frac{L}{s}$ |
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| mass to Planck Length | $G_P$ | $1.62871\times10^8 \frac{L}{kg}$ |
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| energy to Planck Length | $E_P$ | $1.81219\times10^9 \frac{L}{J}$ |
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| momentum to Planck Length | $P_P$ | $5.43280\times10^{-1} \frac{L\cdot s}{kg \cdot m}$ |
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| temperature to Planck Length | $k_P$ | $2.501998\times10^{-14} \frac{L}{K}$ |
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| charge to Planck Length | $C_P$ | $1.89007\times10^{18} \frac{L}{C}$ |
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### Physical Constants
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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:
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$c = 299792458 \frac{m}{s} = 299792458 \frac{m}{s} \cdot 1.74542\times10^{34} \frac{L}{m} \cdot \frac{1}{5.23264\times10^{42} \frac{L}{s}} = 1.00000$
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| Quantity | Symbol | SI | $\nu$ |
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| ------------------------------ | ------------ | ------------------------------------------------------------ | ------------------------------ |
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| Speed of Light | $c$ | $299792458 \frac{m}{s}$ | 1 |
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| Reduced Gravitational Constant | $G_0$ | $8.38659\times10^{-10} \frac{m^3}{kg \cdot s^2}$ | 1 |
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| Boltzmann's Constant | $k$ | $k=1.380649\times10^-23 \frac{J}{K}$ | 1 |
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| Permittivity of Free Space | $\epsilon_o$ | $8.854187817620\times10^{-12} \frac{C^{2}s^2}{kg \cdot m^3}$ | 1 |
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| Permeability of Free Space | $\mu_o$ | $\huge{\frac{1}{\epsilon_{o} \cdot c^{2}}}$ | 1 |
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| Reduced Planck's Constant | $\hbar$ | $1.054571726\times10^-34 \frac{kg \cdot m^2}{s}$ | $1 L^2$ |
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| Mass of the Electron | $m_e$ | $9.10938\times10^{-31} kg$ | $1.48366\times10^{-22} L$ |
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| Charge of the Electron | $e^-$ | $-1.60218\times10^{-19} C$ | $-3.02822\times10^{-1} L$ |
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| Unit Cycle | $\Theta$ | $2\pi = 6.28318...\ Radians$ | $1 \tau = 6.28318...\ Radians$ |
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## Fine Structure Constant
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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.
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$\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar c}=\frac{e^2}{2\tau}=0.00729735≈\frac{1}{137}}$
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#### Dimensional Analysis
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The reader should be familiar with high school physics and chemistry [dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis).
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- $1\ meter\ (m) = 100\ centimeters\ (cm)$
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- $1\ kilometer\ (km) = 1000\ meters\ (m)$
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- $1\ mile = 5280\ feet\ (ft\ or\ ')$
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- $1\ foot\ (ft\ or\ ') = 12\ inches\ (in\ or\ ")$
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- $1\ inch\ (") = 2.54\ centimeters\ (cm)$
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How many kilometers are in 1 mile?
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$1\ mile = 1\ mile \times \frac{5280\ ft}{mile} \times \frac{12\ in}{ft} \times \frac{2.54\ cm}{in}\times \frac{1\ m}{100 cm} \times \frac{1\ km}{1000 m} = \frac{5280 \times 12 \times 2.54}{100 \times 1000}\ km = \frac{160934.40}{100000}\ km = 1.6\ km$
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Note that each unit in the denominator cancels with one if the numerator until we are left with only km.
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## Newton's Law of Gravity
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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):
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$F_{g} = G \frac{m_{1} m_{2}}{r^{2}}$
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Where $G$, is the [Gravitational Constant](http://en.wikipedia.org/wiki/Gravitational_Constant).
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$G = 6.67430 \times 10^{-11}\ N\frac{m^2}{kg^2}$
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The strength of gravitational force follow the inverse square law distributing gravitational flux over the surface area of a sphere ($4\pi r^2$).
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#### Inverse Square Law
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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:
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$$I(r) = \frac{S_0}{4 \pi r^{2}}=\frac{S_0}{2 \tau r^{2}}$$
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#### $R\nu$ Reduced Gravitational Constant
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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.
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$F_g =G \frac{m_{1} m_{2}}{r^{2}}= G_0 \frac{m_{1} m_{2}}{4 \pi r^{2}}=G_0 \frac{m_{1} m_{2}}{2 \tau r^{2}}$
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Where:
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$G = \frac{G_{0}}{2\tau} = 6.67384 \times 10^{-11} \frac{N \cdot m^2}{kg^2}$
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Analyzing the units:
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$$\frac{N \cdot m^2}{kg^2} = \left( \frac{\left( kg \cdot \frac{m}{s^2} \right) \cdot m^2}{kg^2} \right)=\frac{m^3}{s^2 kg}$$
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Converting seconds to meters with the SI speed of light as a conversion factor:
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$$\frac{m^3}{s^2 kg}\cdot\frac{1}{c^2}=\frac{m^3}{s^2 kg}\cdot\frac{s^2}{m^2}=\frac{m}{kg}$$
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Thus where space and time are measured in units of meters, the reduced gravitational constant, is:
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$$\boxed{G_0=\frac{2\tau G}{c^2}=\frac{2\tau \cdot 6.67384 \times 10^{-11}}{299792458^2} \frac{m}{kg} = 9.33135 \times 10^-27 \frac{m}{kg}}$$
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> Observation
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> This implies that not only can space an time be measure in units of meters, but so can mass.
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### Relativistic Energy Momentum Relation
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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.
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$E^{2} = (m_0 \cdot c^2)^2 + (p \cdot c)^2$
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Where space and time are both measure in units of meters, c=1.
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$E^2=(m_0)^2+(p)^2$
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From this we can see that Energy, Momentum and Mass have equivalent units.
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>
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> _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._
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##### Objects of mass at rest
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For an object at rest with no momentum ($p = 0$) we see Einstein's famous equations:
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$E = m_{0} \cdot c^2$
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Or, with $c=1$, this is much simpler to understand. Energy = Mass
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$E = m_0$
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##### Zero mass objects moving at the speed of light
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And for objects with no mass, like photos, ($m_{0}= 0$):
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$E=pc$
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Or, with $c=1$, this is much simpler to understand. Energy = Momentum
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$E=p$
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## Planck's Constant
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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:
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$E=\hbar\omega$
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Where:
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$\hbar=1.054571726 \times 10^{−34} J \cdot s$
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and
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$J \cdot s = {kg}\cdot\frac{m^2}{s}$
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> Reduced Planck's Constant
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> $\hbar = \frac{h}{2\pi} = \frac{h}{\tau}$
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Simplifying our units by converting time and mass to units of meters:
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$$\boxed{\hbar=1.054571726 \times 10^{−34} {kg}\cdot\frac{m^2}{s}\cdot\frac{G_0}{c}=3.282462\times10^{-69}m^2}$$
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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:
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$$\boxed{\sqrt{\hbar}=\sqrt{3.282462\times10^{-69}m^2}=5.72928\times10^{-35}m}$$
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#### Planck Area
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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).
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$l_{P}= \sqrt{\frac{\hbar G}{c^3}}$
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and $l_{P}^{2}= \frac{\hbar G}{c^3}$
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In $R\nu$ units both $c$ and $G_o$ are 1.
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$l_{P} = \sqrt{\hbar}$
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and $l_P^{2}=\hbar$
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## Bekenstein's Bound
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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, $A$.
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$S=\frac{1}{4}\cdot\frac{A}{G\hbar}$
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$S=\frac{k c^{3} A}{4 G \hbar}$
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$S \le \frac{2\pi k R E}{\hbar c} = \frac{\tau R k E}{\hbar c}$
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From our new values for $G_0$and $\hbar$ we can likely rewrite this:
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$S=\frac{\pi\cdot A}{\hbar G_0}$
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With the limiting case being at the Plank scale.
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$S=\frac{\pi\cdot \sqrt{\hbar}}{\hbar G_0}$
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## Planck Length
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https://en.wikipedia.org/wiki/Planck_length
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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.
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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):
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Classic Derivation.
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$r_S=\frac{2G m}{c^2}$
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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):
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$\lambda_C=\frac{h}{m c}$
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$\bar{\lambda_C}=\frac{2\pi\hbar}{m c}=\frac{\tau\hbar}{m c}$
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And set the Schwarzschild Radius equal to the Compton Wavelength:
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$r_S=\lambda_C$
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$\frac{2Gm}{c^{2}}=\frac{h}{m c}$
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$m^{2}= \frac{hc}{2G}$
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$m = \sqrt{\frac{hc}{2G}}$
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$l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^2}$
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$l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^{2}}= \sqrt{\frac{2Gh}{c^2}}$
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With reduced Compton Wavelength
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$r_S=\bar{\lambda_C}$
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$\frac{2Gm}{c^{2}}=\frac{\tau\hbar}{m c}$
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$m^2=\frac{\tau\ \hbar\ c}{2G}$
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$m = \sqrt{\frac{\tau\ \hbar\ c}{2G}}$
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$l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\ c}{2G}}}{c^2}$
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$l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\ c}{2G}}}{c^{2}}=\sqrt{\frac{4\ \tau\ G\ \hbar}{c^3}}$
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If we reduce the units in these equation to those of mass and time measured in meters.
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$l_P=\sqrt{4\tau\hbar G_{o}}$
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and
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$\lambda_C=\frac{\hbar}{m}$
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$m=R_s=\lambda_C=\frac{\hbar}{m}$
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This is known as the Planck Mass, $M_P$.
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$M_P=m=\sqrt{\hbar}$
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Solving the Compton Wavelength for distance we find the classic Plank Length:
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$\lambda_C=\frac{\hbar}{\sqrt{\hbar}}=\frac{\sqrt{\hbar}}{\sqrt{\hbar}}\cdot\frac{\hbar}{\sqrt{\hbar}}=\sqrt{\hbar}=L_P$
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Which is in precise agreement with the value we found in above. Thus the Plank Length is:
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$L_P=\sqrt{\hbar}=5.72928\times10^{-35}m$
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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, $M_P$:
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$$\boxed{L_P=T_P=M_P}$$
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Conversion Factor | Symbol | Value
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--- | --- | ----
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meters to Planck Length | $\chi_P$ | $1.74542\times10^{34} \frac{L}{m}$
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seconds to Planck Length | $\tau_p$ | $5.23264\times10^{42} \frac{L}{s}$
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mass to Planck Length | $G_P$ | $1.62871\times10^8 \frac{L}{kg}$
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energy to Planck Length | $E_P$ | $1.81219\times10^9 \frac{L}{J}$
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momentum to Planck Length | $P_P$ | $5.43280\times10^{-1} \frac{L\cdot s}{kg \cdot m}$
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temperature to Planck Length | $k_P$ | $2.501998\times10^{-14} \frac{L}{K}$
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charge to Planck Length | $C_P$| $1.89007\times10^{18} \frac{L}{C}$
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Applying conversion factors from the table above, we can convert SI values to Reduced Natural Units.
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$c=\frac{1}{\sqrt{\epsilon_o \mu_o}}$
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Quantity | Symbol | SI | $\nu$
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--- | --- | --- | ----
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Speed of Light | $c$ | $299792458 \frac{m}{s}$ | 1
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Gravitational Constant | $G_0$ | $8.38659\times10^{-10} \frac{m^3}{kg \cdot s^2}$ | 1
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Boltzmann's Constant | $k$ | $k=1.380649\times10^-23 \frac{J}{K}$ | 1
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Permittivity of Free Space | $\epsilon_o$ | $8.854187817620\times10^{-12} \frac{C^{2}s^2}{kg \cdot m^3}$ | 1
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Permeability of Free Space | $\mu_o$ | $\huge{\frac{1}{\epsilon_{o} \cdot c^{2}}}$ | 1
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Planck's Constant | $\hbar$ | $1.054571726\times10^-34 \frac{kg \cdot m^2}{s}$ | $1 L^2$
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Mass of the Electron | $m_e$ | $9.10938\times10^{-31} kg$ | $1.48366\times10^{-22} L$
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Charge of the Electron | $e^-$ | $-1.60218\times10^{-19} C$ | $-3.02822\times10^{-1} L$
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## Fine Structure Constant
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https://en.wikipedia.org/wiki/Fine-structure_constant
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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.
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$\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar c}=\frac{e^2}{4\pi}=0.00729735≈\frac{1}{137}}$
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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.
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## Sage Code
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Unit Analysis computations have been performed with [Sage Math](https://www.sagemath.org/).
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```bash
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# Define constance
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one = 1.n(digits=6)
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pi = pi.n(digits=6)
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tau = 2 * pi
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t = tau
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# Define the units
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meters = var('m')
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m = one*meters
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seconds = var('s')
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s = seconds
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kilograms = var('kg')
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kg = kilograms
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newtons = kg * m / s^2
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N = newtons
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joules = N * m
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J = joules
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print("pi =", pi)
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print("tau =", t)
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# Speed of light in meters/second
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speed_of_light = 299792458 * meters/seconds
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sol = speed_of_light
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c = sol
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print("si c =", c)
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rnu_c = c / c
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print("R\u03BD c =", rnu_c)
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# Gravitational Constant
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gravitational_constant = 6.67384e-11 * N*(m^2/kg^2)
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G = gravitational_constant
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print("si G =", G)
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rnu_G = 4*pi*G/c^2
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Go = rnu_G
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print("R\u03BD Go =", Go)
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# Planck's Constant
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reduced_plancks_constant = 1.054571726e-34 * J*s
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h_bar = reduced_plancks_constant
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print("si \u210F =", h_bar)
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rnu_h_bar = h_bar * Go / c
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print("R\u03BD "u"\u210F =", rnu_h_bar)
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# Planck Length
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rnu_h_bar_str = str(rnu_h_bar)
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numerical_part_str = rnu_h_bar_str.split('*')[0]
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numerical_part_str = numerical_part_str.strip('()')
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numerical_part = float(numerical_part_str)
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rnu_sqrt_h_bar = numerical_part^(1/2)
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# ^ Sage cannot process sqrt on units... Lame.
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lP = rnu_sqrt_h_bar * m
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print("R\u03BD \u221A\u210F =", lP)
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```
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#### Output
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```
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pi = 3.14159
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tau = 6.28319
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si c = 299792458*m/s
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Rν c = 1
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si G = (6.67384e-11)*m^3/(kg*s^2)
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Rν Go = (9.33135e-27)*m/kg
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si ℏ = (1.05457e-34)*kg*m^2/s
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Rν ℏ = (3.28246e-69)*m^2
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Rν √ℏ = (5.72928e-35)*m
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si lP = (1.61620e-35)*sqrt(m^2)
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Rν lP = (2.77455e-47)*sqrt(m^3/kg)
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```
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