797 lines
29 KiB
TeX
797 lines
29 KiB
TeX
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% =========================================================
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% Concept Boxes
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% =========================================================
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\author{}
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\date{}
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\begin{document}
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\frontmatter
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\renewcommand*\contentsname{Contents}
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{
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}
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\mainmatter
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\hypertarget{introduction}{%
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\chapter{Introduction}\label{introduction}}
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\begin{quote}
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``\emph{If I have seen further it is by standing on the shoulders of
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Giants.}''
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-- Isaac Newton
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\end{quote}
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\hypertarget{conventions}{%
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\section{Conventions}\label{conventions}}
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In this book we'll use a few conventions.
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\hypertarget{new-concepts}{%
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\subsection{New Concepts}\label{new-concepts}}
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As many of the topics discussed in this book are a mix of
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\textbf{established} math and physics, \textbf{proposed} dualistic
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interpretations of established ideas, and also \textbf{speculative}
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ideas that I don't yet know how to address, I wanted a way to clearly
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distinguish these concepts. I've come up with the following convention
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to highlight these classes of concepts to indicate their level of
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mainstream acceptance.
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In this book, established concept may be highlighted in green, and
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represent mainstream physics or math concepts.
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\begin{establishedbox}
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Established Concept Einsteins Relativistic Dynamics Equations
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\[E^2 = (m_{0} \cdot c^2)^2 + (p \cdot c)^2 \]
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\end{establishedbox}
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New ideas proposed by the author which have not been peer reviewed,
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verified or tested, and should be looked at with scrutiny.
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\begin{proposedbox}
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Proposed Concept With the speed of light, \(c = 1\):
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\[E^2 = m_{0}^2 + p^2 \]
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\end{proposedbox}
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Speculative Idea, that the author wonders about, but does not know how
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to demonstrate, or ideas that need further treatment to prove or
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disprove.
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\begin{speculativebox}
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Speculative Concept With the speed of light, \(c = 1\):
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\[E^2 = m_{0}^2 + p^2 \]
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\end{speculativebox}
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\hypertarget{best-words-ever}{%
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\chapter{Best Words Ever}\label{best-words-ever}}
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Blah blah blah
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\hypertarget{example-content}{%
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\chapter{Example Content}\label{example-content}}
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In \(R\nu\) the
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\href{https://en.wikipedia.org/wiki/Planck_units\#Planck_length}{Planck
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Length} is the universal unit for measurement of distance, and is
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defined approximately to be:
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\[\boxed{L_P=\sqrt{\hbar}=5.72928\times10^{-35}m=1 L}\] Where \(1\ L\),
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is 1 Planck Length of distance.
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\hypertarget{si-conversion-factors}{%
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\subsection{SI Conversion Factors}\label{si-conversion-factors}}
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The following conversion factors can be used to convert observable
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quantities of measure from the \emph{SI} system of units to \(R\nu\) to
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\textasciitilde6 significant digits.
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\begin{longtable}[]{@{}
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>{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3256}}
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>{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.0930}}
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>{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.5814}}@{}}
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\toprule
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\begin{minipage}[b]{\linewidth}\raggedright
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Conversion Factor
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\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
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Symbol
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\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
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Value
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\end{minipage} \\
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\midrule
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\endhead
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meters to Planck Length & \(\chi_P\) &
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\(1.74542\times10^{34} \frac{L}{m}\) \\
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seconds to Planck Length & \(\tau_p\) &
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\(5.23264\times10^{42} \frac{L}{s}\) \\
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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 & \(P_P\) &
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\(5.43280\times10^{-1} \frac{L\cdot s}{kg \cdot m}\) \\
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temperature to Planck Length & \(k_P\) &
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\(2.501998\times10^{-14} \frac{L}{K}\) \\
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charge to Planck Length & \(C_P\) &
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\(1.89007\times10^{18} \frac{L}{C}\) \\
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\bottomrule
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\end{longtable}
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\hypertarget{physical-constants}{%
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\subsection{Physical Constants}\label{physical-constants}}
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Applying conversion factors from the table above, we can convert SI
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values to Reduced Natural Units. For example, performing this analysis
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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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\begin{longtable}[]{@{}
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>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2273}}
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>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.0909}}
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>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.4545}}
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>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2273}}@{}}
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\toprule
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\begin{minipage}[b]{\linewidth}\raggedright
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Quantity
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\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
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Symbol
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\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
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SI
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\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
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\(\nu\)
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\end{minipage} \\
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\midrule
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\endhead
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Speed of Light & \(c\) & \(299792458 \frac{m}{s}\) & 1 \\
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Reduced Gravitational Constant & \(G_0\) &
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\(8.38659\times10^{-10} \frac{m^3}{kg \cdot s^2}\) & 1 \\
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Boltzmann's Constant & \(k\) & \(k=1.380649\times10^-23 \frac{J}{K}\) &
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1 \\
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Permittivity of Free Space & \(\epsilon_o\) &
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\(8.854187817620\times10^{-12} \frac{C^{2}s^2}{kg \cdot m^3}\) & 1 \\
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Permeability of Free Space & \(\mu_o\) &
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\(\huge{\frac{1}{\epsilon_{o} \cdot c^{2}}}\) & 1 \\
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Reduced Planck's Constant & \(\hbar\) &
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\(1.054571726\times10^-34 \frac{kg \cdot m^2}{s}\) & \(1 L^2\) \\
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Mass of the Electron & \(m_e\) & \(9.10938\times10^{-31} kg\) &
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\(1.48366\times10^{-22} L\) \\
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Charge of the Electron & \(e^-\) & \(-1.60218\times10^{-19} C\) &
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\(-3.02822\times10^{-1} L\) \\
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Unit Cycle & \(\Theta\) & \(2\pi = 6.28318...\ Radians\) &
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\(1 \tau = 6.28318...\ Radians\) \\
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\bottomrule
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\end{longtable}
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\hypertarget{fine-structure-constant}{%
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\section{Fine Structure Constant}\label{fine-structure-constant}}
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As a consistency check, we compute the
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\emph{\href{https://en.wikipedia.org/wiki/Fine-structure_constant}{Fine
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Structure Constant}} using Reduced Natural Units which is a unit less
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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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\hypertarget{dimensional-analysis}{%
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\subsubsection{Dimensional Analysis}\label{dimensional-analysis}}
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The reader should be familiar with high school physics and chemistry
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\href{https://en.wikipedia.org/wiki/Dimensional_analysis}{dimensional
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analysis}.
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\begin{itemize}
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\tightlist
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\item
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\(1\ meter\ (m) = 100\ centimeters\ (cm)\)
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\item
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\(1\ kilometer\ (km) = 1000\ meters\ (m)\)
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\item
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\(1\ mile = 5280\ feet\ (ft\ or\ ')\)
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\item
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\(1\ foot\ (ft\ or\ ') = 12\ inches\ (in\ or\ ")\)
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\item
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\(1\ inch\ (") = 2.54\ centimeters\ (cm)\)
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\end{itemize}
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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
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until we are left with only km.
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\hypertarget{newtons-law-of-gravity}{%
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\section{Newton's Law of Gravity}\label{newtons-law-of-gravity}}
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The force of gravity (\(F_g\)) between 2 masses, \(m1\) and \(m2\)
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separated by distance \(r\) is given by
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\href{https://en.wikipedia.org/wiki/Newton\%27s_law_of_universal_gravitation}{Newton's
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Law of Gravity}:
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\(F_{g} = G \frac{m_{1} m_{2}}{r^{2}}\)
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Where \(G\), is the
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\href{http://en.wikipedia.org/wiki/Gravitational_Constant}{Gravitational
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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
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distributing gravitational flux over the surface area of a sphere
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(\(4\pi r^2\)).
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\hypertarget{inverse-square-law}{%
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\subsubsection{Inverse Square Law}\label{inverse-square-law}}
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Any source of a signal strength (\(S_0\)) that radiates isotropically in
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3-dimensional space will distribute that signal strength (\(S_0\)) over
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the surface area of a sphere (\(SA = 4 \pi r\)) of radius (\(r\)). Such
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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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\includegraphics{lib/img/Inverse_square_law.svg.png} \#\#\#\# \(R\nu\)
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Reduced Gravitational Constant In this version of Newton's Law of
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Gravity we introduce a new constant \(G_0\), the reduced gravitational
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constant to accommodate for the factor of \(4\pi = 2\tau\) which is has
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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
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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:
|
||
|
||
\[\boxed{G_0=\frac{2\tau G}{c^2}=\frac{2\tau \cdot 6.67384 \times 10^{-11}}{299792458^2} \frac{m}{kg} = 9.33135 \times 10^-27 \frac{m}{kg}}\]
|
||
|
||
\begin{quote}
|
||
Observation This implies that not only can space an time be measure in
|
||
units of meters, but so can mass.
|
||
\end{quote}
|
||
|
||
\hypertarget{relativistic-energy-momentum-relation}{%
|
||
\subsection{Relativistic Energy Momentum
|
||
Relation}\label{relativistic-energy-momentum-relation}}
|
||
|
||
Einsteins
|
||
\href{https://en.wikipedia.org/wiki/Energy\%E2\%80\%93momentum_relation}{Relativistic
|
||
Energy Momentum} 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.
|
||
|
||
\begin{quote}
|
||
\emph{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.}
|
||
\end{quote}
|
||
|
||
\hypertarget{objects-of-mass-at-rest}{%
|
||
\paragraph{Objects of mass at rest}\label{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
|
||
|
||
\(E=p\)
|
||
|
||
\hypertarget{plancks-constant}{%
|
||
\section{Planck's Constant}\label{plancks-constant}}
|
||
|
||
The \href{https://en.wikipedia.org/wiki/Planck_constant}{Reduced 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
|
||
|
||
\(J \cdot s = {kg}\cdot\frac{m^2}{s}\)
|
||
|
||
\begin{quote}
|
||
Reduced Planck's Constant \(\hbar = \frac{h}{2\pi} = \frac{h}{\tau}\)
|
||
\end{quote}
|
||
|
||
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:
|
||
|
||
\[\boxed{\sqrt{\hbar}=\sqrt{3.282462\times10^{-69}m^2}=5.72928\times10^{-35}m}\]
|
||
|
||
\hypertarget{planck-area}{%
|
||
\subsubsection{Planck Area}\label{planck-area}}
|
||
|
||
The
|
||
\href{https://en.wikipedia.org/wiki/Planck_units\#Derived_units}{Planck
|
||
Area} is the square of the
|
||
\href{https://en.wikipedia.org/wiki/Planck_units\#Planck_length}{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
|
||
\emph{Three Roads to Quantum Gravity} by Lee Smolin, I now suspect the
|
||
meaning of this areas is related to the
|
||
\href{https://en.wikipedia.org/wiki/Bekenstein_bound}{Bekensteins Law}
|
||
as applied to a surface areas surrounding a mass. Where the
|
||
\href{https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)}{thermodynamic
|
||
entropy}, \emph{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.
|
||
|
||
\(S=\frac{\pi\cdot \sqrt{\hbar}}{\hbar G_0}\)
|
||
|
||
\hypertarget{planck-length}{%
|
||
\section{Planck Length}\label{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
|
||
\href{https://simple.wikipedia.org/wiki/Schwarzschild_radius}{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
|
||
\href{https://en.wikipedia.org/wiki/Compton_wavelength}{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, \(M_P\):
|
||
|
||
\[\boxed{L_P=T_P=M_P}\]
|
||
|
||
\begin{longtable}[]{@{}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3000}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3000}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.4000}}@{}}
|
||
\toprule
|
||
\begin{minipage}[b]{\linewidth}\raggedright
|
||
Conversion Factor
|
||
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
|
||
Symbol
|
||
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
|
||
Value
|
||
\end{minipage} \\
|
||
\midrule
|
||
\endhead
|
||
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 & \(C_P\) &
|
||
\(1.89007\times10^{18} \frac{L}{C}\) \\
|
||
\bottomrule
|
||
\end{longtable}
|
||
|
||
Applying conversion factors from the table above, we can convert SI
|
||
values to Reduced Natural Units. \(c=\frac{1}{\sqrt{\epsilon_o \mu_o}}\)
|
||
|
||
\begin{longtable}[]{@{}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2308}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2308}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2308}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.3077}}@{}}
|
||
\toprule
|
||
\begin{minipage}[b]{\linewidth}\raggedright
|
||
Quantity
|
||
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
|
||
Symbol
|
||
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
|
||
SI
|
||
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
|
||
\(\nu\)
|
||
\end{minipage} \\
|
||
\midrule
|
||
\endhead
|
||
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 & \(e^-\) & \(-1.60218\times10^{-19} C\) &
|
||
\(-3.02822\times10^{-1} L\) \\
|
||
\bottomrule
|
||
\end{longtable}
|
||
|
||
\hypertarget{fine-structure-constant-1}{%
|
||
\section{Fine Structure Constant}\label{fine-structure-constant-1}}
|
||
|
||
https://en.wikipedia.org/wiki/Fine-structure\_constant As a consistency
|
||
check, we compute the \emph{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.
|
||
|
||
\hypertarget{sage-code}{%
|
||
\section{Sage Code}\label{sage-code}}
|
||
|
||
Unit Analysis computations have been performed with
|
||
\href{https://www.sagemath.org/}{Sage Math}.
|
||
|
||
\begin{Shaded}
|
||
\begin{Highlighting}[]
|
||
\CommentTok{\# Define constance}
|
||
\ExtensionTok{one}\NormalTok{ = 1.n}\ErrorTok{(}\VariableTok{digits}\OperatorTok{=}\NormalTok{6}\KeywordTok{)}
|
||
\ExtensionTok{pi}\NormalTok{ = pi.n}\ErrorTok{(}\VariableTok{digits}\OperatorTok{=}\NormalTok{6}\KeywordTok{)}
|
||
\ExtensionTok{tau}\NormalTok{ = 2 }\PreprocessorTok{*}\NormalTok{ pi}
|
||
\ExtensionTok{t}\NormalTok{ = tau}
|
||
|
||
\CommentTok{\# Define the units}
|
||
\ExtensionTok{meters}\NormalTok{ = var}\ErrorTok{(}\StringTok{\textquotesingle{}m\textquotesingle{}}\KeywordTok{)}
|
||
\ExtensionTok{m}\NormalTok{ = one}\PreprocessorTok{*}\NormalTok{meters}
|
||
|
||
\ExtensionTok{seconds}\NormalTok{ = var}\ErrorTok{(}\StringTok{\textquotesingle{}s\textquotesingle{}}\KeywordTok{)}
|
||
\ExtensionTok{s}\NormalTok{ = seconds}
|
||
|
||
\ExtensionTok{kilograms}\NormalTok{ = var}\ErrorTok{(}\StringTok{\textquotesingle{}kg\textquotesingle{}}\KeywordTok{)}
|
||
\ExtensionTok{kg}\NormalTok{ = kilograms}
|
||
|
||
\ExtensionTok{newtons}\NormalTok{ = kg }\PreprocessorTok{*}\NormalTok{ m / s\^{}2}
|
||
\ExtensionTok{N}\NormalTok{ = newtons}
|
||
|
||
\ExtensionTok{joules}\NormalTok{ = N }\PreprocessorTok{*}\NormalTok{ m}
|
||
\ExtensionTok{J}\NormalTok{ = joules}
|
||
|
||
\ExtensionTok{print}\ErrorTok{(}\StringTok{"pi ="}\ExtensionTok{,}\NormalTok{ pi}\KeywordTok{)}
|
||
\ExtensionTok{print}\ErrorTok{(}\StringTok{"tau ="}\ExtensionTok{,}\NormalTok{ t}\KeywordTok{)}
|
||
|
||
\CommentTok{\# Speed of light in meters/second}
|
||
\ExtensionTok{speed\_of\_light}\NormalTok{ = 299792458 }\PreprocessorTok{*}\NormalTok{ meters/seconds}
|
||
\ExtensionTok{sol}\NormalTok{ = speed\_of\_light}
|
||
\ExtensionTok{c}\NormalTok{ = sol}
|
||
\ExtensionTok{print}\ErrorTok{(}\StringTok{"si c ="}\ExtensionTok{,}\NormalTok{ c}\KeywordTok{)}
|
||
|
||
\ExtensionTok{rnu\_c}\NormalTok{ = c / c}
|
||
\ExtensionTok{print}\ErrorTok{(}\StringTok{"R\textbackslash{}u03BD c ="}\ExtensionTok{,}\NormalTok{ rnu\_c}\KeywordTok{)}
|
||
|
||
\CommentTok{\# Gravitational Constant}
|
||
\ExtensionTok{gravitational\_constant}\NormalTok{ = 6.67384e{-}11 }\PreprocessorTok{*}\NormalTok{ N}\PreprocessorTok{*(}\NormalTok{m\^{}2/kg\^{}2}\PreprocessorTok{)}
|
||
\ExtensionTok{G}\NormalTok{ = gravitational\_constant}
|
||
\ExtensionTok{print}\ErrorTok{(}\StringTok{"si G ="}\ExtensionTok{,}\NormalTok{ G}\KeywordTok{)}
|
||
|
||
\ExtensionTok{rnu\_G}\NormalTok{ = 4}\PreprocessorTok{*}\NormalTok{pi}\PreprocessorTok{*}\NormalTok{G/c\^{}2}
|
||
\ExtensionTok{Go}\NormalTok{ = rnu\_G}
|
||
\ExtensionTok{print}\ErrorTok{(}\StringTok{"R\textbackslash{}u03BD Go ="}\ExtensionTok{,}\NormalTok{ Go}\KeywordTok{)}
|
||
|
||
|
||
\CommentTok{\# Planck\textquotesingle{}s Constant}
|
||
\ExtensionTok{reduced\_plancks\_constant}\NormalTok{ = 1.054571726e{-}34 }\PreprocessorTok{*}\NormalTok{ J}\PreprocessorTok{*}\NormalTok{s}
|
||
\ExtensionTok{h\_bar}\NormalTok{ = reduced\_plancks\_constant}
|
||
\ExtensionTok{print}\ErrorTok{(}\StringTok{"si \textbackslash{}u210F ="}\ExtensionTok{,}\NormalTok{ h\_bar}\KeywordTok{)}
|
||
|
||
\ExtensionTok{rnu\_h\_bar}\NormalTok{ = h\_bar }\PreprocessorTok{*}\NormalTok{ Go / c}
|
||
\ExtensionTok{print}\ErrorTok{(}\StringTok{"R\textbackslash{}u03BD "}\ExtensionTok{u}\StringTok{"\textbackslash{}u210F ="}\ExtensionTok{,}\NormalTok{ rnu\_h\_bar}\KeywordTok{)}
|
||
|
||
\CommentTok{\# Planck Length}
|
||
\ExtensionTok{rnu\_h\_bar\_str}\NormalTok{ = str}\ErrorTok{(}\ExtensionTok{rnu\_h\_bar}\KeywordTok{)}
|
||
\ExtensionTok{numerical\_part\_str}\NormalTok{ = rnu\_h\_bar\_str.split}\ErrorTok{(}\StringTok{\textquotesingle{}*\textquotesingle{}}\KeywordTok{)}\ExtensionTok{[0]}
|
||
\ExtensionTok{numerical\_part\_str}\NormalTok{ = numerical\_part\_str.strip}\ErrorTok{(}\StringTok{\textquotesingle{}()\textquotesingle{}}\KeywordTok{)}
|
||
\ExtensionTok{numerical\_part}\NormalTok{ = float}\ErrorTok{(}\ExtensionTok{numerical\_part\_str}\KeywordTok{)}
|
||
\ExtensionTok{rnu\_sqrt\_h\_bar}\NormalTok{ = numerical\_part\^{}}\ErrorTok{(}\ExtensionTok{1/2}\KeywordTok{)}
|
||
\CommentTok{\# \^{} Sage cannot process sqrt on units... Lame.}
|
||
\ExtensionTok{lP}\NormalTok{ = rnu\_sqrt\_h\_bar }\PreprocessorTok{*}\NormalTok{ m}
|
||
\ExtensionTok{print}\ErrorTok{(}\StringTok{"R\textbackslash{}u03BD \textbackslash{}u221A\textbackslash{}u210F ="}\ExtensionTok{,}\NormalTok{ lP}\KeywordTok{)}
|
||
\end{Highlighting}
|
||
\end{Shaded}
|
||
|
||
\hypertarget{output}{%
|
||
\subsubsection{Output}\label{output}}
|
||
|
||
\begin{verbatim}
|
||
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)
|
||
\end{verbatim}
|
||
|
||
\hypertarget{terminology}{%
|
||
\chapter{Terminology}\label{terminology}}
|
||
|
||
\hypertarget{citations}{%
|
||
\chapter{Citations}\label{citations}}
|
||
|
||
\backmatter
|
||
\end{document}
|