1230 lines
39 KiB
TeX
1230 lines
39 KiB
TeX
% Options for packages loaded elsewhere
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}{% if KOMA class
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pdfcreator={LaTeX via pandoc}}
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\usepackage{longtable,booktabs,array}
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\usepackage{calc} % for calculating minipage widths
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% Correct order of tables after \paragraph or \subparagraph
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\usepackage{etoolbox}
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\makeatletter
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\patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{}
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\makeatother
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||
% Allow footnotes in longtable head/foot
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\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}}
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\makesavenoteenv{longtable}
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\usepackage{graphicx}
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||
\makeatletter
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||
\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi}
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||
\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi}
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||
\makeatother
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% Scale images if necessary, so that they will not overflow the page
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% margins by default, and it is still possible to overwrite the defaults
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% using explicit options in \includegraphics[width, height, ...]{}
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\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio}
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% Set default figure placement to htbp
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||
\makeatletter
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||
\def\fps@figure{htbp}
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\makeatother
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\setlength{\emergencystretch}{3em} % prevent overfull lines
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\providecommand{\tightlist}{%
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\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
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\setcounter{secnumdepth}{5}
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\newlength{\cslhangindent}
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\setlength{\cslhangindent}{1.5em}
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\newlength{\csllabelwidth}
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\setlength{\csllabelwidth}{3em}
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||
\newlength{\cslentryspacingunit} % times entry-spacing
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\setlength{\cslentryspacingunit}{\parskip}
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\newenvironment{CSLReferences}[2] % #1 hanging-ident, #2 entry spacing
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{% don't indent paragraphs
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||
\setlength{\parindent}{0pt}
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% turn on hanging indent if param 1 is 1
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||
\ifodd #1
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\let\oldpar\par
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\def\par{\hangindent=\cslhangindent\oldpar}
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\fi
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% set entry spacing
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\setlength{\parskip}{#2\cslentryspacingunit}
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||
}%
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{}
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\usepackage{calc}
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\newcommand{\CSLBlock}[1]{#1\hfill\break}
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||
\newcommand{\CSLLeftMargin}[1]{\parbox[t]{\csllabelwidth}{#1}}
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\newcommand{\CSLRightInline}[1]{\parbox[t]{\linewidth - \csllabelwidth}{#1}\break}
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||
\newcommand{\CSLIndent}[1]{\hspace{\cslhangindent}#1}
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||
\usepackage{etoolbox}
|
||
\usepackage[most]{tcolorbox}
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||
\usepackage{graphicx}
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||
\usepackage{xparse}
|
||
\tcbuselibrary{breakable}
|
||
|
||
% --- Disable floating figures globally ---
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||
\usepackage{float}
|
||
\let\origfigure\figure
|
||
\let\endorigfigure\endfigure
|
||
\renewenvironment{figure}[1][H]{\origfigure[H]}{\endorigfigure}
|
||
|
||
% --- Colors ---
|
||
\usepackage{xcolor}
|
||
\definecolor{speculativeback}{RGB}{247,232,255} % #f7e8ff
|
||
\definecolor{speculativeframe}{RGB}{139,43,226} % #8b2be2
|
||
|
||
% --- Left-align all Pandoc tables ---
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||
\usepackage{longtable}
|
||
\usepackage{etoolbox}
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||
\makeatletter
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||
\patchcmd\longtable{\par}{\par\raggedright}{}{}
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||
\setlength{\LTleft}{0pt}
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||
\setlength{\LTright}{0pt}
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||
\makeatother
|
||
|
||
|
||
% Helper to show a blank title if empty
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||
\newcommand{\blanktitle}{\mbox{}}
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||
|
||
% =========================================================
|
||
% Concept Boxes
|
||
% =========================================================
|
||
\newtcolorbox{infobox}[1][Information]{%
|
||
title=\ifstrempty{#1}{\blanktitle}{#1},
|
||
colback=blue!3!white,
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||
colframe=blue!50!black,
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||
colbacktitle=blue!10!white,
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||
fonttitle=\bfseries,
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||
coltitle=black,
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||
enhanced, sharp corners, breakable,
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||
after=\par\vspace{6pt}
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||
}
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||
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||
\newtcolorbox{proposedbox}[1][Proposed Concept]{%
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||
title=\ifstrempty{#1}{\blanktitle}{#1},
|
||
colback=blue!5!white,
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||
colframe=blue!75!black,
|
||
colbacktitle=blue!15!white,
|
||
fonttitle=\bfseries,
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||
coltitle=black,
|
||
enhanced, sharp corners, breakable,
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||
after=\par\vspace{6pt}
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||
}
|
||
|
||
\newtcolorbox{establishedbox}[1][Established Concept]{%
|
||
title=\ifstrempty{#1}{\blanktitle}{#1},
|
||
colback=green!5!white,
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||
colframe=green!75!black,
|
||
colbacktitle=green!15!white,
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||
fonttitle=\bfseries,
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||
coltitle=black,
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||
enhanced, sharp corners, breakable,
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||
after=\par\vspace{6pt}
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||
}
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||
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||
\newtcolorbox{speculativebox}[1][Speculative Concept]{%
|
||
title=\ifstrempty{#1}{\blanktitle}{#1},
|
||
colback=speculativeback,
|
||
colframe=speculativeframe,
|
||
colbacktitle=speculativeback!80!white, % lighter variant for title bar
|
||
fonttitle=\bfseries,
|
||
coltitle=black,
|
||
enhanced, sharp corners, breakable,
|
||
after=\par\vspace{6pt}
|
||
}
|
||
|
||
|
||
\newtcolorbox{cautionbox}[1][Caution]{%
|
||
title=\ifstrempty{#1}{\blanktitle}{#1},
|
||
colback=orange!5!white,
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||
colframe=orange!75!black,
|
||
colbacktitle=orange!15!white,
|
||
fonttitle=\bfseries,
|
||
coltitle=black,
|
||
enhanced, sharp corners, breakable,
|
||
after=\par\vspace{6pt}
|
||
}
|
||
|
||
\newtcolorbox{warningbox}[1][Warning]{%
|
||
title=\ifstrempty{#1}{\blanktitle}{#1},
|
||
colback=orange!5!white,
|
||
colframe=orange!75!black,
|
||
colbacktitle=orange!15!white,
|
||
fonttitle=\bfseries,
|
||
coltitle=black,
|
||
enhanced, sharp corners, breakable,
|
||
after=\par\vspace{6pt}
|
||
}
|
||
|
||
\newtcolorbox{dangerbox}[1][Danger]{%
|
||
title=\ifstrempty{#1}{\blanktitle}{#1},
|
||
colback=red!5!white,
|
||
colframe=red!75!black,
|
||
colbacktitle=red!15!white,
|
||
fonttitle=\bfseries,
|
||
coltitle=black,
|
||
enhanced, sharp corners, breakable,
|
||
after=\par\vspace{6pt}
|
||
}
|
||
\ifLuaTeX
|
||
\usepackage{selnolig} % disable illegal ligatures
|
||
\fi
|
||
|
||
\author{}
|
||
\date{}
|
||
|
||
\begin{document}
|
||
|
||
\renewcommand*\contentsname{Contents}
|
||
{
|
||
\hypersetup{linkcolor=}
|
||
\setcounter{tocdepth}{3}
|
||
\tableofcontents
|
||
}
|
||
\hypertarget{introduction}{%
|
||
\section{Introduction}\label{introduction}}
|
||
|
||
This is a basic book (or article) builder template based on a Pandoc
|
||
build process in conjunction with a number of other tools to generate
|
||
PDF, ODT, HTML, LaTex, Markdown, and Epub book output formats from
|
||
Markdown source content , which can optionally be edited as an Obsidian
|
||
vault.
|
||
|
||
This book builder template has been curated by John
|
||
Haverlack.(\protect\hyperlink{ref-JohnHaverlackACEP}{{``John {Haverlack}
|
||
\textbar{} {ACEP}''} n.d.})
|
||
|
||
\hypertarget{conventions}{%
|
||
\subsection{Conventions}\label{conventions}}
|
||
|
||
A few callout box styles have been added to easily highlight content.
|
||
|
||
\begin{establishedbox}[Established Concept]
|
||
Einsteins Relativistic Dynamics Equations
|
||
\[E^2 = (m_{0} \cdot c^2)^2 + (p \cdot c)^2 \]
|
||
\end{establishedbox}
|
||
|
||
\begin{proposedbox}[Proposed Concept]
|
||
With the speed of light, \(c = 1\): \[E^2 = m_{0}^2 + p^2 \]
|
||
\end{proposedbox}
|
||
|
||
\begin{speculativebox}[Speculative Concept]
|
||
With the speed of light, \(c = 1\): \[E^2 = m_{0}^2 + p^2 \]
|
||
\end{speculativebox}
|
||
|
||
\begin{cautionbox}[Caution Note]
|
||
Beware of this section.
|
||
\end{cautionbox}
|
||
|
||
\begin{warningbox}[Warning Note]
|
||
Beware of this section.
|
||
\end{warningbox}
|
||
|
||
\begin{dangerbox}[Alerts]
|
||
Extreme Highlight
|
||
\end{dangerbox}
|
||
|
||
\hypertarget{getting-started}{%
|
||
\section{Getting Started}\label{getting-started}}
|
||
|
||
\hypertarget{installing-pre-requisites}{%
|
||
\subsection{Installing Pre-Requisites}\label{installing-pre-requisites}}
|
||
|
||
For Debian / ZorinOS and likely Ubuntu based systems.
|
||
|
||
\begin{cautionbox}[TODO]
|
||
It would be nice to roll a setup script to take care of this.
|
||
\end{cautionbox}
|
||
|
||
\hypertarget{required}{%
|
||
\subsubsection{Required}\label{required}}
|
||
|
||
\hypertarget{pandoc}{%
|
||
\paragraph{Pandoc}\label{pandoc}}
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
https://pandoc.org/
|
||
\item
|
||
\href{https://github.com/jgm/pandoc/releases/tag/3.8.2.1}{Download}
|
||
\end{itemize}
|
||
|
||
\begin{verbatim}
|
||
sudo apt install https://github.com/jgm/pandoc/releases/download/3.8.2.1/pandoc-3.8.2.1-1-amd64.deb
|
||
\end{verbatim}
|
||
|
||
\hypertarget{code-editor}{%
|
||
\paragraph{Code Editor}\label{code-editor}}
|
||
|
||
\begin{establishedbox}[Code Editor]
|
||
\href{https://vscodium.com/}{VSCodium} is recommend for privacy
|
||
(telemetry/tracking) reasons - https://vscodium.com/
|
||
\end{establishedbox}
|
||
|
||
But any text editor will work. \#\#\#\# make
|
||
|
||
\begin{verbatim}
|
||
sudo apt install make
|
||
\end{verbatim}
|
||
|
||
\hypertarget{jq-and-yq}{%
|
||
\paragraph{jq and yq}\label{jq-and-yq}}
|
||
|
||
\begin{verbatim}
|
||
sudo apt install jq yq
|
||
\end{verbatim}
|
||
|
||
\hypertarget{texlive}{%
|
||
\paragraph{texlive}\label{texlive}}
|
||
|
||
\begin{verbatim}
|
||
sudo apt install texlive texlive-xetex texlive-latex-extra texlive-fonts-recommended texlive-fonts-extra
|
||
\end{verbatim}
|
||
|
||
\hypertarget{mathjax}{%
|
||
\paragraph{MathJax}\label{mathjax}}
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
https://www.mathjax.org/
|
||
\end{itemize}
|
||
|
||
\begin{verbatim}
|
||
wget https://registry.npmjs.org/mathjax/-/mathjax-3.2.2.tgz
|
||
tar xzf mathjax-3.2.2.tgz
|
||
mv package/es5/* lib/mathjax
|
||
rm -rf package mathjax-3.2.2.tgz
|
||
\end{verbatim}
|
||
|
||
\begin{cautionbox}[TODO]
|
||
This need to be rolled into a setup script.
|
||
\end{cautionbox}
|
||
|
||
\hypertarget{optional}{%
|
||
\subsubsection{Optional}\label{optional}}
|
||
|
||
The following are not strictly requires to use this book builder
|
||
template.
|
||
|
||
\hypertarget{obsidian}{%
|
||
\paragraph{Obsidian}\label{obsidian}}
|
||
|
||
\begin{establishedbox}[Highly Recommended]
|
||
Editing book chapter content in Obsidian is a very productive means for
|
||
editing Markdown source content.
|
||
\end{establishedbox}
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
https://obsidian.md/
|
||
\item
|
||
\href{https://github.com/obsidianmd/obsidian-releases/releases/download/v1.9.14/obsidian_1.9.14_amd64.deb}{Deb
|
||
Package}
|
||
\end{itemize}
|
||
|
||
\begin{verbatim}
|
||
sudo apt install https://github.com/obsidianmd/obsidian-releases/releases/download/v1.9.14/obsidian_1.9.14_amd64.deb
|
||
\end{verbatim}
|
||
|
||
\hypertarget{zotero}{%
|
||
\paragraph{Zotero}\label{zotero}}
|
||
|
||
\begin{establishedbox}[Highly Recommended]
|
||
If you need to managed citations and references, Zotero integration is
|
||
highly recommended.
|
||
\end{establishedbox}
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
https://www.zotero.org/
|
||
\end{itemize}
|
||
|
||
\begin{verbatim}
|
||
sudo cp ./scripts/deps/zotero.list /etc/apt/sources.list.d/
|
||
\end{verbatim}
|
||
|
||
\begin{verbatim}
|
||
sudo apt update
|
||
\end{verbatim}
|
||
|
||
\begin{verbatim}
|
||
sudo apt install zotero
|
||
\end{verbatim}
|
||
|
||
\hypertarget{better-bibtex-for-zotero}{%
|
||
\subparagraph{Better BibTex for Zotero}\label{better-bibtex-for-zotero}}
|
||
|
||
Install the Better BibTex Plugin for Zotero - Zotero \textgreater{} Tool
|
||
\textgreater{} Plugins
|
||
|
||
\hypertarget{export-citations.bib}{%
|
||
\subparagraph{Export citations.bib}\label{export-citations.bib}}
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
Zotero \textgreater{} File \textgreater{} Export Library
|
||
\textgreater{} Format: Better BibTeX
|
||
\item[$\square$]
|
||
Keep Updated
|
||
\item[$\square$]
|
||
Save to: \textasciitilde/Documents/Lib/zotero.bib
|
||
\item[$\square$]
|
||
Symlink your \textasciitilde/Documents/Lib/Citations.bib to
|
||
basic-book-builder/lib/zotero.bib
|
||
\end{itemize}
|
||
|
||
\hypertarget{zotero-connector-browser-plugin}{%
|
||
\subparagraph{Zotero Connector Browser
|
||
Plugin}\label{zotero-connector-browser-plugin}}
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
https://chromewebstore.google.com/detail/zotero-connector/ekhagklcjbdpajgpjgmbionohlpdbjgc
|
||
\end{itemize}
|
||
|
||
Provides you the ability to auto add Web resources to your Zotero
|
||
citation database.
|
||
|
||
\hypertarget{lmodern}{%
|
||
\paragraph{lmodern}\label{lmodern}}
|
||
|
||
\begin{verbatim}
|
||
sudo apt install lmodern
|
||
\end{verbatim}
|
||
|
||
\hypertarget{epubcheck}{%
|
||
\paragraph{epubcheck}\label{epubcheck}}
|
||
|
||
\begin{verbatim}
|
||
sudo apt install epubcheck
|
||
\end{verbatim}
|
||
|
||
\hypertarget{foliate}{%
|
||
\paragraph{foliate}\label{foliate}}
|
||
|
||
An EPub Reader
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
https://johnfactotum.github.io/foliate/
|
||
\end{itemize}
|
||
|
||
\begin{verbatim}
|
||
sudo apt install https://github.com/johnfactotum/foliate/releases/download/2.6.4/com.github.johnfactotum.foliate_2.6.4_all.deb
|
||
\end{verbatim}
|
||
|
||
\hypertarget{calibre}{%
|
||
\paragraph{calibre}\label{calibre}}
|
||
|
||
An EPub Reader
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
https://calibre-ebook.com
|
||
\end{itemize}
|
||
|
||
\begin{verbatim}
|
||
sudo apt install calibre
|
||
\end{verbatim}
|
||
|
||
\hypertarget{editing-the-configuration}{%
|
||
\subsection{Editing the Configuration}\label{editing-the-configuration}}
|
||
|
||
\hypertarget{editing-the-book}{%
|
||
\section{Editing the Book}\label{editing-the-book}}
|
||
|
||
\hypertarget{configuration}{%
|
||
\subsubsection{Configuration}\label{configuration}}
|
||
|
||
There are a number of other config files for each format:
|
||
|
||
\begin{verbatim}
|
||
conf/
|
||
├── epub-metadata.xml
|
||
├── epub_template.html
|
||
├── epub.yaml
|
||
├── frontmatter_epub.md
|
||
├── frontmatter_epub.xhtml
|
||
├── frontmatter.html
|
||
├── frontmatter.tex
|
||
├── header.tex
|
||
├── html.yaml
|
||
├── latex.yaml
|
||
├── markdown.yaml
|
||
├── metadata.yaml
|
||
├── pandoc.yaml
|
||
├── pdf.yaml
|
||
├── style.css
|
||
└── style_epub.css
|
||
\end{verbatim}
|
||
|
||
\hypertarget{main-config-files}{%
|
||
\paragraph{Main Config Files}\label{main-config-files}}
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
metadata.yaml - Set Title, etc
|
||
\item
|
||
pandoc.yaml - Main Pandoc Config \#\#\#\# Per format Configs
|
||
\item
|
||
\texttt{pdf.yaml}
|
||
\item
|
||
\texttt{html.yaml}
|
||
\item
|
||
\texttt{latex.yaml}
|
||
\item
|
||
\texttt{epub.yaml}
|
||
\end{itemize}
|
||
|
||
\hypertarget{frontmatter-config}{%
|
||
\subsubsection{FrontMatter Config}\label{frontmatter-config}}
|
||
|
||
There are 2 Version of the FrontMatter for PDF, and HTML bases formats
|
||
that set the Title, Author, Verizon, Copyright, etc\ldots{}
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
\texttt{frontmatter.tex}
|
||
\item
|
||
\texttt{frontmatter.html}
|
||
\item
|
||
\texttt{frontmatter\_epub.*} - Work in Progress
|
||
\end{itemize}
|
||
|
||
\begin{quote}
|
||
There is probably a better way to do this.
|
||
\end{quote}
|
||
|
||
\hypertarget{editing-the-content}{%
|
||
\subsection{Editing the Content}\label{editing-the-content}}
|
||
|
||
To edit the book open the \texttt{basic-book-builder} directory as an
|
||
Obsidian Vault.
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
Edit the Markdown content in the \texttt{chapters} directory.
|
||
\end{itemize}
|
||
|
||
\hypertarget{citations}{%
|
||
\subsubsection{Citations}\label{citations}}
|
||
|
||
\begin{quote}
|
||
Note: the Zotero database needs configured to export automatically to
|
||
\texttt{lib/citations.bib}
|
||
\end{quote}
|
||
|
||
To insert a Zotero Citation - Ensure the Zotero App and DB are running
|
||
on you system. - Alt + I (to insert citation) - Search for and select
|
||
citation reference
|
||
|
||
\hypertarget{usage-building-the-book}{%
|
||
\subsection{Usage: Building the Book}\label{usage-building-the-book}}
|
||
|
||
\hypertarget{pdf}{%
|
||
\paragraph{PDF}\label{pdf}}
|
||
|
||
\begin{verbatim}
|
||
make pdf
|
||
\end{verbatim}
|
||
|
||
\hypertarget{html}{%
|
||
\paragraph{HTML}\label{html}}
|
||
|
||
\begin{verbatim}
|
||
make html
|
||
\end{verbatim}
|
||
|
||
\hypertarget{latex}{%
|
||
\paragraph{LaTex}\label{latex}}
|
||
|
||
\begin{verbatim}
|
||
make latex
|
||
\end{verbatim}
|
||
|
||
\hypertarget{markdown}{%
|
||
\paragraph{Markdown}\label{markdown}}
|
||
|
||
\begin{verbatim}
|
||
make markdown
|
||
\end{verbatim}
|
||
|
||
\hypertarget{epub}{%
|
||
\paragraph{EPub}\label{epub}}
|
||
|
||
\begin{quote}
|
||
Note: This ePub configuration still needs tuning.
|
||
\end{quote}
|
||
|
||
\begin{verbatim}
|
||
make epub
|
||
\end{verbatim}
|
||
|
||
\hypertarget{example-content}{%
|
||
\section{Example Content}\label{example-content}}
|
||
|
||
In \(R\nu\) the
|
||
\href{https://en.wikipedia.org/wiki/Planck_units\#Planck_length}{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.
|
||
|
||
\hypertarget{si-conversion-factors}{%
|
||
\subsubsection{SI Conversion Factors}\label{si-conversion-factors}}
|
||
|
||
The following conversion factors can be used to convert observable
|
||
quantities of measure from the \emph{SI} system of units to \(R\nu\) to
|
||
\textasciitilde6 significant digits.
|
||
|
||
\begin{longtable}[]{@{}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3256}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.0930}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.5814}}@{}}
|
||
\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}
|
||
|
||
\hypertarget{physical-constants}{%
|
||
\subsubsection{Physical Constants}\label{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:
|
||
|
||
\(c = 299792458 \frac{m}{s} = 299792458 \frac{m}{s} \cdot 1.74542\times10^{34} \frac{L}{m} \cdot \frac{1}{5.23264\times10^{42} \frac{L}{s}} = 1.00000\)
|
||
|
||
\begin{longtable}[]{@{}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2273}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.0909}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.4545}}
|
||
>{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2273}}@{}}
|
||
\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 \\
|
||
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 & \(\Theta\) & \(2\pi = 6.28318...\ Radians\) &
|
||
\(1 \tau = 6.28318...\ Radians\) \\
|
||
\bottomrule
|
||
\end{longtable}
|
||
|
||
\hypertarget{fine-structure-constant}{%
|
||
\subsection{Fine Structure Constant}\label{fine-structure-constant}}
|
||
|
||
As a consistency check, we compute the
|
||
\emph{\href{https://en.wikipedia.org/wiki/Fine-structure_constant}{Fine
|
||
Structure Constant}} using Reduced Natural Units which is a unit less
|
||
ratio that should be independent of our system of units.
|
||
|
||
\(\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar c}=\frac{e^2}{2\tau}=0.00729735≈\frac{1}{137}}\)
|
||
|
||
\hypertarget{dimensional-analysis}{%
|
||
\paragraph{Dimensional Analysis}\label{dimensional-analysis}}
|
||
|
||
The reader should be familiar with high school physics and chemistry
|
||
\href{https://en.wikipedia.org/wiki/Dimensional_analysis}{dimensional
|
||
analysis}.
|
||
|
||
\begin{itemize}
|
||
\tightlist
|
||
\item
|
||
\(1\ meter\ (m) = 100\ centimeters\ (cm)\)
|
||
\item
|
||
\(1\ kilometer\ (km) = 1000\ meters\ (m)\)
|
||
\item
|
||
\(1\ mile = 5280\ feet\ (ft\ or\ ')\)
|
||
\item
|
||
\(1\ foot\ (ft\ or\ ') = 12\ inches\ (in\ or\ ")\)
|
||
\item
|
||
\(1\ inch\ (") = 2.54\ centimeters\ (cm)\)
|
||
\end{itemize}
|
||
|
||
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.
|
||
|
||
\hypertarget{newtons-law-of-gravity}{%
|
||
\subsection{Newton's Law of Gravity}\label{newtons-law-of-gravity}}
|
||
|
||
The force of gravity (\(F_g\)) between 2 masses, \(m1\) and \(m2\)
|
||
separated by distance \(r\) is given by
|
||
\href{https://en.wikipedia.org/wiki/Newton\%27s_law_of_universal_gravitation}{Newton's
|
||
Law of Gravity}:
|
||
|
||
\(F_{g} = G \frac{m_{1} m_{2}}{r^{2}}\)
|
||
|
||
Where \(G\), is the
|
||
\href{http://en.wikipedia.org/wiki/Gravitational_Constant}{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
|
||
(\(4\pi r^2\)).
|
||
|
||
\hypertarget{inverse-square-law}{%
|
||
\paragraph{Inverse Square Law}\label{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:
|
||
|
||
\[I(r) = \frac{S_0}{4 \pi r^{2}}=\frac{S_0}{2 \tau r^{2}}\]
|
||
\includegraphics{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:
|
||
|
||
\[\boxed{G_0=\frac{2\tau G}{c^2}=\frac{2\tau \cdot 6.67384 \times 10^{-11}}{299792458^2} \frac{m}{kg} = 9.33135 \times 10^-27 \frac{m}{kg}}\]
|
||
|
||
\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}{%
|
||
\subsubsection{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}{%
|
||
\subparagraph{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
|
||
|
||
\(E = m_0\)
|
||
|
||
\hypertarget{zero-mass-objects-moving-at-the-speed-of-light}{%
|
||
\subparagraph{Zero mass objects moving at the speed of
|
||
light}\label{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}{%
|
||
\subsection{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}{%
|
||
\paragraph{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}{%
|
||
\subsection{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}{%
|
||
\subsection{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}{%
|
||
\subsection{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}{%
|
||
\paragraph{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}{%
|
||
\section{Terminology}\label{terminology}}
|
||
|
||
\hypertarget{citations-1}{%
|
||
\section*{Citations}\label{citations-1}}
|
||
\addcontentsline{toc}{section}{Citations}
|
||
|
||
\hypertarget{refs}{}
|
||
\begin{CSLReferences}{1}{0}
|
||
\leavevmode\vadjust pre{\hypertarget{ref-JohnHaverlackACEP}{}}%
|
||
{``John {Haverlack} \textbar{} {ACEP}.''} n.d.
|
||
https://www.uaf.edu/acep/about/our-team/john-haverlack.php. Accessed
|
||
September 30, 2025.
|
||
|
||
\end{CSLReferences}
|
||
|
||
\end{document}
|