1092 lines
50 KiB
HTML
1092 lines
50 KiB
HTML
<!DOCTYPE html>
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<link rel="stylesheet" href="conf/style.css" />
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</head>
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<body>
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<div class="frontmatter">
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|
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<!-- Title Page -->
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<h1 style="margin-top:3em; font-size:2.4em; text-align:center;">Basic Book Builder</h1>
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<h2 style="text-align:center; font-weight:normal;">A Pandoc Template for building books and articles</h2>
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|
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<div style="text-align:center; margin:2em 0;">
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<img src="lib/img/ACEPLogo.png" alt="Cover illustration" style="max-width:240px;">
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</div>
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<h3 style="text-align:center;">John Haverlack</h3>
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<hr style="margin:3em 0;">
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<!-- Metadata Page (PDF analog) -->
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<div style="text-align:center; margin-top:5em;">
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<p><strong>Title</strong>: Basic Book Builder</p>
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<p><strong>Subtitle</strong>: A Pandoc Template for building books and articles</p>
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<p><strong>Affiliation</strong>: Alaska Center for Energy and Power</p>
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<p><strong>Institution</strong>: University of Alaska Fairbanks</p>
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<p><strong>Author</strong>: John Haverlack</p>
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<p><strong>Copyright</strong>: © 2025 Alaska Center for Energy and Power</p>
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<p><strong>License</strong>: CC BY-ND 4.0</p>
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<p><strong>Version</strong>: 0.0.2</p>
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<p><strong>Date</strong>: 2025-11-11</p>
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<p><strong>State</strong>: PRE-RELEASE</p>
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<p><strong>Source</strong>: https://github.com/jehaverlack/basic-book-builder</p>
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</div>
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<hr style="margin:4em 0;">
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</div>
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<script>
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window.MathJax = {
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tex: { inlineMath: [['$', '$'], ['\\(', '\\)']] },
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svg: { fontCache: 'global' }
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};
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</script>
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<script src="lib/mathjax/tex-mml-chtml.js"></script>
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<nav id="TOC" role="doc-toc">
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<h2 id="toc-title">Contents</h2>
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<ul>
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<li><a href="#introduction">Introduction</a>
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<ul>
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<li><a href="#conventions">Conventions</a></li>
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||
</ul></li>
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||
<li><a href="#getting-started">Getting Started</a>
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||
<ul>
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<li><a href="#installing-pre-requisites">Installing Pre-Requisites</a>
|
||
<ul>
|
||
<li><a href="#required">Required</a></li>
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||
<li><a href="#optional">Optional</a></li>
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||
</ul></li>
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<li><a href="#editing-the-configuration">Editing the
|
||
Configuration</a></li>
|
||
</ul></li>
|
||
<li><a href="#editing-the-book">Editing the Book</a>
|
||
<ul>
|
||
<li><a href="#configuration">Configuration</a></li>
|
||
<li><a href="#frontmatter-config">FrontMatter Config</a></li>
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||
<li><a href="#editing-the-content">Editing the Content</a>
|
||
<ul>
|
||
<li><a href="#citations">Citations</a></li>
|
||
</ul></li>
|
||
<li><a href="#usage-building-the-book">Usage: Building the Book</a></li>
|
||
</ul></li>
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||
<li><a href="#example-content">Example Content</a>
|
||
<ul>
|
||
<li><a href="#si-conversion-factors">SI Conversion Factors</a></li>
|
||
<li><a href="#physical-constants">Physical Constants</a></li>
|
||
<li><a href="#fine-structure-constant">Fine Structure Constant</a></li>
|
||
<li><a href="#newtons-law-of-gravity">Newton’s Law of Gravity</a>
|
||
<ul>
|
||
<li><a href="#relativistic-energy-momentum-relation">Relativistic Energy
|
||
Momentum Relation</a></li>
|
||
</ul></li>
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||
<li><a href="#plancks-constant">Planck’s Constant</a></li>
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||
<li><a href="#planck-length">Planck Length</a></li>
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||
<li><a href="#fine-structure-constant-1">Fine Structure
|
||
Constant</a></li>
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||
<li><a href="#sage-code">Sage Code</a></li>
|
||
</ul></li>
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||
<li><a href="#terminology">Terminology</a></li>
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||
<li><a href="#citations-1">Citations</a></li>
|
||
</ul>
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||
</nav>
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<h1 id="introduction">Introduction</h1>
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||
<p>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
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||
PDF, ODT, HTML, LaTex, Markdown, and Epub book output formats from
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||
Markdown source content , which can optionally be edited as an Obsidian
|
||
vault.</p>
|
||
<p>This book builder template has been curated by John Haverlack.<span
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||
class="citation" data-cites="JohnHaverlackACEP">(<a
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||
href="#ref-JohnHaverlackACEP" role="doc-biblioref"><span>“John
|
||
<span>Haverlack</span> <span>ACEP</span>”</span> n.d.</a>)</span></p>
|
||
<h2 id="conventions">Conventions</h2>
|
||
<p>A few callout box styles have been added to easily highlight
|
||
content.</p>
|
||
<div class="callout-established">
|
||
<p><strong>Established Concept</strong></p>
|
||
<p> Einsteins Relativistic Dynamics Equations <span
|
||
class="math display"><em>E</em><sup>2</sup> = (<em>m</em><sub>0</sub>⋅<em>c</em><sup>2</sup>)<sup>2</sup> + (<em>p</em>⋅<em>c</em>)<sup>2</sup></span></p>
|
||
</div>
|
||
<div class="callout-proposed">
|
||
<p><strong>Proposed Concept</strong></p>
|
||
<p> With the speed of light, <span
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||
class="math inline"><em>c</em> = 1</span>: <span
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||
class="math display"><em>E</em><sup>2</sup> = <em>m</em><sub>0</sub><sup>2</sup> + <em>p</em><sup>2</sup></span></p>
|
||
</div>
|
||
<div class="callout-speculative">
|
||
<p><strong>Speculative Concept</strong></p>
|
||
<p> With the speed of light, <span
|
||
class="math inline"><em>c</em> = 1</span>: <span
|
||
class="math display"><em>E</em><sup>2</sup> = <em>m</em><sub>0</sub><sup>2</sup> + <em>p</em><sup>2</sup></span></p>
|
||
</div>
|
||
<div class="callout-caution">
|
||
<p><strong>Caution Note</strong></p>
|
||
<p> Beware of this section.</p>
|
||
</div>
|
||
<div class="callout-warning">
|
||
<p><strong>Warning Note</strong></p>
|
||
<p> Beware of this section.</p>
|
||
</div>
|
||
<div class="callout-danger">
|
||
<p><strong>Alerts</strong></p>
|
||
<p> Extreme Highlight</p>
|
||
</div>
|
||
<h1 id="getting-started">Getting Started</h1>
|
||
<h2 id="installing-pre-requisites">Installing Pre-Requisites</h2>
|
||
<p>For Debian / ZorinOS and likely Ubuntu based systems.</p>
|
||
<div class="callout-caution">
|
||
<p><strong>TODO</strong></p>
|
||
<p> It would be nice to roll a setup script to take care of this.</p>
|
||
</div>
|
||
<h3 id="required">Required</h3>
|
||
<h4 id="pandoc">Pandoc</h4>
|
||
<ul>
|
||
<li>https://pandoc.org/</li>
|
||
<li><a
|
||
href="https://github.com/jgm/pandoc/releases/tag/3.8.2.1">Download</a></li>
|
||
</ul>
|
||
<pre><code>sudo apt install https://github.com/jgm/pandoc/releases/download/3.8.2.1/pandoc-3.8.2.1-1-amd64.deb</code></pre>
|
||
<h4 id="code-editor">Code Editor</h4>
|
||
<div class="callout-established">
|
||
<p><strong>Code Editor</strong></p>
|
||
<p> <a href="https://vscodium.com/">VSCodium</a> is recommend for
|
||
privacy (telemetry/tracking) reasons - https://vscodium.com/</p>
|
||
</div>
|
||
<p>But any text editor will work. #### make</p>
|
||
<pre><code>sudo apt install make</code></pre>
|
||
<h4 id="jq-and-yq">jq and yq</h4>
|
||
<pre><code>sudo apt install jq yq</code></pre>
|
||
<h4 id="texlive">texlive</h4>
|
||
<pre><code>sudo apt install texlive texlive-xetex texlive-latex-extra texlive-fonts-recommended texlive-fonts-extra</code></pre>
|
||
<h4 id="mathjax">MathJax</h4>
|
||
<ul>
|
||
<li>https://www.mathjax.org/</li>
|
||
</ul>
|
||
<pre><code>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</code></pre>
|
||
<div class="callout-caution">
|
||
<p><strong>TODO</strong></p>
|
||
<p> This need to be rolled into a setup script.</p>
|
||
</div>
|
||
<h3 id="optional">Optional</h3>
|
||
<p>The following are not strictly requires to use this book builder
|
||
template.</p>
|
||
<h4 id="obsidian">Obsidian</h4>
|
||
<div class="callout-established">
|
||
<p><strong>Highly Recommended</strong></p>
|
||
<p> Editing book chapter content in Obsidian is a very productive means
|
||
for editing Markdown source content.</p>
|
||
</div>
|
||
<ul>
|
||
<li>https://obsidian.md/</li>
|
||
<li><a
|
||
href="https://github.com/obsidianmd/obsidian-releases/releases/download/v1.9.14/obsidian_1.9.14_amd64.deb">Deb
|
||
Package</a></li>
|
||
</ul>
|
||
<pre><code>sudo apt install https://github.com/obsidianmd/obsidian-releases/releases/download/v1.9.14/obsidian_1.9.14_amd64.deb</code></pre>
|
||
<h4 id="zotero">Zotero</h4>
|
||
<div class="callout-established">
|
||
<p><strong>Highly Recommended</strong></p>
|
||
<p> If you need to managed citations and references, Zotero integration
|
||
is highly recommended.</p>
|
||
</div>
|
||
<ul>
|
||
<li>https://www.zotero.org/</li>
|
||
</ul>
|
||
<pre><code>sudo cp ./scripts/deps/zotero.list /etc/apt/sources.list.d/</code></pre>
|
||
<pre><code>sudo apt update</code></pre>
|
||
<pre><code>sudo apt install zotero</code></pre>
|
||
<h5 id="better-bibtex-for-zotero">Better BibTex for Zotero</h5>
|
||
<p>Install the Better BibTex Plugin for Zotero - Zotero > Tool >
|
||
Plugins</p>
|
||
<h5 id="export-citations.bib">Export citations.bib</h5>
|
||
<ul>
|
||
<li>Zotero > File > Export Library > Format: Better BibTeX</li>
|
||
<li><input type="checkbox" disabled="" />
|
||
Keep Updated</li>
|
||
<li><input type="checkbox" disabled="" />
|
||
Save to: ~/Documents/Lib/zotero.bib</li>
|
||
<li><input type="checkbox" disabled="" />
|
||
Symlink your ~/Documents/Lib/Citations.bib to
|
||
basic-book-builder/lib/zotero.bib</li>
|
||
</ul>
|
||
<h5 id="zotero-connector-browser-plugin">Zotero Connector Browser
|
||
Plugin</h5>
|
||
<ul>
|
||
<li>https://chromewebstore.google.com/detail/zotero-connector/ekhagklcjbdpajgpjgmbionohlpdbjgc</li>
|
||
</ul>
|
||
<p>Provides you the ability to auto add Web resources to your Zotero
|
||
citation database.</p>
|
||
<h4 id="lmodern">lmodern</h4>
|
||
<pre><code>sudo apt install lmodern</code></pre>
|
||
<h4 id="epubcheck">epubcheck</h4>
|
||
<pre><code>sudo apt install epubcheck</code></pre>
|
||
<h4 id="foliate">foliate</h4>
|
||
<p>An EPub Reader</p>
|
||
<ul>
|
||
<li>https://johnfactotum.github.io/foliate/</li>
|
||
</ul>
|
||
<pre><code>sudo apt install https://github.com/johnfactotum/foliate/releases/download/2.6.4/com.github.johnfactotum.foliate_2.6.4_all.deb</code></pre>
|
||
<h4 id="calibre">calibre</h4>
|
||
<p>An EPub Reader</p>
|
||
<ul>
|
||
<li>https://calibre-ebook.com</li>
|
||
</ul>
|
||
<pre><code>sudo apt install calibre</code></pre>
|
||
<h2 id="editing-the-configuration">Editing the Configuration</h2>
|
||
<h1 id="editing-the-book">Editing the Book</h1>
|
||
<h3 id="configuration">Configuration</h3>
|
||
<p>There are a number of other config files for each format:</p>
|
||
<pre><code>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
|
||
</code></pre>
|
||
<h4 id="main-config-files">Main Config Files</h4>
|
||
<ul>
|
||
<li>metadata.yaml - Set Title, etc</li>
|
||
<li>pandoc.yaml - Main Pandoc Config #### Per format Configs</li>
|
||
<li><code>pdf.yaml</code></li>
|
||
<li><code>html.yaml</code></li>
|
||
<li><code>latex.yaml</code></li>
|
||
<li><code>epub.yaml</code></li>
|
||
</ul>
|
||
<h3 id="frontmatter-config">FrontMatter Config</h3>
|
||
<p>There are 2 Version of the FrontMatter for PDF, and HTML bases
|
||
formats that set the Title, Author, Verizon, Copyright, etc…</p>
|
||
<ul>
|
||
<li><code>frontmatter.tex</code></li>
|
||
<li><code>frontmatter.html</code></li>
|
||
<li><code>frontmatter_epub.*</code> - Work in Progress</li>
|
||
</ul>
|
||
<blockquote>
|
||
<p>There is probably a better way to do this.</p>
|
||
</blockquote>
|
||
<h2 id="editing-the-content">Editing the Content</h2>
|
||
<p>To edit the book open the <code>basic-book-builder</code> directory
|
||
as an Obsidian Vault.</p>
|
||
<ul>
|
||
<li>Edit the Markdown content in the <code>chapters</code>
|
||
directory.</li>
|
||
</ul>
|
||
<h3 id="citations">Citations</h3>
|
||
<blockquote>
|
||
<p>Note: the Zotero database needs configured to export automatically to
|
||
<code>lib/citations.bib</code></p>
|
||
</blockquote>
|
||
<p>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</p>
|
||
<h2 id="usage-building-the-book">Usage: Building the Book</h2>
|
||
<h4 id="pdf">PDF</h4>
|
||
<pre><code>make pdf</code></pre>
|
||
<h4 id="html">HTML</h4>
|
||
<pre><code>make html</code></pre>
|
||
<h4 id="latex">LaTex</h4>
|
||
<pre><code>make latex</code></pre>
|
||
<h4 id="markdown">Markdown</h4>
|
||
<pre><code>make markdown</code></pre>
|
||
<h4 id="epub">EPub</h4>
|
||
<blockquote>
|
||
<p>Note: This ePub configuration still needs tuning.</p>
|
||
</blockquote>
|
||
<pre><code>make epub</code></pre>
|
||
<h1 id="example-content">Example Content</h1>
|
||
<p>In <span class="math inline"><em>R</em><em>ν</em></span> the <a
|
||
href="https://en.wikipedia.org/wiki/Planck_units#Planck_length">Planck
|
||
Length</a> is the universal unit for measurement of distance, and is
|
||
defined approximately to be: <span
|
||
class="math display">$$\boxed{L_P=\sqrt{\hbar}=5.72928\times10^{-35}m=1
|
||
L}$$</span> Where <span class="math inline">1 <em>L</em></span>, is 1
|
||
Planck Length of distance.</p>
|
||
<h3 id="si-conversion-factors">SI Conversion Factors</h3>
|
||
<p>The following conversion factors can be used to convert observable
|
||
quantities of measure from the <em>SI</em> system of units to <span
|
||
class="math inline"><em>R</em><em>ν</em></span> to ~6 significant
|
||
digits.</p>
|
||
<table>
|
||
<colgroup>
|
||
<col style="width: 32%" />
|
||
<col style="width: 9%" />
|
||
<col style="width: 58%" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr class="header">
|
||
<th>Conversion Factor</th>
|
||
<th>Symbol</th>
|
||
<th>Value</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="odd">
|
||
<td>meters to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>χ</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$1.74542\times10^{34}
|
||
\frac{L}{m}$</span></td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>seconds to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>τ</em><sub><em>p</em></sub></span></td>
|
||
<td><span class="math inline">$5.23264\times10^{42}
|
||
\frac{L}{s}$</span></td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>mass to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>G</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$1.62871\times10^8
|
||
\frac{L}{kg}$</span></td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>energy to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>E</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$1.81219\times10^9
|
||
\frac{L}{J}$</span></td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>momentum to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>P</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$5.43280\times10^{-1} \frac{L\cdot s}{kg
|
||
\cdot m}$</span></td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>temperature to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>k</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$2.501998\times10^{-14}
|
||
\frac{L}{K}$</span></td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>charge to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>C</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$1.89007\times10^{18}
|
||
\frac{L}{C}$</span></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h3 id="physical-constants">Physical Constants</h3>
|
||
<p>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:</p>
|
||
<p><span class="math inline">$c = 299792458 \frac{m}{s} = 299792458
|
||
\frac{m}{s} \cdot 1.74542\times10^{34} \frac{L}{m} \cdot
|
||
\frac{1}{5.23264\times10^{42} \frac{L}{s}} = 1.00000$</span></p>
|
||
<table>
|
||
<colgroup>
|
||
<col style="width: 22%" />
|
||
<col style="width: 9%" />
|
||
<col style="width: 45%" />
|
||
<col style="width: 22%" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr class="header">
|
||
<th>Quantity</th>
|
||
<th>Symbol</th>
|
||
<th>SI</th>
|
||
<th><span class="math inline"><em>ν</em></span></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="odd">
|
||
<td>Speed of Light</td>
|
||
<td><span class="math inline"><em>c</em></span></td>
|
||
<td><span class="math inline">$299792458 \frac{m}{s}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>Reduced Gravitational Constant</td>
|
||
<td><span class="math inline"><em>G</em><sub>0</sub></span></td>
|
||
<td><span class="math inline">$8.38659\times10^{-10} \frac{m^3}{kg \cdot
|
||
s^2}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>Boltzmann’s Constant</td>
|
||
<td><span class="math inline"><em>k</em></span></td>
|
||
<td><span class="math inline">$k=1.380649\times10^-23
|
||
\frac{J}{K}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>Permittivity of Free Space</td>
|
||
<td><span
|
||
class="math inline"><em>ϵ</em><sub><em>o</em></sub></span></td>
|
||
<td><span class="math inline">$8.854187817620\times10^{-12}
|
||
\frac{C^{2}s^2}{kg \cdot m^3}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>Permeability of Free Space</td>
|
||
<td><span
|
||
class="math inline"><em>μ</em><sub><em>o</em></sub></span></td>
|
||
<td><span class="math inline">$\huge{\frac{1}{\epsilon_{o} \cdot
|
||
c^{2}}}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>Reduced Planck’s Constant</td>
|
||
<td><span class="math inline">ℏ</span></td>
|
||
<td><span class="math inline">$1.054571726\times10^-34 \frac{kg \cdot
|
||
m^2}{s}$</span></td>
|
||
<td><span class="math inline">1<em>L</em><sup>2</sup></span></td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>Mass of the Electron</td>
|
||
<td><span
|
||
class="math inline"><em>m</em><sub><em>e</em></sub></span></td>
|
||
<td><span
|
||
class="math inline">9.10938 × 10<sup>−31</sup><em>k</em><em>g</em></span></td>
|
||
<td><span
|
||
class="math inline">1.48366 × 10<sup>−22</sup><em>L</em></span></td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>Charge of the Electron</td>
|
||
<td><span class="math inline"><em>e</em><sup>−</sup></span></td>
|
||
<td><span
|
||
class="math inline"> − 1.60218 × 10<sup>−19</sup><em>C</em></span></td>
|
||
<td><span
|
||
class="math inline"> − 3.02822 × 10<sup>−1</sup><em>L</em></span></td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>Unit Cycle</td>
|
||
<td><span class="math inline"><em>Θ</em></span></td>
|
||
<td><span
|
||
class="math inline">2<em>π</em> = 6.28318... <em>R</em><em>a</em><em>d</em><em>i</em><em>a</em><em>n</em><em>s</em></span></td>
|
||
<td><span
|
||
class="math inline">1<em>τ</em> = 6.28318... <em>R</em><em>a</em><em>d</em><em>i</em><em>a</em><em>n</em><em>s</em></span></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h2 id="fine-structure-constant">Fine Structure Constant</h2>
|
||
<p>As a consistency check, we compute the <em><a
|
||
href="https://en.wikipedia.org/wiki/Fine-structure_constant">Fine
|
||
Structure Constant</a></em> using Reduced Natural Units which is a unit
|
||
less ratio that should be independent of our system of units.</p>
|
||
<p><span
|
||
class="math inline">$\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar
|
||
c}=\frac{e^2}{2\tau}=0.00729735≈\frac{1}{137}}$</span></p>
|
||
<h4 id="dimensional-analysis">Dimensional Analysis</h4>
|
||
<p>The reader should be familiar with high school physics and chemistry
|
||
<a href="https://en.wikipedia.org/wiki/Dimensional_analysis">dimensional
|
||
analysis</a>.</p>
|
||
<ul>
|
||
<li><span
|
||
class="math inline">1 <em>m</em><em>e</em><em>t</em><em>e</em><em>r</em> (<em>m</em>) = 100 <em>c</em><em>e</em><em>n</em><em>t</em><em>i</em><em>m</em><em>e</em><em>t</em><em>e</em><em>r</em><em>s</em> (<em>c</em><em>m</em>)</span></li>
|
||
<li><span
|
||
class="math inline">1 <em>k</em><em>i</em><em>l</em><em>o</em><em>m</em><em>e</em><em>t</em><em>e</em><em>r</em> (<em>k</em><em>m</em>) = 1000 <em>m</em><em>e</em><em>t</em><em>e</em><em>r</em><em>s</em> (<em>m</em>)</span></li>
|
||
<li><span
|
||
class="math inline">1 <em>m</em><em>i</em><em>l</em><em>e</em> = 5280 <em>f</em><em>e</em><em>e</em><em>t</em> (<em>f</em><em>t</em> <em>o</em><em>r</em> ′)</span></li>
|
||
<li><span class="math inline">$1\ foot\ (ft\ or\ ') = 12\ inches\ (in\
|
||
or\ ")$</span></li>
|
||
<li><span class="math inline">$1\ inch\ (") = 2.54\ centimeters\
|
||
(cm)$</span></li>
|
||
</ul>
|
||
<p>How many kilometers are in 1 mile? <span class="math inline">$1\ mile
|
||
= 1\ mile \times \frac{5280\ ft}{mile} \times \frac{12\ in}{ft} \times
|
||
\frac{2.54\ cm}{in}\times \frac{1\ m}{100 cm} \times \frac{1\ km}{1000
|
||
m} = \frac{5280 \times 12 \times 2.54}{100 \times 1000}\ km =
|
||
\frac{160934.40}{100000}\ km = 1.6\ km$</span> Note that each unit in
|
||
the denominator cancels with one if the numerator until we are left with
|
||
only km.</p>
|
||
<h2 id="newtons-law-of-gravity">Newton’s Law of Gravity</h2>
|
||
<p>The force of gravity (<span
|
||
class="math inline"><em>F</em><sub><em>g</em></sub></span>) between 2
|
||
masses, <span class="math inline"><em>m</em>1</span> and <span
|
||
class="math inline"><em>m</em>2</span> separated by distance <span
|
||
class="math inline"><em>r</em></span> is given by <a
|
||
href="https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation">Newton’s
|
||
Law of Gravity</a>:</p>
|
||
<p><span class="math inline">$F_{g} = G \frac{m_{1}
|
||
m_{2}}{r^{2}}$</span></p>
|
||
<p>Where <span class="math inline"><em>G</em></span>, is the <a
|
||
href="http://en.wikipedia.org/wiki/Gravitational_Constant">Gravitational
|
||
Constant</a>.</p>
|
||
<p><span class="math inline">$G = 6.67430 \times 10^{-11}\
|
||
N\frac{m^2}{kg^2}$</span></p>
|
||
<p>The strength of gravitational force follow the inverse square law
|
||
distributing gravitational flux over the surface area of a sphere (<span
|
||
class="math inline">4<em>π</em><em>r</em><sup>2</sup></span>).</p>
|
||
<h4 id="inverse-square-law">Inverse Square Law</h4>
|
||
<p>Any source of a signal strength (<span
|
||
class="math inline"><em>S</em><sub>0</sub></span>) that radiates
|
||
isotropically in 3-dimensional space will distribute that signal
|
||
strength (<span class="math inline"><em>S</em><sub>0</sub></span>) over
|
||
the surface area of a sphere (<span
|
||
class="math inline"><em>S</em><em>A</em> = 4<em>π</em><em>r</em></span>)
|
||
of radius (<span class="math inline"><em>r</em></span>). Such that the
|
||
intensity (<span class="math inline"><em>I</em></span>) at distance
|
||
(<span class="math inline"><em>r</em></span>) is:</p>
|
||
<p><span class="math display">$$I(r) = \frac{S_0}{4 \pi
|
||
r^{2}}=\frac{S_0}{2 \tau r^{2}}$$</span> <img
|
||
src="lib/img/Inverse_square_law.svg.png" alt="inverse square law" />
|
||
#### <span class="math inline"><em>R</em><em>ν</em></span> Reduced
|
||
Gravitational Constant In this version of Newton’s Law of Gravity we
|
||
introduce a new constant <span
|
||
class="math inline"><em>G</em><sub>0</sub></span>, the reduced
|
||
gravitational constant to accommodate for the factor of <span
|
||
class="math inline">4<em>π</em> = 2<em>τ</em></span> which is has been
|
||
integrated in the SI version of the gravitational constant.</p>
|
||
<p><span class="math inline">$F_g =G \frac{m_{1} m_{2}}{r^{2}}= G_0
|
||
\frac{m_{1} m_{2}}{4 \pi r^{2}}=G_0 \frac{m_{1} m_{2}}{2 \tau
|
||
r^{2}}$</span></p>
|
||
<p>Where:</p>
|
||
<p><span class="math inline">$G = \frac{G_{0}}{2\tau} = 6.67384 \times
|
||
10^{-11} \frac{N \cdot m^2}{kg^2}$</span></p>
|
||
<p>Analyzing the units: <span class="math display">$$\frac{N \cdot
|
||
m^2}{kg^2} = \left( \frac{\left( kg \cdot \frac{m}{s^2} \right) \cdot
|
||
m^2}{kg^2} \right)=\frac{m^3}{s^2 kg}$$</span> Converting seconds to
|
||
meters with the SI speed of light as a conversion factor: <span
|
||
class="math display">$$\frac{m^3}{s^2
|
||
kg}\cdot\frac{1}{c^2}=\frac{m^3}{s^2
|
||
kg}\cdot\frac{s^2}{m^2}=\frac{m}{kg}$$</span></p>
|
||
<p>Thus where space and time are measured in units of meters, the
|
||
reduced gravitational constant, is:</p>
|
||
<p><span class="math display">$$\boxed{G_0=\frac{2\tau
|
||
G}{c^2}=\frac{2\tau \cdot 6.67384 \times 10^{-11}}{299792458^2}
|
||
\frac{m}{kg} = 9.33135 \times 10^-27 \frac{m}{kg}}$$</span></p>
|
||
<blockquote>
|
||
<p>Observation This implies that not only can space an time be measure
|
||
in units of meters, but so can mass.</p>
|
||
</blockquote>
|
||
<h3 id="relativistic-energy-momentum-relation">Relativistic Energy
|
||
Momentum Relation</h3>
|
||
<p>Einsteins <a
|
||
href="https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation">Relativistic
|
||
Energy Momentum</a> relationship shows a Pythagorean relation between
|
||
the total energy (<span class="math inline"><em>E</em></span>), rest
|
||
mass (<span class="math inline"><em>m</em><sub>0</sub></span>) and
|
||
momentum (<span class="math inline"><em>p</em></span>) of a system.</p>
|
||
<p><span
|
||
class="math inline"><em>E</em><sup>2</sup> = (<em>m</em><sub>0</sub>⋅<em>c</em><sup>2</sup>)<sup>2</sup> + (<em>p</em>⋅<em>c</em>)<sup>2</sup></span></p>
|
||
<p>Where space and time are both measure in units of meters, c=1.</p>
|
||
<p><span
|
||
class="math inline"><em>E</em><sup>2</sup> = (<em>m</em><sub>0</sub>)<sup>2</sup> + (<em>p</em>)<sup>2</sup></span></p>
|
||
<p>From this we can see that Energy, Momentum and Mass have equivalent
|
||
units.</p>
|
||
<blockquote>
|
||
<p><em>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.</em></p>
|
||
</blockquote>
|
||
<h5 id="objects-of-mass-at-rest">Objects of mass at rest</h5>
|
||
<p>For an object at rest with no momentum (<span
|
||
class="math inline"><em>p</em> = 0</span>) we see Einstein’s famous
|
||
equations:</p>
|
||
<p><span
|
||
class="math inline"><em>E</em> = <em>m</em><sub>0</sub> ⋅ <em>c</em><sup>2</sup></span></p>
|
||
<p>Or, with <span class="math inline"><em>c</em> = 1</span>, this is
|
||
much simpler to understand. Energy = Mass</p>
|
||
<p><span
|
||
class="math inline"><em>E</em> = <em>m</em><sub>0</sub></span></p>
|
||
<h5 id="zero-mass-objects-moving-at-the-speed-of-light">Zero mass
|
||
objects moving at the speed of light</h5>
|
||
<p>And for objects with no mass, like photos, (<span
|
||
class="math inline"><em>m</em><sub>0</sub> = 0</span>):</p>
|
||
<p><span
|
||
class="math inline"><em>E</em> = <em>p</em><em>c</em></span></p>
|
||
<p>Or, with <span class="math inline"><em>c</em> = 1</span>, this is
|
||
much simpler to understand. Energy = Momentum</p>
|
||
<p><span class="math inline"><em>E</em> = <em>p</em></span></p>
|
||
<h2 id="plancks-constant">Planck’s Constant</h2>
|
||
<p>The <a href="https://en.wikipedia.org/wiki/Planck_constant">Reduced
|
||
Planck constant</a> , ħ, represents a conversion factor for relating the
|
||
frequency, <span class="math inline"><em>ω</em></span> (in <span
|
||
class="math inline">2<em>π</em></span> radians per second), of a photon
|
||
to the energy of that photon. This can easily be seen from the simple
|
||
but profound relationship:</p>
|
||
<p><span class="math inline"><em>E</em> = ℏ<em>ω</em></span></p>
|
||
<p>Where:</p>
|
||
<p><span
|
||
class="math inline">ℏ = 1.054571726 × 10<sup>−34</sup><em>J</em> ⋅ <em>s</em></span></p>
|
||
<p>and</p>
|
||
<p><span class="math inline">$J \cdot s =
|
||
{kg}\cdot\frac{m^2}{s}$</span></p>
|
||
<blockquote>
|
||
<p>Reduced Planck’s Constant <span class="math inline">$\hbar =
|
||
\frac{h}{2\pi} = \frac{h}{\tau}$</span></p>
|
||
</blockquote>
|
||
<p>Simplifying our units by converting time and mass to units of meters:
|
||
<span class="math display">$$\boxed{\hbar=1.054571726 \times 10^{−34}
|
||
{kg}\cdot\frac{m^2}{s}\cdot\frac{G_0}{c}=3.282462\times10^{-69}m^2}$$</span></p>
|
||
<p>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:</p>
|
||
<p><span
|
||
class="math display">$$\boxed{\sqrt{\hbar}=\sqrt{3.282462\times10^{-69}m^2}=5.72928\times10^{-35}m}$$</span></p>
|
||
<h4 id="planck-area">Planck Area</h4>
|
||
<p>The <a
|
||
href="https://en.wikipedia.org/wiki/Planck_units#Derived_units">Planck
|
||
Area</a> is the square of the <a
|
||
href="https://en.wikipedia.org/wiki/Planck_units#Planck_length">Planck
|
||
Length</a>.</p>
|
||
<p><span class="math inline">$l_{P}= \sqrt{\frac{\hbar
|
||
G}{c^3}}$</span></p>
|
||
<p>and <span class="math inline">$l_{P}^{2}= \frac{\hbar
|
||
G}{c^3}$</span></p>
|
||
<p>In <span class="math inline"><em>R</em><em>ν</em></span> units both
|
||
<span class="math inline"><em>c</em></span> and <span
|
||
class="math inline"><em>G</em><sub><em>o</em></sub></span> are 1.</p>
|
||
<p><span class="math inline">$l_{P} = \sqrt{\hbar}$</span></p>
|
||
<p>and <span
|
||
class="math inline"><em>l</em><sub><em>P</em></sub><sup>2</sup> = ℏ</span>
|
||
## Bekenstein’s Bound After having recently read <em>Three Roads to
|
||
Quantum Gravity</em> by Lee Smolin, I now suspect the meaning of this
|
||
areas is related to the <a
|
||
href="https://en.wikipedia.org/wiki/Bekenstein_bound">Bekensteins
|
||
Law</a> as applied to a surface areas surrounding a mass. Where the <a
|
||
href="https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)">thermodynamic
|
||
entropy</a>, <em>S</em>, is proportional to the the enclosed surface
|
||
area, <span class="math inline"><em>A</em></span>.</p>
|
||
<p><span
|
||
class="math inline">$S=\frac{1}{4}\cdot\frac{A}{G\hbar}$</span></p>
|
||
<p><span class="math inline">$S=\frac{k c^{3} A}{4 G \hbar}$</span></p>
|
||
<p><span class="math inline">$S \le \frac{2\pi k R E}{\hbar c} =
|
||
\frac{\tau R k E}{\hbar c}$</span></p>
|
||
<p>From our new values for <span
|
||
class="math inline"><em>G</em><sub>0</sub></span>and <span
|
||
class="math inline">ℏ</span> we can likely rewrite this:</p>
|
||
<p><span class="math inline">$S=\frac{\pi\cdot A}{\hbar G_0}$</span></p>
|
||
<p>With the limiting case being at the Plank scale.</p>
|
||
<p><span class="math inline">$S=\frac{\pi\cdot \sqrt{\hbar}}{\hbar
|
||
G_0}$</span></p>
|
||
<h2 id="planck-length">Planck Length</h2>
|
||
<p>https://en.wikipedia.org/wiki/Planck_length</p>
|
||
<p>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.</p>
|
||
<p>The <a
|
||
href="https://simple.wikipedia.org/wiki/Schwarzschild_radius">Schwarzschild
|
||
Radius</a> is defined as the distance at which light cannot escape from
|
||
the gravitational field of a mass (m):</p>
|
||
<p>Classic Derivation.</p>
|
||
<p><span class="math inline">$r_S=\frac{2G m}{c^2}$</span></p>
|
||
<p>The reduced <a
|
||
href="https://en.wikipedia.org/wiki/Compton_wavelength">Compton
|
||
Wavelength</a> represents a lower limit on the wavelength for quanta
|
||
that can interact with a quantum particle with mass (m):</p>
|
||
<p><span class="math inline">$\lambda_C=\frac{h}{m c}$</span></p>
|
||
<p><span class="math inline">$\bar{\lambda_C}=\frac{2\pi\hbar}{m
|
||
c}=\frac{\tau\hbar}{m c}$</span></p>
|
||
<p>And set the Schwarzschild Radius equal to the Compton Wavelength:
|
||
<span
|
||
class="math inline"><em>r</em><sub><em>S</em></sub> = <em>λ</em><sub><em>C</em></sub></span></p>
|
||
<p><span class="math inline">$\frac{2Gm}{c^{2}}=\frac{h}{m
|
||
c}$</span></p>
|
||
<p><span class="math inline">$m^{2}= \frac{hc}{2G}$</span></p>
|
||
<p><span class="math inline">$m = \sqrt{\frac{hc}{2G}}$</span></p>
|
||
<p><span
|
||
class="math inline">$l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^2}$</span>
|
||
<span class="math inline">$l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^{2}}=
|
||
\sqrt{\frac{2Gh}{c^2}}$</span></p>
|
||
<p>With reduced Compton Wavelength <span
|
||
class="math inline">$r_S=\bar{\lambda_C}$</span></p>
|
||
<p><span class="math inline">$\frac{2Gm}{c^{2}}=\frac{\tau\hbar}{m
|
||
c}$</span></p>
|
||
<p><span class="math inline">$m^2=\frac{\tau\ \hbar\ c}{2G}$</span></p>
|
||
<p><span class="math inline">$m = \sqrt{\frac{\tau\ \hbar\
|
||
c}{2G}}$</span></p>
|
||
<p><span class="math inline">$l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\
|
||
c}{2G}}}{c^2}$</span></p>
|
||
<p><span class="math inline">$l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\
|
||
c}{2G}}}{c^{2}}=\sqrt{\frac{4\ \tau\ G\ \hbar}{c^3}}$</span></p>
|
||
<p>If we reduce the units in these equation to those of mass and time
|
||
measured in meters.</p>
|
||
<p><span class="math inline">$l_P=\sqrt{4\tau\hbar G_{o}}$</span></p>
|
||
<p>and</p>
|
||
<p><span class="math inline">$\lambda_C=\frac{\hbar}{m}$</span></p>
|
||
<p><span
|
||
class="math inline">$m=R_s=\lambda_C=\frac{\hbar}{m}$</span></p>
|
||
<p>This is known as the Planck Mass, <span
|
||
class="math inline"><em>M</em><sub><em>P</em></sub></span>. <span
|
||
class="math inline">$M_P=m=\sqrt{\hbar}$</span></p>
|
||
<p>Solving the Compton Wavelength for distance we find the classic Plank
|
||
Length:</p>
|
||
<p><span
|
||
class="math inline">$\lambda_C=\frac{\hbar}{\sqrt{\hbar}}=\frac{\sqrt{\hbar}}{\sqrt{\hbar}}\cdot\frac{\hbar}{\sqrt{\hbar}}=\sqrt{\hbar}=L_P$</span></p>
|
||
<p>Which is in precise agreement with the value we found in above. Thus
|
||
the Plank Length is:</p>
|
||
<p><span
|
||
class="math inline">$L_P=\sqrt{\hbar}=5.72928\times10^{-35}m$</span></p>
|
||
<p>When we measure distance, time, and mass in units of distance, c=1,
|
||
and the Plank Time, <span
|
||
class="math inline"><em>T</em><sub><em>P</em></sub></span>, is equal to
|
||
Plank Length, <span
|
||
class="math inline"><em>L</em><sub><em>P</em></sub></span>, which is
|
||
equal to the Plank Mass, <span
|
||
class="math inline"><em>M</em><sub><em>P</em></sub></span>:</p>
|
||
<p><span class="math display">$$\boxed{L_P=T_P=M_P}$$</span></p>
|
||
<table>
|
||
<colgroup>
|
||
<col style="width: 30%" />
|
||
<col style="width: 30%" />
|
||
<col style="width: 40%" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr class="header">
|
||
<th>Conversion Factor</th>
|
||
<th>Symbol</th>
|
||
<th>Value</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="odd">
|
||
<td>meters to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>χ</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$1.74542\times10^{34}
|
||
\frac{L}{m}$</span></td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>seconds to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>τ</em><sub><em>p</em></sub></span></td>
|
||
<td><span class="math inline">$5.23264\times10^{42}
|
||
\frac{L}{s}$</span></td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>mass to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>G</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$1.62871\times10^8
|
||
\frac{L}{kg}$</span></td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>energy to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>E</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$1.81219\times10^9
|
||
\frac{L}{J}$</span></td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>momentum to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>P</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$5.43280\times10^{-1} \frac{L\cdot s}{kg
|
||
\cdot m}$</span></td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>temperature to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>k</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$2.501998\times10^{-14}
|
||
\frac{L}{K}$</span></td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>charge to Planck Length</td>
|
||
<td><span
|
||
class="math inline"><em>C</em><sub><em>P</em></sub></span></td>
|
||
<td><span class="math inline">$1.89007\times10^{18}
|
||
\frac{L}{C}$</span></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<p>Applying conversion factors from the table above, we can convert SI
|
||
values to Reduced Natural Units. <span
|
||
class="math inline">$c=\frac{1}{\sqrt{\epsilon_o \mu_o}}$</span></p>
|
||
<table>
|
||
<colgroup>
|
||
<col style="width: 23%" />
|
||
<col style="width: 23%" />
|
||
<col style="width: 23%" />
|
||
<col style="width: 30%" />
|
||
</colgroup>
|
||
<thead>
|
||
<tr class="header">
|
||
<th>Quantity</th>
|
||
<th>Symbol</th>
|
||
<th>SI</th>
|
||
<th><span class="math inline"><em>ν</em></span></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr class="odd">
|
||
<td>Speed of Light</td>
|
||
<td><span class="math inline"><em>c</em></span></td>
|
||
<td><span class="math inline">$299792458 \frac{m}{s}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>Gravitational Constant</td>
|
||
<td><span class="math inline"><em>G</em><sub>0</sub></span></td>
|
||
<td><span class="math inline">$8.38659\times10^{-10} \frac{m^3}{kg \cdot
|
||
s^2}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>Boltzmann’s Constant</td>
|
||
<td><span class="math inline"><em>k</em></span></td>
|
||
<td><span class="math inline">$k=1.380649\times10^-23
|
||
\frac{J}{K}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>Permittivity of Free Space</td>
|
||
<td><span
|
||
class="math inline"><em>ϵ</em><sub><em>o</em></sub></span></td>
|
||
<td><span class="math inline">$8.854187817620\times10^{-12}
|
||
\frac{C^{2}s^2}{kg \cdot m^3}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>Permeability of Free Space</td>
|
||
<td><span
|
||
class="math inline"><em>μ</em><sub><em>o</em></sub></span></td>
|
||
<td><span class="math inline">$\huge{\frac{1}{\epsilon_{o} \cdot
|
||
c^{2}}}$</span></td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>Planck’s Constant</td>
|
||
<td><span class="math inline">ℏ</span></td>
|
||
<td><span class="math inline">$1.054571726\times10^-34 \frac{kg \cdot
|
||
m^2}{s}$</span></td>
|
||
<td><span class="math inline">1<em>L</em><sup>2</sup></span></td>
|
||
</tr>
|
||
<tr class="odd">
|
||
<td>Mass of the Electron</td>
|
||
<td><span
|
||
class="math inline"><em>m</em><sub><em>e</em></sub></span></td>
|
||
<td><span
|
||
class="math inline">9.10938 × 10<sup>−31</sup><em>k</em><em>g</em></span></td>
|
||
<td><span
|
||
class="math inline">1.48366 × 10<sup>−22</sup><em>L</em></span></td>
|
||
</tr>
|
||
<tr class="even">
|
||
<td>Charge of the Electron</td>
|
||
<td><span class="math inline"><em>e</em><sup>−</sup></span></td>
|
||
<td><span
|
||
class="math inline"> − 1.60218 × 10<sup>−19</sup><em>C</em></span></td>
|
||
<td><span
|
||
class="math inline"> − 3.02822 × 10<sup>−1</sup><em>L</em></span></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h2 id="fine-structure-constant-1">Fine Structure Constant</h2>
|
||
<p>https://en.wikipedia.org/wiki/Fine-structure_constant As a
|
||
consistency check, we compute the <em>Fine Structure Constant</em> using
|
||
Reduced Natural Units which is a unit less ratio that should be
|
||
independent of our system of units.</p>
|
||
<p><span
|
||
class="math inline">$\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar
|
||
c}=\frac{e^2}{4\pi}=0.00729735≈\frac{1}{137}}$</span></p>
|
||
<p>This check confirms that our system of Reduced Natural Units has
|
||
internally consistent values for <span
|
||
class="math inline"><em>c</em></span>, <span
|
||
class="math inline"><em>ϵ</em><sub><em>o</em></sub></span>, <span
|
||
class="math inline">ℏ</span> and <span
|
||
class="math inline"><em>e</em>−</span>. And also <span
|
||
class="math inline"><em>G</em><sub><em>o</em></sub></span> which was
|
||
used to computer prior values is also consistent.</p>
|
||
<h2 id="sage-code">Sage Code</h2>
|
||
<p>Unit Analysis computations have been performed with <a
|
||
href="https://www.sagemath.org/">Sage Math</a>.</p>
|
||
<div class="sourceCode" id="cb20"><pre
|
||
class="sourceCode bash"><code class="sourceCode bash"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Define constance</span></span>
|
||
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a><span class="ex">one</span> = 1.n<span class="er">(</span><span class="va">digits</span><span class="op">=</span>6<span class="kw">)</span></span>
|
||
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a><span class="ex">pi</span> = pi.n<span class="er">(</span><span class="va">digits</span><span class="op">=</span>6<span class="kw">)</span></span>
|
||
<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a><span class="ex">tau</span> = 2 <span class="pp">*</span> pi</span>
|
||
<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a><span class="ex">t</span> = tau</span>
|
||
<span id="cb20-6"><a href="#cb20-6" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-7"><a href="#cb20-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Define the units</span></span>
|
||
<span id="cb20-8"><a href="#cb20-8" aria-hidden="true" tabindex="-1"></a><span class="ex">meters</span> = var<span class="er">(</span><span class="st">'m'</span><span class="kw">)</span></span>
|
||
<span id="cb20-9"><a href="#cb20-9" aria-hidden="true" tabindex="-1"></a><span class="ex">m</span> = one<span class="pp">*</span>meters</span>
|
||
<span id="cb20-10"><a href="#cb20-10" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-11"><a href="#cb20-11" aria-hidden="true" tabindex="-1"></a><span class="ex">seconds</span> = var<span class="er">(</span><span class="st">'s'</span><span class="kw">)</span></span>
|
||
<span id="cb20-12"><a href="#cb20-12" aria-hidden="true" tabindex="-1"></a><span class="ex">s</span> = seconds</span>
|
||
<span id="cb20-13"><a href="#cb20-13" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-14"><a href="#cb20-14" aria-hidden="true" tabindex="-1"></a><span class="ex">kilograms</span> = var<span class="er">(</span><span class="st">'kg'</span><span class="kw">)</span></span>
|
||
<span id="cb20-15"><a href="#cb20-15" aria-hidden="true" tabindex="-1"></a><span class="ex">kg</span> = kilograms</span>
|
||
<span id="cb20-16"><a href="#cb20-16" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-17"><a href="#cb20-17" aria-hidden="true" tabindex="-1"></a><span class="ex">newtons</span> = kg <span class="pp">*</span> m / s^2</span>
|
||
<span id="cb20-18"><a href="#cb20-18" aria-hidden="true" tabindex="-1"></a><span class="ex">N</span> = newtons</span>
|
||
<span id="cb20-19"><a href="#cb20-19" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-20"><a href="#cb20-20" aria-hidden="true" tabindex="-1"></a><span class="ex">joules</span> = N <span class="pp">*</span> m</span>
|
||
<span id="cb20-21"><a href="#cb20-21" aria-hidden="true" tabindex="-1"></a><span class="ex">J</span> = joules</span>
|
||
<span id="cb20-22"><a href="#cb20-22" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-23"><a href="#cb20-23" aria-hidden="true" tabindex="-1"></a><span class="ex">print</span><span class="er">(</span><span class="st">"pi ="</span><span class="ex">,</span> pi<span class="kw">)</span></span>
|
||
<span id="cb20-24"><a href="#cb20-24" aria-hidden="true" tabindex="-1"></a><span class="ex">print</span><span class="er">(</span><span class="st">"tau ="</span><span class="ex">,</span> t<span class="kw">)</span></span>
|
||
<span id="cb20-25"><a href="#cb20-25" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-26"><a href="#cb20-26" aria-hidden="true" tabindex="-1"></a><span class="co"># Speed of light in meters/second</span></span>
|
||
<span id="cb20-27"><a href="#cb20-27" aria-hidden="true" tabindex="-1"></a><span class="ex">speed_of_light</span> = 299792458 <span class="pp">*</span> meters/seconds</span>
|
||
<span id="cb20-28"><a href="#cb20-28" aria-hidden="true" tabindex="-1"></a><span class="ex">sol</span> = speed_of_light</span>
|
||
<span id="cb20-29"><a href="#cb20-29" aria-hidden="true" tabindex="-1"></a><span class="ex">c</span> = sol</span>
|
||
<span id="cb20-30"><a href="#cb20-30" aria-hidden="true" tabindex="-1"></a><span class="ex">print</span><span class="er">(</span><span class="st">"si c ="</span><span class="ex">,</span> c<span class="kw">)</span></span>
|
||
<span id="cb20-31"><a href="#cb20-31" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-32"><a href="#cb20-32" aria-hidden="true" tabindex="-1"></a><span class="ex">rnu_c</span> = c / c</span>
|
||
<span id="cb20-33"><a href="#cb20-33" aria-hidden="true" tabindex="-1"></a><span class="ex">print</span><span class="er">(</span><span class="st">"R\u03BD c ="</span><span class="ex">,</span> rnu_c<span class="kw">)</span></span>
|
||
<span id="cb20-34"><a href="#cb20-34" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-35"><a href="#cb20-35" aria-hidden="true" tabindex="-1"></a><span class="co"># Gravitational Constant</span></span>
|
||
<span id="cb20-36"><a href="#cb20-36" aria-hidden="true" tabindex="-1"></a><span class="ex">gravitational_constant</span> = 6.67384e-11 <span class="pp">*</span> N<span class="pp">*(</span>m^2/kg^2<span class="pp">)</span></span>
|
||
<span id="cb20-37"><a href="#cb20-37" aria-hidden="true" tabindex="-1"></a><span class="ex">G</span> = gravitational_constant</span>
|
||
<span id="cb20-38"><a href="#cb20-38" aria-hidden="true" tabindex="-1"></a><span class="ex">print</span><span class="er">(</span><span class="st">"si G ="</span><span class="ex">,</span> G<span class="kw">)</span></span>
|
||
<span id="cb20-39"><a href="#cb20-39" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-40"><a href="#cb20-40" aria-hidden="true" tabindex="-1"></a><span class="ex">rnu_G</span> = 4<span class="pp">*</span>pi<span class="pp">*</span>G/c^2</span>
|
||
<span id="cb20-41"><a href="#cb20-41" aria-hidden="true" tabindex="-1"></a><span class="ex">Go</span> = rnu_G</span>
|
||
<span id="cb20-42"><a href="#cb20-42" aria-hidden="true" tabindex="-1"></a><span class="ex">print</span><span class="er">(</span><span class="st">"R\u03BD Go ="</span><span class="ex">,</span> Go<span class="kw">)</span></span>
|
||
<span id="cb20-43"><a href="#cb20-43" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-44"><a href="#cb20-44" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-45"><a href="#cb20-45" aria-hidden="true" tabindex="-1"></a><span class="co"># Planck's Constant</span></span>
|
||
<span id="cb20-46"><a href="#cb20-46" aria-hidden="true" tabindex="-1"></a><span class="ex">reduced_plancks_constant</span> = 1.054571726e-34 <span class="pp">*</span> J<span class="pp">*</span>s</span>
|
||
<span id="cb20-47"><a href="#cb20-47" aria-hidden="true" tabindex="-1"></a><span class="ex">h_bar</span> = reduced_plancks_constant</span>
|
||
<span id="cb20-48"><a href="#cb20-48" aria-hidden="true" tabindex="-1"></a><span class="ex">print</span><span class="er">(</span><span class="st">"si \u210F ="</span><span class="ex">,</span> h_bar<span class="kw">)</span></span>
|
||
<span id="cb20-49"><a href="#cb20-49" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-50"><a href="#cb20-50" aria-hidden="true" tabindex="-1"></a><span class="ex">rnu_h_bar</span> = h_bar <span class="pp">*</span> Go / c</span>
|
||
<span id="cb20-51"><a href="#cb20-51" aria-hidden="true" tabindex="-1"></a><span class="ex">print</span><span class="er">(</span><span class="st">"R\u03BD "</span><span class="ex">u</span><span class="st">"\u210F ="</span><span class="ex">,</span> rnu_h_bar<span class="kw">)</span></span>
|
||
<span id="cb20-52"><a href="#cb20-52" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb20-53"><a href="#cb20-53" aria-hidden="true" tabindex="-1"></a><span class="co"># Planck Length</span></span>
|
||
<span id="cb20-54"><a href="#cb20-54" aria-hidden="true" tabindex="-1"></a><span class="ex">rnu_h_bar_str</span> = str<span class="er">(</span><span class="ex">rnu_h_bar</span><span class="kw">)</span> </span>
|
||
<span id="cb20-55"><a href="#cb20-55" aria-hidden="true" tabindex="-1"></a><span class="ex">numerical_part_str</span> = rnu_h_bar_str.split<span class="er">(</span><span class="st">'*'</span><span class="kw">)</span><span class="ex">[0]</span> </span>
|
||
<span id="cb20-56"><a href="#cb20-56" aria-hidden="true" tabindex="-1"></a><span class="ex">numerical_part_str</span> = numerical_part_str.strip<span class="er">(</span><span class="st">'()'</span><span class="kw">)</span></span>
|
||
<span id="cb20-57"><a href="#cb20-57" aria-hidden="true" tabindex="-1"></a><span class="ex">numerical_part</span> = float<span class="er">(</span><span class="ex">numerical_part_str</span><span class="kw">)</span></span>
|
||
<span id="cb20-58"><a href="#cb20-58" aria-hidden="true" tabindex="-1"></a><span class="ex">rnu_sqrt_h_bar</span> = numerical_part^<span class="er">(</span><span class="ex">1/2</span><span class="kw">)</span></span>
|
||
<span id="cb20-59"><a href="#cb20-59" aria-hidden="true" tabindex="-1"></a><span class="co"># ^ Sage cannot process sqrt on units... Lame.</span></span>
|
||
<span id="cb20-60"><a href="#cb20-60" aria-hidden="true" tabindex="-1"></a><span class="ex">lP</span> = rnu_sqrt_h_bar <span class="pp">*</span> m</span>
|
||
<span id="cb20-61"><a href="#cb20-61" aria-hidden="true" tabindex="-1"></a><span class="ex">print</span><span class="er">(</span><span class="st">"R\u03BD \u221A\u210F ="</span><span class="ex">,</span> lP<span class="kw">)</span></span></code></pre></div>
|
||
<h4 id="output">Output</h4>
|
||
<pre><code>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)</code></pre>
|
||
<h1 id="terminology">Terminology</h1>
|
||
<h1 class="unnumbered" id="citations-1">Citations</h1>
|
||
<div id="refs" class="references csl-bib-body hanging-indent"
|
||
role="doc-bibliography">
|
||
<div id="ref-JohnHaverlackACEP" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<span>“John <span>Haverlack</span> <span>ACEP</span>.”</span> n.d.
|
||
https://www.uaf.edu/acep/about/our-team/john-haverlack.php. Accessed
|
||
September 30, 2025.
|
||
</div>
|
||
</div>
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||
</body>
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</html>
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