840 lines
27 KiB
Markdown
840 lines
27 KiB
Markdown
---
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bibliography:
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- lib/zotero.bib
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link-citations: true
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mainfont: Linux Libertine O
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monofont: DejaVu Sans Mono
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sansfont: Linux Biolinum O
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title: Clear Skies - A Reference Architecture for Resilient Alaskan
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Microgrid Cyberinfrastructure
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title-block: false
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---
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<div class="frontmatter">
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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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<div style="text-align:center; margin:2em 0;">
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<img src="lib/img/UAF-ACEP.png" alt="Cover illustration" style="max-height:100px;">
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</div>
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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:left; margin-top:5em;">
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<!-- Metadata Page (PDF analog) -->
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<table style="margin-top:5em; width:100%; border-collapse:collapse;">
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<tbody>
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<tr><td style="width:30%; font-weight:bold;">Title</td><td>Basic Book Builder</td></tr>
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<tr><td style="font-weight:bold;">Subtitle</td><td>A Pandoc Template for building books and articles</td></tr>
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<tr><td style="font-weight:bold;">Affiliation</td><td>Alaska Center for Energy and Power</td></tr>
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<tr><td style="font-weight:bold;">Institution</td><td>University of Alaska Fairbanks</td></tr>
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<tr><td style="font-weight:bold;">Author</td><td>John Haverlack</td></tr>
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<tr><td style="font-weight:bold;">Copyright</td><td>© 2025 Alaska Center for Energy and Power</td></tr>
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<tr><td style="font-weight:bold;">License</td><td>CC BY-ND 4.0</td></tr>
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<tr><td style="font-weight:bold;">Version</td><td>1.0.0</td></tr>
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<tr><td style="font-weight:bold;">Date</td><td>2025-11-13</td></tr>
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<tr><td style="font-weight:bold;">State</td><td>BETA</td></tr>
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<tr><td style="font-weight:bold;">Source</td><td>https://github.com/jehaverlack/basic-book-builder</td></tr>
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</tbody>
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</table>
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</div>
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<hr style="margin:4em 0;">
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</div>
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- [Introduction](#introduction)
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- [Conventions](#conventions)
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- [Getting Started](#getting-started)
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- [Installing Pre-Requisites](#installing-pre-requisites)
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- [Required](#required)
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- [Optional](#optional)
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- [Editing the Configuration](#editing-the-configuration)
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- [Editing the Book](#editing-the-book)
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- [Configuration](#configuration)
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- [FrontMatter Config](#frontmatter-config)
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- [Editing the Content](#editing-the-content)
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- [Citations](#citations)
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- [Usage: Building the Book](#usage-building-the-book)
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- [Example Content](#example-content)
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- [SI Conversion Factors](#si-conversion-factors)
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- [Physical Constants](#physical-constants)
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- [Fine Structure Constant](#fine-structure-constant)
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- [Newton’s Law of Gravity](#newtons-law-of-gravity)
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- [Relativistic Energy Momentum
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Relation](#relativistic-energy-momentum-relation)
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- [Planck’s Constant](#plancks-constant)
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- [Planck Length](#planck-length)
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- [Fine Structure Constant](#fine-structure-constant-1)
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- [Sage Code](#sage-code)
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- [Terminology](#terminology)
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- [Citations](#citations-1)
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# Introduction
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This is a basic book (or article) builder template based on a Pandoc
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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
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vault.
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This book builder template has been curated by John Haverlack.([“John
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Haverlack ACEP” n.d.](#ref-JohnHaverlackACEP))
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## Conventions
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A few callout box styles have been added to easily highlight content.
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<div class="callout-established">
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**Established Concept**
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Einsteins Relativistic Dynamics Equations
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*E*<sup>2</sup> = (*m*<sub>0</sub>⋅*c*<sup>2</sup>)<sup>2</sup> + (*p*⋅*c*)<sup>2</sup>
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</div>
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<div class="callout-proposed">
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**Proposed Concept**
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With the speed of light, *c* = 1:
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*E*<sup>2</sup> = *m*<sub>0</sub><sup>2</sup> + *p*<sup>2</sup>
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</div>
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<div class="callout-speculative">
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**Speculative Concept**
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With the speed of light, *c* = 1:
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*E*<sup>2</sup> = *m*<sub>0</sub><sup>2</sup> + *p*<sup>2</sup>
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</div>
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<div class="callout-caution">
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**Caution Note**
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Beware of this section.
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</div>
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<div class="callout-warning">
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**Warning Note**
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Beware of this section.
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</div>
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<div class="callout-danger">
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**Alerts**
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Extreme Highlight
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</div>
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# Getting Started
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## Installing Pre-Requisites
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For Debian / ZorinOS and likely Ubuntu based systems.
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<div class="callout-caution">
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**TODO**
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It would be nice to roll a setup script to take care of this.
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</div>
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### Required
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#### Pandoc
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- https://pandoc.org/
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- [Download](https://github.com/jgm/pandoc/releases/tag/3.8.2.1)
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<!-- -->
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sudo apt install https://github.com/jgm/pandoc/releases/download/3.8.2.1/pandoc-3.8.2.1-1-amd64.deb
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#### Code Editor
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<div class="callout-established">
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**Code Editor**
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[VSCodium](https://vscodium.com/) is recommend for privacy
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(telemetry/tracking) reasons - https://vscodium.com/
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</div>
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But any text editor will work. \#### make
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sudo apt install make
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#### jq and yq
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sudo apt install jq yq
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#### texlive
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sudo apt install texlive texlive-xetex texlive-latex-extra texlive-fonts-recommended texlive-fonts-extra
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#### MathJax
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- https://www.mathjax.org/
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<!-- -->
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wget https://registry.npmjs.org/mathjax/-/mathjax-3.2.2.tgz
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tar xzf mathjax-3.2.2.tgz
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mv package/es5/* lib/mathjax
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rm -rf package mathjax-3.2.2.tgz
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<div class="callout-caution">
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**TODO**
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This need to be rolled into a setup script.
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</div>
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### Optional
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The following are not strictly requires to use this book builder
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template.
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#### Obsidian
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<div class="callout-established">
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**Highly Recommended**
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Editing book chapter content in Obsidian is a very productive means for
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editing Markdown source content.
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</div>
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- https://obsidian.md/
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- [Deb
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Package](https://github.com/obsidianmd/obsidian-releases/releases/download/v1.9.14/obsidian_1.9.14_amd64.deb)
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<!-- -->
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sudo apt install https://github.com/obsidianmd/obsidian-releases/releases/download/v1.9.14/obsidian_1.9.14_amd64.deb
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#### Zotero
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<div class="callout-established">
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**Highly Recommended**
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If you need to managed citations and references, Zotero integration is
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highly recommended.
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</div>
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- https://www.zotero.org/
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<!-- -->
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sudo cp ./scripts/deps/zotero.list /etc/apt/sources.list.d/
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sudo apt update
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sudo apt install zotero
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##### Better BibTex for Zotero
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Install the Better BibTex Plugin for Zotero - Zotero \> Tool \> Plugins
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##### Export citations.bib
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- Zotero \> File \> Export Library \> Format: Better BibTeX
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- [ ] Keep Updated
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- [ ] Save to: \~/Documents/Lib/zotero.bib
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- [ ] Symlink your \~/Documents/Lib/Citations.bib to
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basic-book-builder/lib/zotero.bib
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##### Zotero Connector Browser Plugin
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- https://chromewebstore.google.com/detail/zotero-connector/ekhagklcjbdpajgpjgmbionohlpdbjgc
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Provides you the ability to auto add Web resources to your Zotero
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citation database.
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#### lmodern
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sudo apt install lmodern
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#### epubcheck
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sudo apt install epubcheck
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#### foliate
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An EPub Reader
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- https://johnfactotum.github.io/foliate/
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<!-- -->
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sudo apt install https://github.com/johnfactotum/foliate/releases/download/2.6.4/com.github.johnfactotum.foliate_2.6.4_all.deb
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#### calibre
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An EPub Reader
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- https://calibre-ebook.com
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<!-- -->
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sudo apt install calibre
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## Editing the Configuration
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# Editing the Book
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### Configuration
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There are a number of other config files for each format:
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conf/
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├── epub-metadata.xml
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├── epub_template.html
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├── epub.yaml
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├── frontmatter_epub.md
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├── frontmatter_epub.xhtml
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├── frontmatter.html
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├── frontmatter.tex
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├── header.tex
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├── html.yaml
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├── latex.yaml
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├── markdown.yaml
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├── metadata.yaml
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├── pandoc.yaml
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├── pdf.yaml
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├── style.css
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└── style_epub.css
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#### Main Config Files
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- metadata.yaml - Set Title, etc
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- pandoc.yaml - Main Pandoc Config \#### Per format Configs
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- `pdf.yaml`
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- `html.yaml`
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- `latex.yaml`
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- `epub.yaml`
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### FrontMatter Config
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There are 2 Version of the FrontMatter for PDF, and HTML bases formats
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that set the Title, Author, Verizon, Copyright, etc…
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- `frontmatter.tex`
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- `frontmatter.html`
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- `frontmatter_epub.*` - Work in Progress
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> There is probably a better way to do this.
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## Editing the Content
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To edit the book open the `basic-book-builder` directory as an Obsidian
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Vault.
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- Edit the Markdown content in the `chapters` directory.
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### Citations
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> Note: the Zotero database needs configured to export automatically to
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> `lib/citations.bib`
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To insert a Zotero Citation - Ensure the Zotero App and DB are running
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on you system. - Alt + I (to insert citation) - Search for and select
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citation reference
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## Usage: Building the Book
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#### PDF
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make pdf
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#### HTML
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make html
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#### LaTex
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make latex
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#### Markdown
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make markdown
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#### EPub
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> Note: This ePub configuration still needs tuning.
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make epub
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# Example Content
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In *R**ν* the [Planck
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Length](https://en.wikipedia.org/wiki/Planck_units#Planck_length) is the
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universal unit for measurement of distance, and is defined approximately
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to be:
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$$\\boxed{L_P=\\sqrt{\\hbar}=5.72928\\times10^{-35}m=1 L}$$
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Where 1 *L*, is 1 Planck Length of distance.
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### SI Conversion Factors
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The following conversion factors can be used to convert observable
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quantities of measure from the *SI* system of units to *R**ν* to \~6
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significant digits.
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| Conversion Factor | Symbol | Value |
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|------------------------------|-------------------|--------------------------------------------------------|
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| meters to Planck Length | *χ*<sub>*P*</sub> | $1.74542\\times10^{34} \\frac{L}{m}$ |
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| seconds to Planck Length | *τ*<sub>*p*</sub> | $5.23264\\times10^{42} \\frac{L}{s}$ |
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| mass to Planck Length | *G*<sub>*P*</sub> | $1.62871\\times10^8 \\frac{L}{kg}$ |
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| energy to Planck Length | *E*<sub>*P*</sub> | $1.81219\\times10^9 \\frac{L}{J}$ |
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| momentum to Planck Length | *P*<sub>*P*</sub> | $5.43280\\times10^{-1} \\frac{L\\cdot s}{kg \\cdot m}$ |
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| temperature to Planck Length | *k*<sub>*P*</sub> | $2.501998\\times10^{-14} \\frac{L}{K}$ |
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| charge to Planck Length | *C*<sub>*P*</sub> | $1.89007\\times10^{18} \\frac{L}{C}$ |
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### Physical Constants
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Applying conversion factors from the table above, we can convert SI
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values to Reduced Natural Units. For example, performing this analysis
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on the the speed of light yields a unit-less number with a value of 1:
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$c = 299792458 \\frac{m}{s} = 299792458 \\frac{m}{s} \\cdot 1.74542\\times10^{34} \\frac{L}{m} \\cdot \\frac{1}{5.23264\\times10^{42} \\frac{L}{s}} = 1.00000$
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| Quantity | Symbol | SI | *ν* |
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|--------------------------------|-------------------|-----------------------------------------------------------------|-----------------------------------------|
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| Speed of Light | *c* | $299792458 \\frac{m}{s}$ | 1 |
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| Reduced Gravitational Constant | *G*<sub>0</sub> | $8.38659\\times10^{-10} \\frac{m^3}{kg \\cdot s^2}$ | 1 |
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| Boltzmann’s Constant | *k* | $k=1.380649\\times10^-23 \\frac{J}{K}$ | 1 |
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| Permittivity of Free Space | *ϵ*<sub>*o*</sub> | $8.854187817620\\times10^{-12} \\frac{C^{2}s^2}{kg \\cdot m^3}$ | 1 |
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| Permeability of Free Space | *μ*<sub>*o*</sub> | $\\huge{\\frac{1}{\\epsilon\_{o} \\cdot c^{2}}}$ | 1 |
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| Reduced Planck’s Constant | ℏ | $1.054571726\\times10^-34 \\frac{kg \\cdot m^2}{s}$ | 1*L*<sup>2</sup> |
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| Mass of the Electron | *m*<sub>*e*</sub> | 9.10938 × 10<sup>−31</sup>*k**g* | 1.48366 × 10<sup>−22</sup>*L* |
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| Charge of the Electron | *e*<sup>−</sup> | − 1.60218 × 10<sup>−19</sup>*C* | − 3.02822 × 10<sup>−1</sup>*L* |
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| Unit Cycle | *Θ* | 2*π* = 6.28318... *R**a**d**i**a**n**s* | 1*τ* = 6.28318... *R**a**d**i**a**n**s* |
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## Fine Structure Constant
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As a consistency check, we compute the *[Fine Structure
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Constant](https://en.wikipedia.org/wiki/Fine-structure_constant)* using
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Reduced Natural Units which is a unit less ratio that should be
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independent of our system of units.
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$\\huge{\\alpha=\\frac{e^2}{4\\pi\\epsilon_o\\hbar c}=\\frac{e^2}{2\\tau}=0.00729735≈\\frac{1}{137}}$
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#### Dimensional Analysis
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The reader should be familiar with high school physics and chemistry
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[dimensional
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analysis](https://en.wikipedia.org/wiki/Dimensional_analysis).
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- 1 *m**e**t**e**r* (*m*) = 100 *c**e**n**t**i**m**e**t**e**r**s* (*c**m*)
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- 1 *k**i**l**o**m**e**t**e**r* (*k**m*) = 1000 *m**e**t**e**r**s* (*m*)
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- 1 *m**i**l**e* = 5280 *f**e**e**t* (*f**t* *o**r* ′)
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- $1\\ foot\\ (ft\\ or\\ ') = 12\\ inches\\ (in\\ or\\ ")$
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- $1\\ inch\\ (") = 2.54\\ centimeters\\ (cm)$
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How many kilometers are in 1 mile?
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$1\\ mile = 1\\ mile \\times \\frac{5280\\ ft}{mile} \\times \\frac{12\\ in}{ft} \\times \\frac{2.54\\ cm}{in}\\times \\frac{1\\ m}{100 cm} \\times \\frac{1\\ km}{1000 m} = \\frac{5280 \\times 12 \\times 2.54}{100 \\times 1000}\\ km = \\frac{160934.40}{100000}\\ km = 1.6\\ km$
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Note that each unit in the denominator cancels with one if the numerator
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until we are left with only km.
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## Newton’s Law of Gravity
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The force of gravity (*F*<sub>*g*</sub>) between 2 masses, *m*1 and *m*2
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separated by distance *r* is given by [Newton’s Law of
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Gravity](https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation):
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$F\_{g} = G \\frac{m\_{1} m\_{2}}{r^{2}}$
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Where *G*, is the [Gravitational
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Constant](http://en.wikipedia.org/wiki/Gravitational_Constant).
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$G = 6.67430 \\times 10^{-11}\\ N\\frac{m^2}{kg^2}$
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The strength of gravitational force follow the inverse square law
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distributing gravitational flux over the surface area of a sphere
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(4*π**r*<sup>2</sup>).
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#### Inverse Square Law
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Any source of a signal strength (*S*<sub>0</sub>) that radiates
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isotropically in 3-dimensional space will distribute that signal
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strength (*S*<sub>0</sub>) over the surface area of a sphere
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(*S**A* = 4*π**r*) of radius (*r*). Such that the intensity (*I*) at
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distance (*r*) is:
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$$I(r) = \\frac{S_0}{4 \\pi r^{2}}=\\frac{S_0}{2 \\tau r^{2}}$$
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 \#### *R**ν*
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Reduced Gravitational Constant In this version of Newton’s Law of
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Gravity we introduce a new constant *G*<sub>0</sub>, the reduced
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gravitational constant to accommodate for the factor of 4*π* = 2*τ*
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which is has been integrated in the SI version of the gravitational
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constant.
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$F_g =G \\frac{m\_{1} m\_{2}}{r^{2}}= G_0 \\frac{m\_{1} m\_{2}}{4 \\pi r^{2}}=G_0 \\frac{m\_{1} m\_{2}}{2 \\tau r^{2}}$
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Where:
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|
||
$G = \\frac{G\_{0}}{2\\tau} = 6.67384 \\times 10^{-11} \\frac{N \\cdot m^2}{kg^2}$
|
||
|
||
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:
|
||
|
||
$$\\boxed{G_0=\\frac{2\\tau G}{c^2}=\\frac{2\\tau \\cdot 6.67384 \\times 10^{-11}}{299792458^2} \\frac{m}{kg} = 9.33135 \\times 10^-27 \\frac{m}{kg}}$$
|
||
|
||
> Observation This implies that not only can space an time be measure in
|
||
> units of meters, but so can mass.
|
||
|
||
### Relativistic Energy Momentum Relation
|
||
|
||
Einsteins [Relativistic Energy
|
||
Momentum](https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation)
|
||
relationship shows a Pythagorean relation between the total energy
|
||
(*E*), rest mass (*m*<sub>0</sub>) and momentum (*p*) of a system.
|
||
|
||
*E*<sup>2</sup> = (*m*<sub>0</sub>⋅*c*<sup>2</sup>)<sup>2</sup> + (*p*⋅*c*)<sup>2</sup>
|
||
|
||
Where space and time are both measure in units of meters, c=1.
|
||
|
||
*E*<sup>2</sup> = (*m*<sub>0</sub>)<sup>2</sup> + (*p*)<sup>2</sup>
|
||
|
||
From this we can see that Energy, Momentum and Mass have equivalent
|
||
units.
|
||
|
||
> *While we do not really know what energy, mass and momentum are we
|
||
> know that they are fundamentally “made” out of the same stuff because
|
||
> they have the same units.*
|
||
|
||
##### Objects of mass at rest
|
||
|
||
For an object at rest with no momentum (*p* = 0) we see Einstein’s
|
||
famous equations:
|
||
|
||
*E* = *m*<sub>0</sub> ⋅ *c*<sup>2</sup>
|
||
|
||
Or, with *c* = 1, this is much simpler to understand. Energy = Mass
|
||
|
||
*E* = *m*<sub>0</sub>
|
||
|
||
##### Zero mass objects moving at the speed of light
|
||
|
||
And for objects with no mass, like photos, (*m*<sub>0</sub> = 0):
|
||
|
||
*E* = *p**c*
|
||
|
||
Or, with *c* = 1, this is much simpler to understand. Energy = Momentum
|
||
|
||
*E* = *p*
|
||
|
||
## Planck’s Constant
|
||
|
||
The [Reduced Planck
|
||
constant](https://en.wikipedia.org/wiki/Planck_constant) , ħ, represents
|
||
a conversion factor for relating the frequency, *ω* (in 2*π* radians per
|
||
second), of a photon to the energy of that photon. This can easily be
|
||
seen from the simple but profound relationship:
|
||
|
||
*E* = ℏ*ω*
|
||
|
||
Where:
|
||
|
||
ℏ = 1.054571726 × 10<sup>−34</sup>*J* ⋅ *s*
|
||
|
||
and
|
||
|
||
$J \\cdot s = {kg}\\cdot\\frac{m^2}{s}$
|
||
|
||
> Reduced Planck’s Constant
|
||
> $\\hbar = \\frac{h}{2\\pi} = \\frac{h}{\\tau}$
|
||
|
||
Simplifying our units by converting time and mass to units of meters:
|
||
$$\\boxed{\\hbar=1.054571726 \\times 10^{−34} {kg}\\cdot\\frac{m^2}{s}\\cdot\\frac{G_0}{c}=3.282462\\times10^{-69}m^2}$$
|
||
|
||
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:
|
||
|
||
$$\\boxed{\\sqrt{\\hbar}=\\sqrt{3.282462\\times10^{-69}m^2}=5.72928\\times10^{-35}m}$$
|
||
|
||
#### Planck Area
|
||
|
||
The [Planck
|
||
Area](https://en.wikipedia.org/wiki/Planck_units#Derived_units) is the
|
||
square of the [Planck
|
||
Length](https://en.wikipedia.org/wiki/Planck_units#Planck_length).
|
||
|
||
$l\_{P}= \\sqrt{\\frac{\\hbar G}{c^3}}$
|
||
|
||
and $l\_{P}^{2}= \\frac{\\hbar G}{c^3}$
|
||
|
||
In *R**ν* units both *c* and *G*<sub>*o*</sub> are 1.
|
||
|
||
$l\_{P} = \\sqrt{\\hbar}$
|
||
|
||
and *l*<sub>*P*</sub><sup>2</sup> = ℏ \## Bekenstein’s Bound After
|
||
having recently read *Three Roads to Quantum Gravity* by Lee Smolin, I
|
||
now suspect the meaning of this areas is related to the [Bekensteins
|
||
Law](https://en.wikipedia.org/wiki/Bekenstein_bound) as applied to a
|
||
surface areas surrounding a mass. Where the [thermodynamic
|
||
entropy](https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)),
|
||
*S*, is proportional to the the enclosed surface area, *A*.
|
||
|
||
$S=\\frac{1}{4}\\cdot\\frac{A}{G\\hbar}$
|
||
|
||
$S=\\frac{k c^{3} A}{4 G \\hbar}$
|
||
|
||
$S \\le \\frac{2\\pi k R E}{\\hbar c} = \\frac{\\tau R k E}{\\hbar c}$
|
||
|
||
From our new values for *G*<sub>0</sub>and ℏ we can likely rewrite this:
|
||
|
||
$S=\\frac{\\pi\\cdot A}{\\hbar G_0}$
|
||
|
||
With the limiting case being at the Plank scale.
|
||
|
||
$S=\\frac{\\pi\\cdot \\sqrt{\\hbar}}{\\hbar G_0}$
|
||
|
||
## Planck Length
|
||
|
||
https://en.wikipedia.org/wiki/Planck_length
|
||
|
||
The concept of the Planck Length comes from exploring the limits of
|
||
Quantum Mechanics and General Relativity. The limits of General
|
||
Relativity can be seen a the event horizon of a black hole, described by
|
||
the Schwarzschild Radius. And the limits of Quantum Mechanics can be
|
||
found in the Compton Wavelength for a given quanta.
|
||
|
||
The [Schwarzschild
|
||
Radius](https://simple.wikipedia.org/wiki/Schwarzschild_radius) is
|
||
defined as the distance at which light cannot escape from the
|
||
gravitational field of a mass (m):
|
||
|
||
Classic Derivation.
|
||
|
||
$r_S=\\frac{2G m}{c^2}$
|
||
|
||
The reduced [Compton
|
||
Wavelength](https://en.wikipedia.org/wiki/Compton_wavelength) represents
|
||
a lower limit on the wavelength for quanta that can interact with a
|
||
quantum particle with mass (m):
|
||
|
||
$\\lambda_C=\\frac{h}{m c}$
|
||
|
||
$\\bar{\\lambda_C}=\\frac{2\\pi\\hbar}{m c}=\\frac{\\tau\\hbar}{m c}$
|
||
|
||
And set the Schwarzschild Radius equal to the Compton Wavelength:
|
||
*r*<sub>*S*</sub> = *λ*<sub>*C*</sub>
|
||
|
||
$\\frac{2Gm}{c^{2}}=\\frac{h}{m c}$
|
||
|
||
$m^{2}= \\frac{hc}{2G}$
|
||
|
||
$m = \\sqrt{\\frac{hc}{2G}}$
|
||
|
||
$l_P=\\frac{2G\\sqrt{\\frac{hc}{2G}}}{c^2}$
|
||
$l_P=\\frac{2G\\sqrt{\\frac{hc}{2G}}}{c^{2}}= \\sqrt{\\frac{2Gh}{c^2}}$
|
||
|
||
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*<sub>*P*</sub>.
|
||
$M_P=m=\\sqrt{\\hbar}$
|
||
|
||
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*<sub>*P*</sub>, is equal to Plank Length,
|
||
*L*<sub>*P*</sub>, which is equal to the Plank Mass, *M*<sub>*P*</sub>:
|
||
|
||
$$\\boxed{L_P=T_P=M_P}$$
|
||
|
||
| Conversion Factor | Symbol | Value |
|
||
|------------------------------|-------------------|--------------------------------------------------------|
|
||
| meters to Planck Length | *χ*<sub>*P*</sub> | $1.74542\\times10^{34} \\frac{L}{m}$ |
|
||
| seconds to Planck Length | *τ*<sub>*p*</sub> | $5.23264\\times10^{42} \\frac{L}{s}$ |
|
||
| mass to Planck Length | *G*<sub>*P*</sub> | $1.62871\\times10^8 \\frac{L}{kg}$ |
|
||
| energy to Planck Length | *E*<sub>*P*</sub> | $1.81219\\times10^9 \\frac{L}{J}$ |
|
||
| momentum to Planck Length | *P*<sub>*P*</sub> | $5.43280\\times10^{-1} \\frac{L\\cdot s}{kg \\cdot m}$ |
|
||
| temperature to Planck Length | *k*<sub>*P*</sub> | $2.501998\\times10^{-14} \\frac{L}{K}$ |
|
||
| charge to Planck Length | *C*<sub>*P*</sub> | $1.89007\\times10^{18} \\frac{L}{C}$ |
|
||
|
||
Applying conversion factors from the table above, we can convert SI
|
||
values to Reduced Natural Units.
|
||
$c=\\frac{1}{\\sqrt{\\epsilon_o \\mu_o}}$
|
||
|
||
| Quantity | Symbol | SI | *ν* |
|
||
|----------------------------|-------------------|-----------------------------------------------------------------|---------------------------------|
|
||
| Speed of Light | *c* | $299792458 \\frac{m}{s}$ | 1 |
|
||
| Gravitational Constant | *G*<sub>0</sub> | $8.38659\\times10^{-10} \\frac{m^3}{kg \\cdot s^2}$ | 1 |
|
||
| Boltzmann’s Constant | *k* | $k=1.380649\\times10^-23 \\frac{J}{K}$ | 1 |
|
||
| Permittivity of Free Space | *ϵ*<sub>*o*</sub> | $8.854187817620\\times10^{-12} \\frac{C^{2}s^2}{kg \\cdot m^3}$ | 1 |
|
||
| Permeability of Free Space | *μ*<sub>*o*</sub> | $\\huge{\\frac{1}{\\epsilon\_{o} \\cdot c^{2}}}$ | 1 |
|
||
| Planck’s Constant | ℏ | $1.054571726\\times10^-34 \\frac{kg \\cdot m^2}{s}$ | 1*L*<sup>2</sup> |
|
||
| Mass of the Electron | *m*<sub>*e*</sub> | 9.10938 × 10<sup>−31</sup>*k**g* | 1.48366 × 10<sup>−22</sup>*L* |
|
||
| Charge of the Electron | *e*<sup>−</sup> | − 1.60218 × 10<sup>−19</sup>*C* | − 3.02822 × 10<sup>−1</sup>*L* |
|
||
|
||
## Fine Structure Constant
|
||
|
||
https://en.wikipedia.org/wiki/Fine-structure_constant As a consistency
|
||
check, we compute the *Fine Structure Constant* using Reduced Natural
|
||
Units which is a unit less ratio that should be independent of our
|
||
system of units.
|
||
|
||
$\\huge{\\alpha=\\frac{e^2}{4\\pi\\epsilon_o\\hbar c}=\\frac{e^2}{4\\pi}=0.00729735≈\\frac{1}{137}}$
|
||
|
||
This check confirms that our system of Reduced Natural Units has
|
||
internally consistent values for *c*, *ϵ*<sub>*o*</sub>, ℏ and *e*−. And
|
||
also *G*<sub>*o*</sub> which was used to computer prior values is also
|
||
consistent.
|
||
|
||
## Sage Code
|
||
|
||
Unit Analysis computations have been performed with [Sage
|
||
Math](https://www.sagemath.org/).
|
||
|
||
``` bash
|
||
# Define constance
|
||
one = 1.n(digits=6)
|
||
pi = pi.n(digits=6)
|
||
tau = 2 * pi
|
||
t = tau
|
||
|
||
# Define the units
|
||
meters = var('m')
|
||
m = one*meters
|
||
|
||
seconds = var('s')
|
||
s = seconds
|
||
|
||
kilograms = var('kg')
|
||
kg = kilograms
|
||
|
||
newtons = kg * m / s^2
|
||
N = newtons
|
||
|
||
joules = N * m
|
||
J = joules
|
||
|
||
print("pi =", pi)
|
||
print("tau =", t)
|
||
|
||
# Speed of light in meters/second
|
||
speed_of_light = 299792458 * meters/seconds
|
||
sol = speed_of_light
|
||
c = sol
|
||
print("si c =", c)
|
||
|
||
rnu_c = c / c
|
||
print("R\u03BD c =", rnu_c)
|
||
|
||
# Gravitational Constant
|
||
gravitational_constant = 6.67384e-11 * N*(m^2/kg^2)
|
||
G = gravitational_constant
|
||
print("si G =", G)
|
||
|
||
rnu_G = 4*pi*G/c^2
|
||
Go = rnu_G
|
||
print("R\u03BD Go =", Go)
|
||
|
||
|
||
# Planck's Constant
|
||
reduced_plancks_constant = 1.054571726e-34 * J*s
|
||
h_bar = reduced_plancks_constant
|
||
print("si \u210F =", h_bar)
|
||
|
||
rnu_h_bar = h_bar * Go / c
|
||
print("R\u03BD "u"\u210F =", rnu_h_bar)
|
||
|
||
# Planck Length
|
||
rnu_h_bar_str = str(rnu_h_bar)
|
||
numerical_part_str = rnu_h_bar_str.split('*')[0]
|
||
numerical_part_str = numerical_part_str.strip('()')
|
||
numerical_part = float(numerical_part_str)
|
||
rnu_sqrt_h_bar = numerical_part^(1/2)
|
||
# ^ Sage cannot process sqrt on units... Lame.
|
||
lP = rnu_sqrt_h_bar * m
|
||
print("R\u03BD \u221A\u210F =", lP)
|
||
```
|
||
|
||
#### Output
|
||
|
||
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)
|
||
|
||
# Terminology
|
||
|
||
# Citations
|
||
|
||
<div id="refs" class="references csl-bib-body hanging-indent">
|
||
|
||
<div id="ref-JohnHaverlackACEP" class="csl-entry">
|
||
|
||
“John Haverlack ACEP.” n.d.
|
||
https://www.uaf.edu/acep/about/our-team/john-haverlack.php. Accessed
|
||
September 30, 2025.
|
||
|
||
</div>
|
||
|
||
</div>
|