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Update 2022-02-09 07:44

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Jean-Sébastien
2022-02-09 07:44:58 +01:00
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commit 3c40f5bfe8
209 changed files with 3663 additions and 1367 deletions
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-08 Tue 17:21 -->
<!-- 2022-02-09 Wed 07:31 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,7 +1616,7 @@ Table of contents
</ul>
</details>
</nav>
<ul class="navigation-links"><li>Prev:&nbsp;<a href="red_sr_Lt.html">Lorentz Transformations&emsp;<small>[red.sr.Lt]</small></a></li><li>Next:&nbsp;<a href="red_rm.html">Relativistic Mechanics&emsp;<small>[red.rm]</small></a></li><li>Up:&nbsp;<a href="red_sr.html">Special Relativity&emsp;<small>[red.sr]</small></a></li></ul><div id="outline-container-red_sr_4v" class="outline-4">
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="red.html">Relativistic Electrodynamics</a></li><li><a class="breadcrumb-link"href="red_sr.html">Special Relativity</a></li><li>Covariant and Contravariant Four-Vectors</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="red_sr_Lt.html">Lorentz Transformations&emsp;<small>[red.sr.Lt]</small></a></li><li>Next:&nbsp;<a href="red_rm.html">Relativistic Mechanics&emsp;<small>[red.rm]</small></a></li><li>Up:&nbsp;<a href="red_sr.html">Special Relativity&emsp;<small>[red.sr]</small></a></li></ul><div id="outline-container-red_sr_4v" class="outline-4">
<h4 id="red_sr_4v">Covariant and Contravariant Four-Vectors<a class="headline-permalink" href="./red_sr_4v.html#red_sr_4v"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
@@ -1630,7 +1630,7 @@ Table of contents
\]
The Lorentz transformation then reads
</p>
<div class="core div" id="org1a15489">
<div class="core div" id="org07b8623">
<p>
{\bf Lorentz transformation (motion along \(x\) at velocity \(v\))}
\[
@@ -1666,7 +1666,7 @@ upper index are called {\it contravariant}. Their lower-index
counterparts are called {\it covariant} vectors and are obtained by
using the Minkowski metric \(g_{\mu \nu}\) according to
</p>
<div class="main div" id="orgebe8ad6">
<div class="main div" id="org21803ce">
<p>
\[
a_\mu = \sum_{\nu = 0}^3 g_{\mu \nu} a^\nu, \hspace{10mm}
@@ -1682,7 +1682,7 @@ using the Minkowski metric \(g_{\mu \nu}\) according to
<p>
\paragraph{Scalar products} are defined as the in-product of covariant/contravariant four-vectors,
</p>
<div class="main div" id="org6c9220f">
<div class="main div" id="org3f848ec">
<p>
\[
\sum_{\mu = 0}^3 a^\mu b_\mu \equiv a^\mu b_\mu
@@ -1740,10 +1740,21 @@ on the same hyperboloid.
</div>
<hr><div id="postamble" class="status">
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
target="_blank" class="m-2">
<img alt="Creative Commons License" style="border-width:0"
src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
</a>
Except where otherwise noted, all content is licensed under a
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-08 Tue 17:21</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
<p class="date">Created: 2022-02-09 Wed 07:31</p>
<p class="validation"></p>
</div>