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Update 2022-03-24 08:43

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Jean-Sébastien
2022-03-24 08:43:21 +01:00
parent 1d852e7213
commit 50704eba07
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build/red_rem_Fmunu.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-22 Tue 10:52 -->
<!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary>
<ul>
<li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li>
@@ -1622,12 +1622,16 @@ in which \(\Lambda\) is a matrix representing the Lorentz transformation.
The form this matrix takes depends on the actual transformation: for the specific
case of motion in the \(x\) direction with velocity \(v\),
\[
\Lambda = \left( \begin{array}{cccc}
\Lambda^\mu{}_\nu = \left( \begin{array}{cccc}
\gamma &amp; -\gamma \beta &amp; 0 &amp; 0 \\
-\gamma \beta &amp; \gamma &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 1 \end{array} \right).
0 &amp; 0 &amp; 0 &amp; 1 \end{array} \right)
\]
(in which \(\mu\) is the "row" index and \(\nu\) the "column" one).
</p>
<p>
A four-vector is synonymous to a <b>rank-one tensor</b>.
Higher-rank tensors are simply objects carrying more indices.
For example, a <b>rank-two tensor</b> transforms as
@@ -1683,18 +1687,18 @@ how the nonvanishing elements of an antisymmetric tensor transform:
<p>
Comparing with the transformation rules for the electromagnetic field which we obtained in <a href="./red_rem_Ltf.html#EMtr">EMtr</a>, we can define the
</p>
<div class="core div" id="org117de8f">
<div class="core div" id="orgf4b9c4f">
<p>
<b>Electromagnetic Field Tensor</b>
</p>
<div class="eqlabel" id="org358ea94">
<div class="eqlabel" id="org81eea70">
<p>
<a id="Fmunu"></a><a href="./red_rem_Fmunu.html#Fmunu"><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"/>
</svg></a>
</p>
<div class="alteqlabels" id="org597a4d9">
<div class="alteqlabels" id="org185605b">
</div>
@@ -1714,18 +1718,18 @@ F^{\mu \nu} = \left( \begin{array}{cccc}
<p>
together with the handy
</p>
<div class="main div" id="orgc56f47d">
<div class="main div" id="orgc181a2e">
<p>
<b>Dual Field Tensor</b>
</p>
<div class="eqlabel" id="orgda1a353">
<div class="eqlabel" id="orgc4536aa">
<p>
<a id="Gmunu"></a><a href="./red_rem_Fmunu.html#Gmunu"><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"/>
</svg></a>
</p>
<div class="alteqlabels" id="org31f1240">
<div class="alteqlabels" id="orgbf03668">
</div>
@@ -1753,18 +1757,18 @@ obtained from the field tensor by the substitution
<p>
Our electromagnetic field transformation laws then become the simple
</p>
<div class="core div" id="org0e51dab">
<div class="core div" id="org0260e5e">
<p>
<b>Lorentz Transformation Rules for EM Fields</b>
</p>
<div class="eqlabel" id="org41f872c">
<div class="eqlabel" id="org75fce95">
<p>
<a id="LorF"></a><a href="./red_rem_Fmunu.html#LorF"><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"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgbfcae60">
<div class="alteqlabels" id="org854087d">
</div>
@@ -1798,7 +1802,7 @@ 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: 2022-03-22 Tue 10:52</p>
<p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p>
</div>