jscaux/pqed_html
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update 2022-03-15 10:07

Browse Source
This commit is contained in:
Jean-Sébastien
2022-03-15 10:07:27 +01:00
parent 4808df71e6
commit 55f0de8197
193 changed files with 2416 additions and 2082 deletions
Download Patch File Download Diff File Expand all files Collapse all files
build/red_rem_Fmunu.html
+90 -45
View File
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-07 Mon 20:38 -->
<!-- 2022-03-15 Tue 08:10 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1310,10 +1310,6 @@ Table of contents
</summary>
<ul>
<li>
<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
</li>
<li>
<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
</li>
@@ -1352,6 +1348,10 @@ Table of contents
<li>
<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
</li>
<li>
<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
</li>
</ul>
@@ -1616,7 +1616,7 @@ Table of contents
<p>
We have seen that a four-vector transforms according to
\[
\bar{a}^\mu = \Lambda_\nu^\mu a^\nu
\bar{a}^\mu = \Lambda^\mu{}_\nu a^\nu
\]
in which \(\Lambda\) is a matrix representing the Lorentz transformation.
The form this matrix takes depends on the actual transformation: for the specific
@@ -1628,11 +1628,11 @@ case of motion in the \(x\) direction with velocity \(v\),
0 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 1 \end{array} \right).
\]
A four-vector is synonymous to a {\bf rank-one tensor}.
A four-vector is synonymous to a <b>rank-one tensor</b>.
Higher-rank tensors are simply objects carrying more indices.
For example, a {\bf rank-two tensor} transforms as
For example, a <b>rank-two tensor</b> transforms as
\[
\bar{t}^{\mu \nu} = \Lambda^\mu_\lambda \Lambda^\nu_\sigma t^{\lambda \sigma}.
\bar{t}^{\mu \nu} = \Lambda^\mu{}_\lambda \Lambda^\nu{}_\sigma t^{\lambda \sigma}.
\]
Such a rank-two tensor can be represented similarly to a matrix:
\[
@@ -1656,59 +1656,89 @@ Under the Lorentz transformation along \(x\) defined above, we can work out
how the nonvanishing elements of an antisymmetric tensor transform:
</p>
\begin{align}
\bar{t}_a^{01} &amp;= \Lambda^0_\mu \Lambda^1_\nu t_a^{\mu \nu}
= \Lambda^0_0 \Lambda^1_0 t_a^{00} + \Lambda^0_1 \Lambda^1_0 t_a^{10}
+ \Lambda^0_0 \Lambda^1_1 t_a^{01} + \Lambda^0_1 \Lambda^1_1 t_a^{11} \\
&amp; = (\Lambda^0_0 \Lambda^1_1 - \Lambda^0_1 \Lambda^1_0) t_a^{01}
\bar{t}_a^{01} &amp;= \Lambda^0{}_\mu \Lambda^1{}_\nu t_a^{\mu \nu}
= \Lambda^0{}_0 \Lambda^1{}_0 t_a^{00} + \Lambda^0{}_1 \Lambda^1{}_0 t_a^{10}
+ \Lambda^0{}_0 \Lambda^1{}_1 t_a^{01} + \Lambda^0{}_1 \Lambda^1{}_1 t_a^{11} \\
&amp; = (\Lambda^0{}_0 \Lambda^1{}_1 - \Lambda^0{}_1 \Lambda^1{}_0) t_a^{01}
= \gamma^2 (1 - \beta^2) t_a^{01} = t_a^{01}, \\
\bar{t}_a^{02} &amp;= \Lambda^0_\mu \Lambda^2_\nu t_a^{\mu \nu}
= \Lambda^0_0 \Lambda^2_2 t_a^{02} + \Lambda^0_1 \Lambda^2_2 t_a^{12}
\bar{t}_a^{02} &amp;= \Lambda^0{}_\mu \Lambda^2{}_\nu t_a^{\mu \nu}
= \Lambda^0{}_0 \Lambda^2{}_2 t_a^{02} + \Lambda^0{}_1 \Lambda^2{}_2 t_a^{12}
= \gamma (t_a^{02} - \beta t_a^{12}), \\
\bar{t}_a^{03} &amp;= \Lambda^0_\mu \Lambda^3_\nu t_a^{\mu \nu}
= \Lambda^0_0 \Lambda^3_3 t_a^{03} + \Lambda^0_1 \Lambda^3_3 t_a^{13}
\bar{t}_a^{03} &amp;= \Lambda^0{}_\mu \Lambda^3{}_\nu t_a^{\mu \nu}
= \Lambda^0{}_0 \Lambda^3{}_3 t_a^{03} + \Lambda^0{}_1 \Lambda^3{}_3 t_a^{13}
= \gamma (t_a^{03} - \beta t_a^{13}), \\
\bar{t}_a^{12} &amp;= \Lambda^1_\mu \Lambda^2_\nu t_a^{\mu \nu}
= \Lambda^1_0 \Lambda^2_2 t_a^{02} + \Lambda^1_1 \Lambda^2_2 t_a^{12}
\bar{t}_a^{12} &amp;= \Lambda^1{}_\mu \Lambda^2{}_\nu t_a^{\mu \nu}
= \Lambda^1{}_0 \Lambda^2{}_2 t_a^{02} + \Lambda^1{}_1 \Lambda^2{}_2 t_a^{12}
= \gamma (t_a^{12} - \beta t_a^{02}), \\
\bar{t}_a^{13} &amp;= \Lambda^1_\mu \Lambda^3_\nu t_a^{\mu \nu}
= \Lambda^1_0 \Lambda^3_3 t_a^{03} + \Lambda^1_1 \Lambda^3_3 t_a^{13}
\bar{t}_a^{13} &amp;= \Lambda^1{}_\mu \Lambda^3{}_\nu t_a^{\mu \nu}
= \Lambda^1{}_0 \Lambda^3{}_3 t_a^{03} + \Lambda^1{}_1 \Lambda^3{}_3 t_a^{13}
= \gamma (t_a^{13} - \beta t_a^{03}), \\
\bar{t}_a^{23} &amp;= \Lambda^2_\mu \Lambda^3_\nu t_a^{\mu \nu}
= \Lambda^2_2 \Lambda^3_3 t_a^{23}
\bar{t}_a^{23} &amp;= \Lambda^2{}_\mu \Lambda^3{}_\nu t_a^{\mu \nu}
= \Lambda^2{}_2 \Lambda^3{}_3 t_a^{23}
= t_a^{23}.
\end{align}
<p>
Comparing with the transformation rules for the electromagnetic field which we obtained in (\ref{eq:EMFieldsLorentzTransfo}), we can define the
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="orge72c66d">
<div class="core div" id="org5361587">
<p>
<b>Electromagnetic Field Tensor</b>
</p>
<div class="eqlabel" id="orgdddfe8b">
<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="org9e15d6e">
</div>
</div>
<p>
{\bf Electromagnetic Field Tensor}
\[
F^{\mu \nu} = \left( \begin{array}{cccc}
0 &amp; E_x/c &amp; E_y/c &amp; E_z/c \\
-E_x/c &amp; 0 &amp; B_z &amp; -B_y \\
-E_y/c &amp; -B_z &amp; 0 &amp; B_x \\
-E_z/c &amp; B_y &amp; -B_x &amp; 0 \end{array} \right)
\]
F^{\mu \nu} = \left( \begin{array}{cccc}
0 &amp; E_x/c &amp; E_y/c &amp; E_z/c \\
-E_x/c &amp; 0 &amp; B_z &amp; -B_y \\
-E_y/c &amp; -B_z &amp; 0 &amp; B_x \\
-E_z/c &amp; B_y &amp; -B_x &amp; 0 \end{array} \right)
\tag{Fmunu}\label{Fmunu}
\]
</p>
</div>
<p>
together with the handy
</p>
<div class="main div" id="org5c6516a">
<div class="main div" id="org53973f7">
<p>
<b>Dual Field Tensor</b>
</p>
<div class="eqlabel" id="orgae9a2b2">
<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="org9c07047">
</div>
</div>
<p>
{\bf Dual Field Tensor}
\[
G^{\mu \nu} = \left( \begin{array}{cccc}
0 &amp; B_x &amp; B_y &amp; B_z \\
-B_x &amp; 0 &amp; -E_z/c &amp; E_y/c \\
-B_y &amp; E_z/c &amp; 0 &amp; -E_x/c \\
-B_z &amp; -E_y/c &amp; E_x/c &amp; 0 \end{array} \right)
\]
G^{\mu \nu} = \left( \begin{array}{cccc}
0 &amp; B_x &amp; B_y &amp; B_z \\
-B_x &amp; 0 &amp; -E_z/c &amp; E_y/c \\
-B_y &amp; E_z/c &amp; 0 &amp; -E_x/c \\
-B_z &amp; -E_y/c &amp; E_x/c &amp; 0 \end{array} \right)
\tag{Gmunu}\label{Gmunu}
\]
</p>
</div>
@@ -1723,12 +1753,27 @@ obtained from the field tensor by the substitution
<p>
Our electromagnetic field transformation laws then become the simple
</p>
<div class="core div" id="org7f5ef81">
<div class="core div" id="org840afc2">
<p>
<b>Lorentz Transformation Rules for EM Fields</b>
</p>
<div class="eqlabel" id="orga8672c3">
<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="org49ca23d">
</div>
</div>
<p>
{\bf Lorentz Transformation Rules for EM Fields}
\[
\bar{F}^{\mu \nu} = \Lambda^\mu_\lambda \Lambda^\nu_\sigma F^{\lambda \sigma}
\]
\bar{F}^{\mu \nu} = \Lambda^\mu{}_\lambda \Lambda^\nu{}_\sigma F^{\lambda \sigma}
\tag{LorF}\label{LorF}
\]
</p>
</div>
@@ -1753,7 +1798,7 @@ 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: 2022-03-07 Mon 20:38</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p>
<p class="validation"></p>
</div>