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Jean-Sébastien
2022-03-15 10:07:27 +01:00
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@@ -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>
@@ -1614,24 +1614,24 @@ Table of contents
</svg></a><span class="headline-id">red.sr.4v</span></h4>
<div class="outline-text-4" id="text-red_sr_4v">
<p>
\paragraph{Four-vectors.} Let's introduce the standard notations
<b>Four-vectors</b>: let's introduce the standard notations
\[
x^0 \equiv ct, \hspace{10mm} \beta \equiv \frac{v}{c},
\hspace{10mm} x^1 = x, ~~x^2 = y, ~~x^3 = z.
\]
The Lorentz transformation then reads
</p>
<div class="core div" id="org5cbcae9">
<div class="core div" id="orgf4d6e6e">
<p>
{\bf Lorentz transformation (motion along \(x\) at velocity \(v\))}
\[
<b>Lorentz transformation</b> (motion along \(x\) at velocity \(v\))
\[
\bar{x}^0 = \gamma \left( x^0 - \beta x^1 \right),
~~~~\bar{x}^1 = \gamma \left( x^1 - \beta x^0 \right),
~~~~\bar{x}^2 = x^2,
~~~~\bar{x}^3 = x^3
\]
or in matrix form
\[
or in matrix form
\[
\left( \begin{array}{c} \bar{x}^0 \\ \bar{x}^1 \\ \bar{x}^2 \\ \bar{x}^3
\end{array} \right)
= \left( \begin{array}{cccc}
@@ -1647,17 +1647,17 @@ or in matrix form
<p>
This can be compactly written as
\[
\bar{x}^\mu = \sum_{\nu = 0}^3 \Lambda^\mu_\nu x^\nu.
\bar{x}^\mu = \sum_{\nu = 0}^3 \Lambda^\mu{}_\nu x^\nu.
\]
</p>
<p>
\paragraph{Covariant and contravariant vectors.} Four-vectors with
upper index are called {\it contravariant}. Their lower-index
counterparts are called {\it covariant} vectors and are obtained by
<b>Covariant and contravariant vectors</b>: four-vectors with
upper index are called <b>contravariant</b>. Their lower-index
counterparts are called <b>covariant</b> vectors and are obtained by
using the Minkowski metric \(g_{\mu \nu}\) according to
</p>
<div class="main div" id="org6643ea5">
<div class="main div" id="orge005c2f">
<p>
\[
a_\mu = \sum_{\nu = 0}^3 g_{\mu \nu} a^\nu, \hspace{10mm}
@@ -1671,9 +1671,9 @@ using the Minkowski metric \(g_{\mu \nu}\) according to
</div>
<p>
\paragraph{Scalar products} are defined as the in-product of covariant/contravariant four-vectors,
<b>Scalar products</b> are defined as the in-product of covariant/contravariant four-vectors,
</p>
<div class="main div" id="orge05e08a">
<div class="main div" id="org411b956">
<p>
\[
\sum_{\mu = 0}^3 a^\mu b_\mu \equiv a^\mu b_\mu
@@ -1683,7 +1683,7 @@ using the Minkowski metric \(g_{\mu \nu}\) according to
</div>
<p>
where in the right-hand side we have introduced the
{\bf Einstein summation convention}, namely that any repeated index
<b>Einstein summation convention</b>, namely that any repeated index
is implicitly summed over. As you can trivially check, it doesn't matter
which vector is co/contravariant: \(a^\mu b_\mu = a_\mu b^\mu\).
Scalar products are Lorentz-invariant and thus take the same value in
@@ -1691,25 +1691,45 @@ all inertial systems.
</p>
<p>
\paragraph{Invariant intervals.} Generalizing the notion of the norm of
<b>Invariant intervals</b>: generalizing the notion of the norm of
a vector, the scalar product of a four-vector with itself is known as
the invariant interval. Because of the geometry of spacetime, the invariant
can take positive or negative values. The nomenclature goes as follows:
</p>
\begin{center}
\begin{tabular}{cc}
$a^\mu a_\mu &gt; 0$ &amp; $a^\mu$ is {\it spacelike} \\
$a^\mu a_\mu &lt; 0$ &amp; $a^\mu$ is {\it timelike} \\
$a^\mu a_\mu = 0$ &amp; $a^\mu$ is {\it lightlike}
\end{tabular}
\end{center}
<table>
<colgroup>
<col class="org-left">
<col class="org-left">
</colgroup>
<tbody>
<tr>
<td class="org-left">\(a^\mu a_\mu &gt; 0\)</td>
<td class="org-left">\(a^\mu\) is {\it spacelike}</td>
</tr>
<tr>
<td class="org-left">\(a^\mu a_\mu &lt; 0\)</td>
<td class="org-left">\(a^\mu\) is {\it timelike}</td>
</tr>
<tr>
<td class="org-left">\(a^\mu a_\mu = 0\)</td>
<td class="org-left">\(a^\mu\) is {\it lightlike}</td>
</tr>
</tbody>
</table>
<p>
For two events \(A\) and \(B\), the difference
\[
\Delta x^\mu \equiv x_A^\mu - x_B^\mu
\]
is called the {\bf displacement four-vector} and its self-scalar product
is the {\bf invariant interval} between the two events:
is called the <b>displacement four-vector</b> and its self-scalar product
is the <b>invariant interval</b> between the two events:
\[
I \equiv \Delta x^\mu \Delta x_\mu = -c^2 \Delta t^2 + |{\boldsymbol x}|^2
\]
@@ -1718,12 +1738,12 @@ is their spatial separation vector.
</p>
<p>
\paragraph{Spacetime diagrams.} These are also know as
{\it Minkowski diagrams}. Time is on the vertical axis, space on the
<b>Spacetime diagrams</b>: these are also know as
<b>Minkowski diagrams</b>. Time is on the vertical axis, space on the
horizontal one. The trajectory of a particle is known as its
{\bf world line}. Light is represented as propagating at lines at
45 degrees, defining the {\bf forward} and
{\bf backward light cones}. Lorentz transformations, which preserve
<b>world line</b>. Light is represented as propagating at lines at
45 degrees, defining the <i>forward</i> and
<i>backward light cones</i>. Lorentz transformations, which preserve
all invariant intervals, move spacetime points around but leave them
on the same hyperboloid.
</p>
@@ -1746,7 +1766,7 @@ 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>