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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>
@@ -1631,66 +1631,80 @@ This has a number of important consequences.
</p>
<p>
\paragraph{Relativity of simultaneity:} two events which are simultaneous in one reference frame, are not necessarily simultaneous in another one.
<b>Relativity of simultaneity</b>: two events which are simultaneous in one reference frame, are not necessarily simultaneous in another one.
</p>
<p>
\paragraph{Time dilation:} example of a light ray in a travelling train car.
For the observer inside the car: \(\Delta t_{\mbox{\tiny car}} = h/c\).
<b>Time dilation</b>: example of a light ray in a travelling train car.
For the observer inside the car: \(\Delta t_{\mbox{car}} = h/c\).
For an observer on the ground, if the train is moving at velocity \(v\),
then \(\Delta t_{\mbox{\tiny gr}} = \sqrt{h^2 + v^2 \Delta t_{\mbox{\tiny gr}}^2}/c\) so
then \(\Delta t_{\mbox{gr}} = \sqrt{h^2 + v^2 \Delta t_{\mbox{gr}}^2}/c\) so
\[
\Delta t_{\mbox{\tiny gr}} = \frac{h}{c} \frac{1}{\sqrt{1 - v^2/c^2}}
\Delta t_{\mbox{gr}} = \frac{h}{c} \frac{1}{\sqrt{1 - v^2/c^2}}
\]
and we get
\[
\Delta t_{\mbox{\tiny tr}} = \sqrt{1 - v^2/c^2}~ \Delta t_{\mbox{\tiny gr}}
\Delta t_{\mbox{tr}} = \sqrt{1 - v^2/c^2}~ \Delta t_{\mbox{gr}}
\]
so the time interval in the train is shorter, namely there is a
</p>
<div class="core div" id="orgc12be33">
<div class="core div" id="org2b8c90f">
<p>
<b>Time dilation factor</b>
</p>
<div class="eqlabel" id="orgf672abf">
<p>
<a id="gamma"></a><a href="./red_sr_p.html#gamma"><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="org0f03bec">
</div>
</div>
<p>
{\bf Time dilation factor}
\[
\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}
\label{eq:Gamma}
\]
\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}
\tag{gamma}\label{gamma}
\]
</p>
</div>
<p>
\paragraph{Lorentz contraction:} lengths are also modified.
<b>Lorentz contraction</b>: lengths are also modified.
Back to our train, with a mirror on one end. A light signal
is sent from the opposite end, and the time for the round-trip
of the light is measured. For the observer on the
train, the time is \(\Delta t_{\mbox{\tiny tr}} = 2 \Delta x_{\mbox{\tiny tr}}/c\)
with \(\Delta x_{\mbox{\tiny tr}}\) being the length of the train car.
train, the time is \(\Delta t_{\mbox{tr}} = 2 \Delta x_{\mbox{tr}}/c\)
with \(\Delta x_{\mbox{tr}}\) being the length of the train car.
For the observer on the ground, the total time is made up of the
back and forth journey of the light, with times
\[
\Delta t_{\mbox{\tiny gr,1}} = \frac{\Delta x_{\mbox{\tiny gr}} + v \Delta t_{\mbox{\tiny gr,1}}}{c}, \hspace{10mm}
\Delta t_{\mbox{\tiny gr,2}} = \frac{\Delta x_{\mbox{\tiny gr}} - v \Delta t_{\mbox{\tiny gr,2}}}{c}
\Delta t_{\mbox{gr,1}} = \frac{\Delta x_{\mbox{gr}} + v \Delta t_{\mbox{gr,1}}}{c}, \hspace{10mm}
\Delta t_{\mbox{gr,2}} = \frac{\Delta x_{\mbox{gr}} - v \Delta t_{\mbox{gr,2}}}{c}
\]
so
\[
\Delta t_{\mbox{\tiny gr,1}} = \frac{\Delta x_{\mbox{\tiny gr}}}{c-v}, \hspace{10mm}
\Delta t_{\mbox{\tiny gr,2}} = \frac{\Delta x_{\mbox{\tiny gr}}}{c+v}
\Delta t_{\mbox{gr,1}} = \frac{\Delta x_{\mbox{gr}}}{c-v}, \hspace{10mm}
\Delta t_{\mbox{gr,2}} = \frac{\Delta x_{\mbox{gr}}}{c+v}
\]
and thus
\[
\Delta t_{\mbox{\tiny gr}} =
\Delta t_{\mbox{\tiny gr,1}} + \Delta t_{\mbox{\tiny gr,2}}
= \frac{2 \Delta x_{\mbox{\tiny gr}}}{c} \frac{1}{1 - v^2/c^2}.
\Delta t_{\mbox{gr}} =
\Delta t_{\mbox{gr,1}} + \Delta t_{\mbox{gr,2}}
= \frac{2 \Delta x_{\mbox{gr}}}{c} \frac{1}{1 - v^2/c^2}.
\]
Using the time dilation relation then gives
</p>
<div class="core div" id="orgb3ffc56">
<div class="core div" id="orge1bda84">
<p>
{\bf Lorentz contraction}
<b>Lorentz contraction</b>
\[
\Delta x_{\mbox{\tiny tr}} = \frac{1}{\sqrt{1 - v^2/c^2}} \Delta x_{\mbox{\tiny gr}}
\]
\Delta x_{\mbox{tr}} = \frac{1}{\sqrt{1 - v^2/c^2}} \Delta x_{\mbox{gr}}
\]
</p>
</div>
@@ -1718,7 +1732,7 @@ 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>