Mark/handouts
Mark/handouts
1
0
Fork 0
Code Pull Requests 1 Actions Packages Releases Activity

Cleanup

Browse Source
This commit is contained in:
Mark 2025-02-05 13:57:11 -08:00
parent 02d2d97f93
commit 15d334c29f
Signed by: Mark
GPG Key ID: C6D63995FE72FD80
10 changed files with 379 additions and 448 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

141
src/Advanced/Relativity/backups/dilation.tex
View File

@ -1,141 +0,0 @@
\section{Length contraction and time dilation}
Up to now, we've been considering everything as points. We haven't considered how tall or wide you are when drawing these diagrams. Now we are going to.
\problem{}
Draw a stick of length $1$ at rest in a spacetime diagram.
\begin{solution}\begin{center}\begin{tikzpicture}[scale=2.0]
\message{Worldlines^^J}
\def\ymin{0.2}
\def\xmin{1.6}
\def\xmax{2}
\def\Nlines{4} % number of world lines (at constant x/t)
\pgfmathsetmacro\d{0.9*\xmax/\Nlines} % grid size
\coordinate (O) at (0,0);
\coordinate (T) at (0,\xmax+0.2);
% WORLD LINES GRID
\message{ Making world lines...^^J}
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\message{ Running i/N=\i/\Nlines, x=\x...^^J}
\draw[world line] ( \x,-\ymin) -- ( \x,\xmax);
\draw[world line t] (-\xmin, \x) -- (\xmax, \x);
}
\draw[world line] (-\d,-\ymin) -- (-\d,\xmax);
\draw[world line] (-2*\d,-\ymin) -- (-2*\d,\xmax);
\draw[world line] (-3*\d,-\ymin) -- (-3*\d,\xmax);
% AXES
\draw[->,thick] (0,-\ymin) -- (T) node[left=-1] {$ct$};
\draw[->,thick] (-\xmin,0) -- (\xmax+0.2,0) node[below=0] {$x$};
% VECTORS
\draw[vector,myred] (O) -- (0,4*\d)
node[mydarkred,below left=0] {\contour{white}{stick: $x(t)=0$}};
\draw[vector,myred] (\d,0) -- (\d,4*\d);
%\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};
\fill[myred,opacity=1]
(0,0) -- (0,4*\d) -- (\d,4*\d) -- (\d,0) -- cycle;
\end{tikzpicture}\end{center}\end{solution}
\problem{}
Draw a stick of length $1$ sitting on a train moving at speed $c/2$.
{\em Hint: Begin by drawing the train's reference frame.}
\begin{solution}\begin{center}\begin{tikzpicture}[scale=2.0]
\message{Worldlines^^J}
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\def\ymin{0.2}
\def\xmin{1.6}
\def\xmax{2}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{4} % number of world lines (at constant x/t)
\pgfmathsetmacro\d{0.9*\xmax/\Nlines} % grid size
\coordinate (O) at (0,0);
\coordinate (T) at (0,\xmax+0.2);
\def\xmax{2}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{5} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,\xmax+0.2);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);
% WORLD LINES GRID
\message{ Making world lines...^^J}
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\message{ Running i/N=\i/\Nlines, x=\x...^^J}
\draw[world line] ( \x,-\ymin) -- ( \x,\xmax);
\draw[world line t] (-\xmin, \x) -- (\xmax, \x);
}
\draw[world line] (-\d,-\ymin) -- (-\d,\xmax);
\draw[world line] (-2*\d,-\ymin) -- (-2*\d,\xmax);
\draw[world line] (-3*\d,-\ymin) -- (-3*\d,\xmax);
\draw[vector,mydarkred] (O) -- (\ang:\xmaxp)
node[mydarkred,right=0] {\contour{white}{$x'$}};
\draw[vector,mydarkred] (O) -- (90-\ang:\xmaxp)
node[mydarkred,left=0] {\contour{white}{$ct'$}};
% AXES
\draw[->,thick] (0,-\ymin) -- (T) node[left=-1] {$ct$};
\draw[->,thick] (-\xmin,0) -- (\xmax+0.2,0) node[below=0] {$x$};
% BOOSTED WORLD LINE GRID
\message{ Making world lines for boosted frame...^^J}
\fill[mydarkred,opacity=0.05]
(O) --++ (\ang:\xmaxp) --++ (90-\ang:\xmaxp) --++ (\ang:-\xmaxp) -- cycle;
% \fill[mydarkred,opacity=0.05]
% (O) --++ (\ang:-\xmaxp) --++ (90-\ang:-\xmaxp) --++ (\ang:\xmaxp) -- cycle;
\foreach \i [evaluate={\x=\i*\D;}] in {1,...,4}{
\message{ Running i/N=\i/\Nlines, x=\x...^^J}
% \draw[world line'] (\ang:-\x) --++ (90-\ang:-\xmaxp);
% \draw[world line'] (90-\ang:-\x) --++ (\ang:-\xmaxp);
\draw[world line'] (\ang:\x) --++ (90-\ang:\xmaxp);
\draw[world line'] (90-\ang:\x) --++ (\ang:\xmaxp);
}
\fill[myred,opacity=1]
(O) --++ (\ang:\D) --++ (90-\ang:\xmaxp) --++ (\ang:-\D) -- cycle;
\end{tikzpicture}\end{center}\end{solution}
\problem{}
In the rest frame, what is the length of the stick? This should require only some simple trigonometry.
{\em Hint: pick a particular time in the rest frame and look at where the stick is at that particular time.}
Generalize this to a general reference frame moving at speed $ck$.
Suppose that the stick has a length $\ell'$ in the reference frame of the train moving at speed $ck$. What is the length, $\ell$, of the stick in the rest frame?
Physically, what does this mean? What happens to the stick as it moves faster?
\begin{solution}
$\ell$ is the length of the stick on the ground, $\ell'$ is the length of the stick on the train. For a speed of $ck$, the $x'$ axis has a slope of $k$. We then calculate
\begin{align*}
\ell^2 + k^2 \ell^2 & = (\ell')^2\\
\ell & = \frac{\ell'}{\sqrt{1 + k^2}}\\
\end{align*}
\end{solution}
\problem{}
Now suppose that you measure out $t'$ seconds of time on the train which is moving at a speed of $kc$. When you start your clock, you yell "START" and when you stop your clock, you yell "STOP". How long between the two yells would someone at rest measure? What does this imply physically?
\begin{solution}
Let $t'$ be the seconds on the train and $t$ be the seconds in the rest frame. The same calculation as the length contraction implies that
\begin{align*}
(ct)^2 + (ckt)^2 & = (ct')^2 \\
t & = \frac{t'}{\sqrt{1 + k^2}} \\
\end{align*}
\end{solution}

249
src/Advanced/Relativity/diagram.tex Normal file
View File

@ -0,0 +1,249 @@
% Diagrams adapted from https://tikz.net/relativity_minkowski_diagram/
%
% MARK: Matt's tweaks
%
\newcommand{\emptydiagram}[1]{
\begin{center}
\begin{tikzpicture}[scale=2]
\def\xmax{2}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{6} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,\xmax+0.2);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\draw[world line] (-\x,-\xmax) -- (-\x,\xmax);
\draw[world line] ( \x,-\xmax) -- ( \x,\xmax);
\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
% AXES
\draw[->,thick] (0,-\xmax) -- (T) node[left=-1] {#1: $t$};
\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
\end{tikzpicture}
\end{center}
}
\newcommand{\emptydiagramc}[1]{
\begin{center}
\begin{tikzpicture}[scale=2]
\def\xmax{2}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{6} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,\xmax+0.2);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\draw[world line] (-\x,-\xmax) -- (-\x,\xmax);
\draw[world line] ( \x,-\xmax) -- ( \x,\xmax);
\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
% AXES
\draw[->,thick] (0,-\xmax) -- (T) node[left=-1] {#1: $ct$};
\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
\end{tikzpicture}
\end{center}
}
\newcommand{\halfdiagramc}[1]{
\begin{center}
\begin{tikzpicture}[scale=2]
\def\xmax{2}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{6} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,\xmax+0.2);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);
% WORLD LINE GRID
\message{ Making world lines...^^J}
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\message{ Running i/N=\i/\Nlines, x=\x...^^J}
\draw[world line] (-\x,-0.2) -- (-\x,\xmax);
\draw[world line] ( \x,-0.2) -- ( \x,\xmax);
%\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
% AXES
\draw[->,thick] (0,-0.2) -- (T) node[left=-1] {#1: $ct$};
\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
\end{tikzpicture}
\end{center}
}
\newcommand{\halfdiagramcwide}[1]{
\begin{center}
\begin{tikzpicture}[scale=2]
\def\xmax{3.5}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{10} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,1.95);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\draw[world line] (-\x,-0.2) -- (-\x,1.95);
\draw[world line] ( \x,-0.2) -- ( \x,1.95);
%\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
%\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,5}{
% \draw[world line] (-\x,-0.2) -- (-\x,\xmax);
% \draw[world line] ( \x,-0.2) -- ( \x,\xmax);
%\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
% AXES
\draw[->,thick] (0,-0.2) -- (T) node[left=-1] {#1: $ct$};
\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
\end{tikzpicture}
\end{center}
}
%
% MARK: tikz setup
%
\usepackage{tikz}
\usetikzlibrary{patterns}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{caption}
\usepackage{etoolbox} % ifthen
\usepackage[outline]{contour} % glow around text
\usetikzlibrary{calc} % for adding up coordinates
\usetikzlibrary{decorations.markings,decorations.pathmorphing}
\usetikzlibrary{angles,quotes} % for pic (angle labels)
\usetikzlibrary{arrows.meta} % for arrow size
\usepackage{xfp} % higher precision (16 digits?)
\contourlength{1.1pt}
\tikzset{>=latex} % for LaTeX arrow head
\colorlet{myred}{red!85!black}
\colorlet{mydarkred}{red!55!black}
\colorlet{mylightred}{red!85!black!12}
\colorlet{myfieldred}{mydarkred!5} % for S' background
\colorlet{myredhighlight}{myred!20} % highlights simultaneity in ladder paradox
\colorlet{myblue}{blue!80!black}
\colorlet{mydarkblue}{blue!50!black}
\colorlet{mylightblue}{blue!50!black!30}
\colorlet{mylightblue2}{myblue!10}
\colorlet{mygreen}{green!80!black}
\colorlet{mypurple}{blue!40!red!80!black}
\colorlet{mydarkgreen}{green!50!black}
\colorlet{mydarkpurple}{blue!40!red!50!black}
\colorlet{myorange}{orange!40!yellow!95!black}
\colorlet{mydarkorange}{orange!40!yellow!85!black}
\colorlet{mybrown}{brown!20!orange!90!black}
\colorlet{mydarkbrown}{brown!20!orange!55!black}
\colorlet{mypurplehighlight}{mydarkpurple!20} % highlights simultaneity in ladder paradox
\tikzstyle{world line}=[myblue!40,line width=0.3]
\tikzstyle{world line t}=[mypurple!50!myblue!40,line width=0.3]
\tikzstyle{world line'}=[mydarkred!40,line width=0.3]
\tikzstyle{mysmallarr}=[-{Latex[length=3,width=2]},thin]
\tikzstyle{mydashed}=[dash pattern=on 3 off 3]
\tikzstyle{rod}=[mydarkbrown,draw=mydarkbrown,double=mybrown,double distance=2pt,
line width=0.2,line cap=round,shorten >=1pt,shorten <=1pt]
%\tikzstyle{rod'}=[rod,draw=mydarkbrown!80!red!85,double=mybrown!80!red!85]
\tikzstyle{vector}=[->,line width=1,line cap=round]
\tikzstyle{vector'}=[vector,shorten >=1.2]
\tikzstyle{particle}=[mygreen,line width=0.9]
\tikzstyle{photon}=[-{Latex[length=5,width=4]},myorange,line width=0.8,decorate,
decoration={snake,amplitude=1.0,segment length=5,post length=5}]
\def\tick#1#2{\draw[thick] (#1) ++ (#2:0.06) --++ (#2-180:0.12)}
\def\tickp#1#2{\draw[thick,mydarkred] (#1) ++ (#2:0.06) --++ (#2-180:0.12)}
\def\Nsamples{100} % number samples in plot
% COMMON AXES
\pgfdeclarelayer{back} % to draw on background
\pgfsetlayers{back,main} % set order
\def\xmin{0.23}
\def\xmax{2}
\def\Nlines{6} % number of world lines (at constant x/t)
\def\DNxp{0} % difference in number of world lines of x' axis
\def\DNyp{0} % difference in number of world lines of ct' axis
\def\DNy{0} % difference in number of world lines of ct axis
\def\ang{20} % angle between x and x' axes
\def\xplabelang{180} % anchor angle of x' axis label
%\pgfmathsetmacro\ang{atan(0.44)} % angle between x and x' axes
\def\axes{
\pgfmathsetmacro\d{\xmax/(\Nlines+0.4)} % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\pgfmathsetmacro\ymax{\xmax+\DNy*\d} % maximum of y = ct axis
\pgfmathsetmacro\xmaxp{(\xmax/\d+\DNxp)*\D} % maximum of x' axis
\pgfmathsetmacro\ymaxp{(\xmax/\d+\DNyp)*\D} % maximum of y' = ct' axis
\pgfmathsetmacro\Nylines{\Nlines+\DNy} % number of world lines at constant ct'
\pgfmathsetmacro\Nxplines{\Nlines+\DNxp} % number of world lines at constant x'
\pgfmathsetmacro\Nyplines{\Nlines+\DNyp} % number of world lines at constant ct'
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.15,0);
\coordinate (T) at (0,\ymax+0.15);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\ymaxp+0.2);
% FILL
\begin{pgfonlayer}{back} % draw on back
\fill[myfieldred]
(\ang:-\xmin) -- (\ang:\xmaxp) --++ (90-\ang:\ymaxp) --++ (\ang:-\xmaxp)
-- (90-\ang:-\xmin) -- cycle;
\end{pgfonlayer}
% WORLD LINE GRID
\message{ Making world lines...^^J}
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\draw[world line] (\x,0) -- (\x,\ymax);
}
\foreach \i [evaluate={\t=\i*\d;}] in {1,...,\Nylines}{
\draw[world line t] (0,\t) -- (\xmax,\t);
}
% BOOSTED WORLD LINE GRID
\foreach \i [evaluate={\x=\i*\D;}] in {1,...,\Nxplines}{
\draw[world line'] (\ang:\x) --++ (90-\ang:\ymaxp);
}
\foreach \i [evaluate={\t=\i*\D;}] in {1,...,\Nyplines}{
\draw[world line'] (90-\ang:\t) --++ (\ang:\xmaxp);
}
% AXES
\draw[->,thick] (0,-\xmin) -- (T) node[left=-1] {$ct$};
\draw[->,thick] (-\xmin,0) -- (X) node[below=0] {$x$};
\draw[->,thick,mydarkred] (90-\ang:-\xmin) -- (T')
node[right=5,above=-1] {$ct'$};
\draw[->,thick,mydarkred] (\ang:-\xmin) -- (X')
node[anchor=\xplabelang,inner sep=2] {$x'$};
}

122
src/Advanced/Relativity/main.tex
View File

@ -1,129 +1,12 @@
% use [nosolutions] flag to hide solutions. % use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions. % use [solutions] flag to show solutions.
\documentclass[ \documentclass[
solutions, nosolutions,
singlenumbering, singlenumbering,
shortwarning shortwarning
]{../../../lib/tex/ormc_handout} ]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros} \usepackage{../../../lib/tex/macros}
\input{diagram}
\def\emptydiagrams{0} % set this to 1 to generate empty diagrams, 0 removes all empty diagrams
\input{matt}
\usepackage{tikz}
\usetikzlibrary{patterns}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{caption}
\usepackage{etoolbox} % ifthen
\usepackage[outline]{contour} % glow around text
\usetikzlibrary{calc} % for adding up coordinates
\usetikzlibrary{decorations.markings,decorations.pathmorphing}
\usetikzlibrary{angles,quotes} % for pic (angle labels)
\usetikzlibrary{arrows.meta} % for arrow size
\usepackage{xfp} % higher precision (16 digits?)
\contourlength{1.1pt}
\tikzset{>=latex} % for LaTeX arrow head
\colorlet{myred}{red!85!black}
\colorlet{mydarkred}{red!55!black}
\colorlet{mylightred}{red!85!black!12}
\colorlet{myfieldred}{mydarkred!5} % for S' background
\colorlet{myredhighlight}{myred!20} % highlights simultaneity in ladder paradox
\colorlet{myblue}{blue!80!black}
\colorlet{mydarkblue}{blue!50!black}
\colorlet{mylightblue}{blue!50!black!30}
\colorlet{mylightblue2}{myblue!10}
\colorlet{mygreen}{green!80!black}
\colorlet{mypurple}{blue!40!red!80!black}
\colorlet{mydarkgreen}{green!50!black}
\colorlet{mydarkpurple}{blue!40!red!50!black}
\colorlet{myorange}{orange!40!yellow!95!black}
\colorlet{mydarkorange}{orange!40!yellow!85!black}
\colorlet{mybrown}{brown!20!orange!90!black}
\colorlet{mydarkbrown}{brown!20!orange!55!black}
\colorlet{mypurplehighlight}{mydarkpurple!20} % highlights simultaneity in ladder paradox
\tikzstyle{world line}=[myblue!40,line width=0.3]
\tikzstyle{world line t}=[mypurple!50!myblue!40,line width=0.3]
\tikzstyle{world line'}=[mydarkred!40,line width=0.3]
\tikzstyle{mysmallarr}=[-{Latex[length=3,width=2]},thin]
\tikzstyle{mydashed}=[dash pattern=on 3 off 3]
\tikzstyle{rod}=[mydarkbrown,draw=mydarkbrown,double=mybrown,double distance=2pt,
line width=0.2,line cap=round,shorten >=1pt,shorten <=1pt]
%\tikzstyle{rod'}=[rod,draw=mydarkbrown!80!red!85,double=mybrown!80!red!85]
\tikzstyle{vector}=[->,line width=1,line cap=round]
\tikzstyle{vector'}=[vector,shorten >=1.2]
\tikzstyle{particle}=[mygreen,line width=0.9]
\tikzstyle{photon}=[-{Latex[length=5,width=4]},myorange,line width=0.8,decorate,
decoration={snake,amplitude=1.0,segment length=5,post length=5}]
\def\tick#1#2{\draw[thick] (#1) ++ (#2:0.06) --++ (#2-180:0.12)}
\def\tickp#1#2{\draw[thick,mydarkred] (#1) ++ (#2:0.06) --++ (#2-180:0.12)}
\def\Nsamples{100} % number samples in plot
% COMMON AXES
\pgfdeclarelayer{back} % to draw on background
\pgfsetlayers{back,main} % set order
\def\xmin{0.23}
\def\xmax{2}
\def\Nlines{6} % number of world lines (at constant x/t)
\def\DNxp{0} % difference in number of world lines of x' axis
\def\DNyp{0} % difference in number of world lines of ct' axis
\def\DNy{0} % difference in number of world lines of ct axis
\def\ang{20} % angle between x and x' axes
\def\xplabelang{180} % anchor angle of x' axis label
%\pgfmathsetmacro\ang{atan(0.44)} % angle between x and x' axes
\def\axes{
\pgfmathsetmacro\d{\xmax/(\Nlines+0.4)} % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\pgfmathsetmacro\ymax{\xmax+\DNy*\d} % maximum of y = ct axis
\pgfmathsetmacro\xmaxp{(\xmax/\d+\DNxp)*\D} % maximum of x' axis
\pgfmathsetmacro\ymaxp{(\xmax/\d+\DNyp)*\D} % maximum of y' = ct' axis
\pgfmathsetmacro\Nylines{\Nlines+\DNy} % number of world lines at constant ct'
\pgfmathsetmacro\Nxplines{\Nlines+\DNxp} % number of world lines at constant x'
\pgfmathsetmacro\Nyplines{\Nlines+\DNyp} % number of world lines at constant ct'
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.15,0);
\coordinate (T) at (0,\ymax+0.15);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\ymaxp+0.2);
% FILL
\begin{pgfonlayer}{back} % draw on back
\fill[myfieldred]
(\ang:-\xmin) -- (\ang:\xmaxp) --++ (90-\ang:\ymaxp) --++ (\ang:-\xmaxp)
-- (90-\ang:-\xmin) -- cycle;
\end{pgfonlayer}
% WORLD LINE GRID
\message{ Making world lines...^^J}
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\draw[world line] (\x,0) -- (\x,\ymax);
}
\foreach \i [evaluate={\t=\i*\d;}] in {1,...,\Nylines}{
\draw[world line t] (0,\t) -- (\xmax,\t);
}
% BOOSTED WORLD LINE GRID
\foreach \i [evaluate={\x=\i*\D;}] in {1,...,\Nxplines}{
\draw[world line'] (\ang:\x) --++ (90-\ang:\ymaxp);
}
\foreach \i [evaluate={\t=\i*\D;}] in {1,...,\Nyplines}{
\draw[world line'] (90-\ang:\t) --++ (\ang:\xmaxp);
}
% AXES
\draw[->,thick] (0,-\xmin) -- (T) node[left=-1] {$ct$};
\draw[->,thick] (-\xmin,0) -- (X) node[below=0] {$x$};
\draw[->,thick,mydarkred] (90-\ang:-\xmin) -- (T')
node[right=5,above=-1] {$ct'$};
\draw[->,thick,mydarkred] (\ang:-\xmin) -- (X')
node[anchor=\xplabelang,inner sep=2] {$x'$};
}
@ -132,7 +15,6 @@
\title{Special Relativity} \title{Special Relativity}
\subtitle{ \subtitle{
Prepared by Matthew Kowalski on \today{} \\ Prepared by Matthew Kowalski on \today{} \\
All diagrams are adapted from Izaak Neutelings, see \url{https://tikz.net/relativity_minkowski_diagram/}
} }
\begin{document} \begin{document}

133
src/Advanced/Relativity/matt.tex
View File

@ -1,133 +0,0 @@
\newcommand{\emptydiagram}[1]{
\if\emptydiagrams1
\begin{center}
\begin{tikzpicture}[scale=2]
\def\xmax{2}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{6} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,\xmax+0.2);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\draw[world line] (-\x,-\xmax) -- (-\x,\xmax);
\draw[world line] ( \x,-\xmax) -- ( \x,\xmax);
\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
% AXES
\draw[->,thick] (0,-\xmax) -- (T) node[left=-1] {#1: $t$};
\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
\end{tikzpicture}
\end{center}
\fi
}
\newcommand{\emptydiagramc}[1]{
\if\emptydiagrams1
\begin{center}
\begin{tikzpicture}[scale=2]
\def\xmax{2}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{6} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,\xmax+0.2);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\draw[world line] (-\x,-\xmax) -- (-\x,\xmax);
\draw[world line] ( \x,-\xmax) -- ( \x,\xmax);
\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
% AXES
\draw[->,thick] (0,-\xmax) -- (T) node[left=-1] {#1: $ct$};
\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
\end{tikzpicture}
\end{center}
\fi
}
\newcommand{\halfdiagramc}[1]{
\if\emptydiagrams1
\begin{center}
\begin{tikzpicture}[scale=2]
\def\xmax{2}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{6} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,\xmax+0.2);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);
% WORLD LINE GRID
\message{ Making world lines...^^J}
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\message{ Running i/N=\i/\Nlines, x=\x...^^J}
\draw[world line] (-\x,-0.2) -- (-\x,\xmax);
\draw[world line] ( \x,-0.2) -- ( \x,\xmax);
%\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
% AXES
\draw[->,thick] (0,-0.2) -- (T) node[left=-1] {#1: $ct$};
\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
\end{tikzpicture}
\end{center}
\fi
}
\newcommand{\halfdiagramcwide}[1]{
\if\emptydiagrams1
\begin{center}
\begin{tikzpicture}[scale=2]
\def\xmax{3.5}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{10} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,1.95);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\draw[world line] (-\x,-0.2) -- (-\x,1.95);
\draw[world line] ( \x,-0.2) -- ( \x,1.95);
%\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
%\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,5}{
% \draw[world line] (-\x,-0.2) -- (-\x,\xmax);
% \draw[world line] ( \x,-0.2) -- ( \x,\xmax);
%\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}
% AXES
\draw[->,thick] (0,-0.2) -- (T) node[left=-1] {#1: $ct$};
\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
\end{tikzpicture}
\end{center}
\fi
}

96
src/Advanced/Relativity/parts/01 spacetime diagrams.tex
View File

@ -1,7 +1,7 @@
\section{Spacetime Diagrams} \section{Spacetime Diagrams}
We are going to derive the principles and consequences of special relativity using basic geometry. We are going to derive the principles and consequences of special relativity using basic geometry. \\
\\ To help with our visualization, we will be using spacetime diagrams (called {\em Minkowski diagrams}). o help with our visualization, we will be using spacetime diagrams (called {\em Minkowski diagrams}).
To make our models simpler, we will only be considering {\em one spatial dimension}. To make our models simpler, we will only be considering {\em one spatial dimension}.
@ -45,7 +45,6 @@ Your cat walks away from you at speed $1$. We can represent this with a spacetim
node[mydarkblue,below left=0] {\contour{white}{you: $x(t)=0$}}; node[mydarkblue,below left=0] {\contour{white}{you: $x(t)=0$}};
%\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$}; %\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
\note[Note]{ \note[Note]{
@ -53,6 +52,9 @@ Your cat walks away from you at speed $1$. We can represent this with a spacetim
We are only working with one spatial dimension. We are only working with one spatial dimension.
} }
\vfill
\pagebreak
\problem{}<pets scatter> \problem{}<pets scatter>
Suppose that you are standing still at time $t = 0$ and your many pets lie at your feet. Suppose that you are standing still at time $t = 0$ and your many pets lie at your feet.
@ -64,8 +66,12 @@ Suppose that you are standing still at time $t = 0$ and your many pets lie at yo
\end{itemize} \end{itemize}
Draw this situation in the provided spacetime diagram. Draw this situation in the provided spacetime diagram.
\emptydiagram{Alice}
\makeatletter
\if@solutions\else
\emptydiagram{Alice}
\fi
\makeatother
\begin{solution} \begin{solution}
\begin{center} \begin{center}
@ -111,6 +117,9 @@ Draw this situation in the provided spacetime diagram.
\end{center} \end{center}
\end{solution} \end{solution}
\vfill
\problem{Event} \problem{Event}
Any single point $(x,t)$ on a spacetime diagram is considered an {\em event} Any single point $(x,t)$ on a spacetime diagram is considered an {\em event}
because it describes a time and place. For instance, what is the event that because it describes a time and place. For instance, what is the event that
@ -121,7 +130,7 @@ corresponds to your hamster catching up to your dog?
\end{solution} \end{solution}
\vfill \vfill
\pagebreak
\problem{}<pets scatter train> \problem{}<pets scatter train>
Suppose that the situation of \ref{pets scatter} occurred while you were riding on Suppose that the situation of \ref{pets scatter} occurred while you were riding on
@ -130,50 +139,53 @@ Everything occurs relative to you in the same way.
Draw the same diagram in this new situation. Draw the same diagram in this new situation.
Are any of your pets staying still in this new situation? Are any of your pets staying still in this new situation?
\emptydiagram{Train} \makeatletter
\if@solutions\else
\emptydiagram{Train}
\fi
\makeatother
\begin{solution} \begin{solution}
\begin{center} \begin{center}
\begin{tikzpicture}[scale=2.0] \begin{tikzpicture}[scale=2.0]
\def\ymin{0.2}
\def\xmin{2}
\def\xmax{2}
\def\Nlines{4} % number of world lines (at constant x/t)
\pgfmathsetmacro\d{0.9*\xmax/\Nlines} % grid size
\coordinate (O) at (0,0);
\coordinate (T) at (0,\xmax+0.2);
\def\ymin{0.2} % WORLD LINES GRID
\def\xmin{2} \message{ Making world lines...^^J}
\def\xmax{2} \foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\def\Nlines{4} % number of world lines (at constant x/t) \message{ Running i/N=\i/\Nlines, x=\x...^^J}
\pgfmathsetmacro\d{0.9*\xmax/\Nlines} % grid size \draw[world line] ( \x,-\ymin) -- ( \x,\xmax);
\coordinate (O) at (0,0); \draw[world line t] (-\xmin, \x) -- (\xmax, \x);
\coordinate (T) at (0,\xmax+0.2); }
\draw[world line] (-\d,-\ymin) -- (-\d,\xmax);
\draw[world line] (-2*\d,-\ymin) -- (-2*\d,\xmax);
\draw[world line] (-3*\d,-\ymin) -- (-3*\d,\xmax);
\draw[world line] (-4*\d,-\ymin) -- (-4*\d,\xmax);
% WORLD LINES GRID % AXES
\message{ Making world lines...^^J} \draw[->,thick] (0,-\ymin) -- (T) node[left=-1] {$t$};
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{ \draw[->,thick] (-\xmin,0) -- (\xmax+0.2,0) node[below=0] {$x$};
\message{ Running i/N=\i/\Nlines, x=\x...^^J}
\draw[world line] ( \x,-\ymin) -- ( \x,\xmax);
\draw[world line t] (-\xmin, \x) -- (\xmax, \x);
}
\draw[world line] (-\d,-\ymin) -- (-\d,\xmax);
\draw[world line] (-2*\d,-\ymin) -- (-2*\d,\xmax);
\draw[world line] (-3*\d,-\ymin) -- (-3*\d,\xmax);
\draw[world line] (-4*\d,-\ymin) -- (-4*\d,\xmax);
% AXES % VECTORS
\draw[->,thick] (0,-\ymin) -- (T) node[left=-1] {$t$}; \draw[vector,myred] (O) -- (4*\d,4*\d)
\draw[->,thick] (-\xmin,0) -- (\xmax+0.2,0) node[below=0] {$x$}; node[mydarkred,below left=0] {\contour{white}{you: $x(t)=t$}};
\draw[vector,myblue] (O) -- (6*\d,2*\d)
% VECTORS node[mydarkblue,above left=0] {\contour{white}{cat: $x(t)=3t$}};
\draw[vector,myred] (O) -- (4*\d,4*\d) \draw[vector,mygreen] (O) -- (0,4*\d)
node[mydarkred,below left=0] {\contour{white}{you: $x(t)=t$}}; node[mydarkgreen,below left=0] {\contour{white}{dog: $x(t)=0$}};
\draw[vector,myblue] (O) -- (6*\d,2*\d) \draw[vector,black] (O) -- (\d,\d) -- (0, 2*\d) -- (8*\d, 4*\d)
node[mydarkblue,above left=0] {\contour{white}{cat: $x(t)=3t$}}; node[black,below right=0] {\contour{white}{hamster}};
\draw[vector,mygreen] (O) -- (0,4*\d) % \draw[vector,myblue]
node[mydarkgreen,below left=0] {\contour{white}{dog: $x(t)=0$}}; % (O) to[out=35,in=-100] (O)
\draw[vector,black] (O) -- (\d,\d) -- (0, 2*\d) -- (8*\d, 4*\d) % to[out=80,in=-80,looseness=1.5] (0.3*\xmax,4*\d)
node[black,below right=0] {\contour{white}{hamster}}; % node[mydarkblue,above=-3] {\contour{white}{cat: $x(t)$}};
% \draw[vector,myblue] %\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};
% (O) to[out=35,in=-100] (O)
% to[out=80,in=-80,looseness=1.5] (0.3*\xmax,4*\d)
% node[mydarkblue,above=-3] {\contour{white}{cat: $x(t)$}};
%\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}

7
src/Advanced/Relativity/parts/02 galilean.tex
View File

@ -46,7 +46,12 @@ Draw the situation from \ref{pets scatter} in your cat's perspective. \par
What if we drew the situation from \ref{pets scatter train} in your cat's perspective? \par What if we drew the situation from \ref{pets scatter train} in your cat's perspective? \par
Would there be any change when the cat is on the train? Why or why not? Would there be any change when the cat is on the train? Why or why not?
\emptydiagram{Cat}
\makeatletter
\if@solutions\else
\emptydiagram{Cat}
\fi
\makeatother
\begin{solution} \begin{solution}
\begin{center} \begin{center}

26
src/Advanced/Relativity/parts/03 special.tex
View File

@ -11,7 +11,12 @@ We are going to making things easier for ourselves now and change units. Instead
\problem{}<photon diagram> \problem{}<photon diagram>
Suppose you are sitting still and you send one photon to your right. Draw this photon on a spacetime diagram, with horizontal axis $x$ and vertical axis $ct$. Suppose you are sitting still and you send one photon to your right. Draw this photon on a spacetime diagram, with horizontal axis $x$ and vertical axis $ct$.
\halfdiagramc{You}
\makeatletter
\if@solutions\else
\halfdiagramc{You}
\fi
\makeatother
\begin{solution} \begin{solution}
\begin{center} \begin{center}
@ -48,6 +53,8 @@ Suppose you are sitting still and you send one photon to your right. Draw this p
\end{center} \end{center}
\end{solution} \end{solution}
\vfill
\problem{} \problem{}
Suppose you are now sitting on a train that is moving to the right at $c/2$ Suppose you are now sitting on a train that is moving to the right at $c/2$
and again send one photon to your right. and again send one photon to your right.
@ -56,7 +63,11 @@ Draw this diagram in the reference frame of the ground.
Draw your (Galilean) reference frame on top of this diagram Draw your (Galilean) reference frame on top of this diagram
What is the speed of the photon in your reference frame? Is that a problem? What is the speed of the photon in your reference frame? Is that a problem?
\halfdiagramc{Train} \makeatletter
\if@solutions\else
\halfdiagramc{Train}
\fi
\makeatother
\begin{solution} \begin{solution}
\begin{center} \begin{center}
@ -106,6 +117,7 @@ What is the speed of the photon in your reference frame? Is that a problem?
\end{solution} \end{solution}
\vfill
\pagebreak \pagebreak
@ -119,9 +131,15 @@ Can you brainstorm any ways to fix Galilean relativity to account for this absol
{\em \color{gray} Don't worry if you don't have any ideas! It took physicists a while to figure this out. Whenever you want to move on, we have the solution on the new page.} {\em \color{gray} Don't worry if you don't have any ideas! It took physicists a while to figure this out. Whenever you want to move on, we have the solution on the new page.}
\emptydiagramc{Train} \makeatletter
\if@solutions\else
\emptydiagramc{Train}
\fi
\makeatother
\vfill
\pagebreak \pagebreak
@ -196,6 +214,8 @@ Please verify that in the diagram above, if you shoot a photon behind you, it st
This follows by just extending the boosted axes to the second quadrant and drawing the photon's worldline. This follows by just extending the boosted axes to the second quadrant and drawing the photon's worldline.
\end{solution} \end{solution}
\vfill
\pagebreak \pagebreak
% \begin{solution}\begin{center}\begin{tikzpicture}[scale=1.8] % \begin{solution}\begin{center}\begin{tikzpicture}[scale=1.8]

29
src/Advanced/Relativity/parts/04 simultaneity.tex
View File

@ -5,7 +5,11 @@ we'll bring in some professionals: Alice and Bob.
\ref{simultaneity setup} to \ref{Bob overreacted} can all be completed on one spacetime diagram. Feel free to use the one provided below, or draw your own if it gets too crowded. \ref{simultaneity setup} to \ref{Bob overreacted} can all be completed on one spacetime diagram. Feel free to use the one provided below, or draw your own if it gets too crowded.
\emptydiagramc{Alice} \makeatletter
\if@solutions\else
\emptydiagramc{Alice}
\fi
\makeatother
\problem{}<simultaneity setup> \problem{}<simultaneity setup>
Draw a spacetime diagram from Alice's reference frame or use the one provided. Draw a spacetime diagram from Alice's reference frame or use the one provided.
@ -81,7 +85,13 @@ Suppose that Alice can run at an impressive speed of $c/2$ while Bob can only ru
\vspace{20pt} \vspace{20pt}
\item Is there any reference frame in which Bob wins? \item Is there any reference frame in which Bob wins?
\halfdiagramcwide{Ground}
\makeatletter
\if@solutions\else
\halfdiagramcwide{Ground}
\fi
\makeatother
\begin{solution} \begin{solution}
No No
\end{solution} \end{solution}
@ -95,7 +105,12 @@ Suppose that Alice can run at an impressive speed of $c/2$ while Bob can only ru
\end{solution} \end{solution}
\end{enumerate} \end{enumerate}
\halfdiagramcwide{Ground}
\makeatletter
\if@solutions\else
\halfdiagramcwide{Ground}
\fi
\makeatother
\vfill \vfill
@ -201,6 +216,8 @@ Another way to think of this is thinking of causality. In a reference frame, the
The top region is the future, the bottom region is the past, and the left/right regions are the present. The top region is the future, the bottom region is the past, and the left/right regions are the present.
\end{solution} \end{solution}
\vfill
\pagebreak \pagebreak
@ -212,7 +229,11 @@ The spaceships are at rest and then simultaneously accelerate to speed $c/2$.
Draw the spacetime diagram for this situation. Include the reference frame of the spaceships {\em after} they start moving (i.e. the reference frame moving at speed $c/2$).\\ Draw the spacetime diagram for this situation. Include the reference frame of the spaceships {\em after} they start moving (i.e. the reference frame moving at speed $c/2$).\\
What happens to the floss in the boosted frame? Does it break? Why? What happens to the floss in the boosted frame? Does it break? Why?
\emptydiagramc{Rest} \makeatletter
\if@solutions\else
\emptydiagramc{Rest}
\fi
\makeatother
\vfill \vfill
\problem{Bell's Spaceship Paradox (continued)} \problem{Bell's Spaceship Paradox (continued)}

9
src/Advanced/Relativity/parts/05 proper.tex
View File

@ -299,7 +299,14 @@ Let's break this down step by step.
\item[\bf E:] \item[\bf E:]
To help answer this, draw a spacetime diagram of this situation. In Lucas' reference frames (one for going out to the planet and one for coming back), draw lines of constant time. What happens to Mark from Lucas' perspective?\\ To help answer this, draw a spacetime diagram of this situation. In Lucas' reference frames (one for going out to the planet and one for coming back), draw lines of constant time. What happens to Mark from Lucas' perspective?\\
{\em Hint: changing speed is weird. Focus on where Lucas changes speed.} {\em Hint: changing speed is weird. Focus on where Lucas changes speed.}
\halfdiagramc{Mark}
\makeatletter
\if@solutions\else
\halfdiagramc{Mark}
\fi
\makeatother
\begin{solution} \begin{solution}
% SPACETIME DIAGRAM of TWIN PARADOX % SPACETIME DIAGRAM of TWIN PARADOX
\begin{tikzpicture}[scale=2.0] \begin{tikzpicture}[scale=2.0]

15
src/Advanced/Relativity/parts/06 contraction.tex
View File

@ -4,7 +4,11 @@ With proper time and distance done, we can now tackle length contraction easily!
\problem{} \problem{}
Suppose that you (at rest) see a rod of length $L$ moving at speed $v$. Suppose that you (at rest) see a rod of length $L$ moving at speed $v$.
\halfdiagramc{you} \makeatletter
\if@solutions\else
\halfdiagramc{you}
\fi
\makeatother
\begin{enumerate} \begin{enumerate}
\item Using the provided grid, draw a spacetime diagram where the left side of the rod is at $x = 0$ at $t = 0$. \item Using the provided grid, draw a spacetime diagram where the left side of the rod is at $x = 0$ at $t = 0$.
@ -23,7 +27,7 @@ Suppose that you (at rest) see a rod of length $L$ moving at speed $v$.
\begin{solution} \begin{solution}
\begin{center} \begin{center}
\begin{tikzpicture}[scale=1] \begin{tikzpicture}[scale=1.5]
%\def\R{2*\xmax} % radius of clip %\def\R{2*\xmax} % radius of clip
%\clip (-\xmin,\R) |- (\R,-\xmin) arc(0:90:\xmin+\R); %\clip (-\xmin,\R) |- (\R,-\xmin) arc(0:90:\xmin+\R);
@ -143,8 +147,13 @@ Now, Aiden is particular smart, so he gives Matt the ladder and has Matt run at
\vfill\pagebreak \vfill\pagebreak
\item Despite the barn being too short, we know that the ladder has to fit! Using the provided grid, draw a spacetime diagram of the situation. Include Matt's reference frame on your diagram. \item Despite the barn being too short, we know that the ladder has to fit! Using the provided grid, draw a spacetime diagram of the situation. Include Matt's reference frame on your diagram.
\makeatletter
\if@solutions\else
\emptydiagramc{Aiden}
\fi
\makeatother
\emptydiagramc{Aiden}
\vfill \vfill
\item From Matt's perspective, why don't the doors of the barn crush the ladder? \item From Matt's perspective, why don't the doors of the barn crush the ladder?
\end{enumerate} \end{enumerate}