197 lines
6.7 KiB
TeX
197 lines
6.7 KiB
TeX
\section{Spacetime Diagrams}
|
|
|
|
We are going to derive the principles and consequences of special relativity using basic geometry. \\
|
|
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}.
|
|
|
|
We plot space, which we denote by $x$, as the horizontal axis, and time, which we denote by $t$ as the vertical axis. For a given object, we can then plot its position at any given time. \\
|
|
This will give a (potentially curvy) line that we call the object's {\em world line}.
|
|
|
|
\example{}
|
|
Suppose that at time $t = 0$, you are standing still with your cat at your feet.
|
|
Your cat walks away from you at speed $1$. We can represent this with a spacetime diagram:
|
|
|
|
\begin{center}
|
|
% SPACETIME DIAGRAM with WORLD LINES
|
|
\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}{
|
|
\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] {$t$};
|
|
\draw[->,thick] (-\xmin,0) -- (\xmax+0.2,0) node[below=0] {$x$};
|
|
|
|
% VECTORS
|
|
\draw[vector,myred, very thick] (O) -- (4*\d,4*\d)
|
|
node[mydarkred,right=10,above] {\contour{white}{cat: $x(t)=t$}};
|
|
\draw[vector,myblue, very thick] (O) -- (0,4*\d)
|
|
node[mydarkblue,below left=0] {\contour{white}{you: $x(t)=0$}};
|
|
%\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\note[Note]{
|
|
The horizontal axis is space and the vertical axis is time. \par
|
|
We are only working with one spatial dimension.
|
|
}
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\problem{}<pets scatter>
|
|
Suppose that you are standing still at time $t = 0$ and your many pets lie at your feet.
|
|
|
|
\begin{itemize}
|
|
\item Your cat, unhappy that she is not fed, begins walking away to your right at speed $2$.
|
|
\item Your dog, distracted by a squirrel, walks away to your left at speed $1$.
|
|
\item Your hamster, just wanting to feel included, waits a second and then follows the dog at speed $2$.
|
|
After reaching your dog, your hamster turns around and sprints after the cat at speed $3$.
|
|
\end{itemize}
|
|
|
|
Draw this situation in the provided spacetime diagram.
|
|
|
|
\makeatletter
|
|
\if@solutions\else
|
|
\emptydiagram{Alice}
|
|
\fi
|
|
\makeatother
|
|
|
|
\begin{solution}
|
|
\begin{center}
|
|
\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);
|
|
|
|
% WORLD LINES GRID
|
|
\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[world line] (-4*\d,-\ymin) -- (-4*\d,\xmax);
|
|
|
|
% AXES
|
|
\draw[->,thick] (0,-\ymin) -- (T) node[left=-1] {$t$};
|
|
\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}{you: $x(t)=0$}};
|
|
\draw[vector,myblue] (O) -- (4*\d,2*\d)
|
|
node[mydarkblue,above left=0] {\contour{white}{cat: $x(t)=2t$}};
|
|
\draw[vector,mygreen] (O) -- (-4*\d,4*\d)
|
|
node[mydarkgreen,below left=0] {\contour{white}{dog: $x(t)=-t$}};
|
|
\draw[vector,black] (O) -- (0,\d) -- (-2*\d, 2*\d) -- (4*\d, 4*\d)
|
|
node[black,below right=0] {\contour{white}{hamster}};
|
|
% \draw[vector,myblue]
|
|
% (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{center}
|
|
\end{solution}
|
|
|
|
|
|
\vfill
|
|
|
|
\problem{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
|
|
corresponds to your hamster catching up to your dog?
|
|
|
|
\begin{solution}
|
|
$(-2,2)$
|
|
\end{solution}
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\problem{}<pets scatter train>
|
|
Suppose that the situation of \ref{pets scatter} occurred while you were riding on
|
|
a train moving to the right at speed $1$.
|
|
Everything occurs relative to you in the same way.
|
|
Draw the same diagram in this new situation.
|
|
Are any of your pets staying still in this new situation?
|
|
|
|
\makeatletter
|
|
\if@solutions\else
|
|
\emptydiagram{Train}
|
|
\fi
|
|
\makeatother
|
|
|
|
\begin{solution}
|
|
\begin{center}
|
|
\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);
|
|
|
|
% 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[world line] (-4*\d,-\ymin) -- (-4*\d,\xmax);
|
|
|
|
% AXES
|
|
\draw[->,thick] (0,-\ymin) -- (T) node[left=-1] {$t$};
|
|
\draw[->,thick] (-\xmin,0) -- (\xmax+0.2,0) node[below=0] {$x$};
|
|
|
|
% VECTORS
|
|
\draw[vector,myred] (O) -- (4*\d,4*\d)
|
|
node[mydarkred,below left=0] {\contour{white}{you: $x(t)=t$}};
|
|
\draw[vector,myblue] (O) -- (6*\d,2*\d)
|
|
node[mydarkblue,above left=0] {\contour{white}{cat: $x(t)=3t$}};
|
|
\draw[vector,mygreen] (O) -- (0,4*\d)
|
|
node[mydarkgreen,below left=0] {\contour{white}{dog: $x(t)=0$}};
|
|
\draw[vector,black] (O) -- (\d,\d) -- (0, 2*\d) -- (8*\d, 4*\d)
|
|
node[black,below right=0] {\contour{white}{hamster}};
|
|
% \draw[vector,myblue]
|
|
% (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{center}
|
|
|
|
The dog remains stationary in this reference frame.
|
|
\end{solution}
|
|
|
|
\vfill
|
|
\pagebreak
|