\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
