\section{Galilean Relativity}

Much like you can watch your pets scatter from the perspective of a train, we can watch the world from anyone's perspective. When we shift perspective like this, just using our normal intuition, we call this {\em Galilean relativity}.

\example{}
Consider the situation of Example $1$ again, but now from the perspective of your cat.
From your cat's perspective, she's the one staying still and you're the one walking away,
only now you're walking away at speed $1$ to the left. We'll denote our new spatial variable with $x'$.

\begin{center}
	% SPACETIME DIAGRAM with WORLD LINES
	\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}{
			\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, very thick] (O) -- (0,4*\d)
		node[mydarkred,above right] {\contour{white}{cat: $x'(t)=0$}};
		\draw[vector,myblue, very thick] (O) -- (-4*\d,4*\d)
		node[mydarkblue,below left=0] {\contour{white}{you: $x'(t)=-t$}};
		%\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};
	\end{tikzpicture}
\end{center}

\problem{}
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
Would there be any change when the cat is on the train? Why or why not?


\makeatletter
\if@solutions\else
	\emptydiagram{Cat}
\fi
\makeatother

\begin{solution}
	\begin{center}
		\begin{tikzpicture}[scale=1.8]
		\def\ymin{0.2}
		\def\xmin{4}
		\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);
		\draw[world line] (-4*\d,-\ymin) -- (-4*\d,\xmax);
		\draw[world line] (-5*\d,-\ymin) -- (-5*\d,\xmax);
		\draw[world line] (-6*\d,-\ymin) -- (-6*\d,\xmax);
		\draw[world line] (-7*\d,-\ymin) -- (-7*\d,\xmax);
		\draw[world line] (-8*\d,-\ymin) -- (-8*\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) -- (-8*\d,4*\d)
		node[mydarkred,below left=0] {\contour{white}{you: $x(t)=-2t$}};
		\draw[vector,myblue] (O) -- (0,4*\d)
		node[mydarkblue,above left=0] {\contour{white}{cat: $x(t)=0$}};
		\draw[vector,mygreen] (O) -- (-9*\d,3*\d)
		node[mydarkgreen,below left=0] {\contour{white}{dog: $x(t)=-3t$}};
		\draw[vector,black] (O) -- (-2*\d,1*\d) -- (-6*\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
\pagebreak


\definition{Reference Frame}
When we view the world from the perspective of different objects,
we say that we are working in different {\em reference frames}.
The original Example 1, where you are stationary, is your reference frame.
The new plot in Example 5, where your cat is stationary, is your cat's reference frame.

If we want to compare what is happening in multiple reference frames at once, we can graph
multiple spacetime grids on one plot. If we overlay the cat's reference frame onto your reference frame,
we can visualize everything in Example 1 as:

% SPACETIME DIAGRAM - GALILEAN TRANSFORMATION
\begin{center}
	\begin{tikzpicture}[scale=1.8]
		\def\xmax{2}
		\def\xmaxp{2.1} % maximum of rotated axis
		\def\Nlines{4} % number of world lines (at constant x/t)
		\pgfmathsetmacro\d{0.9*\xmax/\Nlines} % grid size
		\pgfmathsetmacro\ang{atan(1)} % angle
		\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
		\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-tan(\ang)*\x},-\x) -- (\xmax,-\x);
			\draw[world line t] (-\xmax,\x) -- ({\xmax+tan(\ang)*\x},\x);
		}

		% AXES
		\draw[->,thick] (0,-\xmax) -- (T) node[left=0] {$t$};
		\draw[->,thick] (-\xmax,0) -- (X) node[right=6,below=-1] {$x={\color{mydarkred}x'}$};
		\draw[->,thick,mydarkred, very thick] (90-\ang:-\xmaxp) -- (T')
		node[left=-1] {$t'$}
		node[right=2,below right=-2] {cat: $x(t) = t$};
		% VECTORS
		\draw[vector,myblue, very thick] (O) -- (0,4*\d)
		node[mydarkblue,below left=0] {\contour{white}{you: $x(t)=0$}};

		% WORLD LINES GRID - BOOSTED
		\message{  Making world lines, boosted...^^J}
		\fill[mydarkred,opacity=0.05]
		(O) --++ (90-\ang:\xmax) --++ (\xmax,0) --++ (90-\ang:-\xmax) -- cycle;
		\fill[mydarkred,opacity=0.05]
		(O) --++ (90-\ang:-\xmax) --++ (-\xmax,0) --++ (90-\ang:\xmax) -- cycle;
		\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
			\draw[world line'] (\x,0) --++ (90-\ang:\xmax);
			\draw[world line'] (-\x,0) --++ (90-\ang:-\xmax);
		}

		\node[right] at (5.6*\d, \d) {$x=\color{mydarkred}x' + 1$};
		%\draw pic[<-,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = T'--O--T};
	\end{tikzpicture}
\end{center}

Here $x,t$ are the spacetimes coordinates in your perspective and $x',t'$ are
the spacetime coordinates in your cat's perspective. Note that $t = t'$ for any point while $x = x' + t$.



\problem{}
What does it mean for two events to lie on the same vertical (blue) line from your perspective? \par
What does it mean for two events to lie on the same slanted (red) line from your cat's perspective?
\vfill

\begin{solution}
	Two events lie on the same vertical line if they occur at the same location in your reference frame. \par
	Two events lie on the same slanted line if they occur at the same location in your cat's reference frame.
\end{solution}


% \remark{}
%     Here we are forcing time in the cat's reference frame to behave the same as in our reference frame. As in, one second for the cat is one second for us and vice versa. However, looking at the plot and measuring distances, it almost looks like one second for us is longer than one second for the cat... Suspicious...


\problem{}
In the situation from Problem 2, when will your hamster catch up to your cat? \\
Choose the most convenient reference frame to work in, you shouldn't have to do much math.

\begin{solution}
	Using the cat's reference frame drawn in Problem 6, the hamster will catch up to the cat at $t = 8$.
\end{solution}

\vfill
\pagebreak
