\section{Special Relativity}
Galilean relativity is nice until we start going really, {\em really} fast.
Since most of us are terribly slow, we can use it without any issues.
However, in reality, things are much weirder. In particular, there is a maximum speed: the speed of light,
$$c = 299,792,458 \tfrac{m}{s}.$$
Nothing can move faster than the speed of light and {\bf in every reference frame, light will move at this speed}.

Let's see if this is consistent with Galilean relativity.
We are going to making things easier for ourselves now and change units. Instead of measuring time $t$, we are now going to measure $ct$.

\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 $ct$.


\makeatletter
\if@solutions\else
	\halfdiagramc{You}
\fi
\makeatother

\begin{solution}
	\begin{center}
		\begin{tikzpicture}[scale=2.0]
			\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] {$ct$};
			\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}{you: $x(t)=0$}};
			\draw[photon,shorten >=2] (O) -- (4*\d, 4*\d)
			node[black, above right] {\contour{white}{photon}};
			%\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};
		\end{tikzpicture}
	\end{center}
\end{solution}

\vfill

\problem{}
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.
Draw this diagram in the reference frame of the ground.

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?

\makeatletter
\if@solutions\else
	\halfdiagramc{Train}
\fi
\makeatother

\begin{solution}
	\begin{center}
		\begin{tikzpicture}[scale=1.5]
			\pgfmathsetmacro\ang{atan(1/2)} % angle
			\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
			\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] {$ct$};
			\draw[->,thick] (-\xmin,0) -- (\xmax+0.2,0) node[below=0] {$x=\color{mydarkred}x'$};

			% VECTORS
			\draw[vector,myred] (O) -- (2*\d,4*\d)
			node[mydarkred,below left=0] {\contour{white}{you: $x(t)=ct/2$}};
			\draw[photon,shorten >=2] (O) -- (4*\d, 4*\d)
			node[black, above right] {\contour{white}{photon}};
			%\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};

			% WORLD LINES GRID - BOOSTED
			\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);
			}
		\end{tikzpicture}
	\end{center}

	In your reference frame, the light is only moving at speed $c/2$. Uh oh.
\end{solution}


\vfill
\pagebreak


\problem{}
Clearly, Galilean relativity and the absolute speed of light do not mix well together. Having noticed this, you are now in the same boat as early 20th century physicists.
Can you brainstorm any ways to fix Galilean relativity to account for this absolute speed of light?

{\em Hint 1}: Try different methods of drawing the axes of your reference frame that would maintain the speed of light in both the rest frame and in your reference frame.

{\em Hint 2}: The photon worldline always bisects the angle between the space and time axes. \\ Is there a way that you can make that happen in your reference frame?

{\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.}

\makeatletter
\if@solutions\else
	\emptydiagramc{Train}
\fi
\makeatother



\vfill
\pagebreak


\definition{Lorentz Boost}
Looking at our spacetime diagram from \ref{photon diagram}, we see that the photon worldline bisects the angle between the $ct$ axis and $x$ axis. So if we want to maintain this speed in all reference frames, we just need to make sure that photons bisect our new time axis $ct'$ and our new space axis $x'$.

In order to do this, we're going to to rotate our space axis $x'$ by the same angle that our $ct'$ axis is rotated. Rotating both axes like this is called a {\em Lorentz boost} and is best visualized in the following diagram:

% SPACETIME DIAGRAM - LORENTZ BOOST
\begin{center}
	\begin{tikzpicture}[scale=1.8]
		\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 LINE 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,-\x) -- (\xmax,-\x);
			\draw[world line t] (-\xmax, \x) -- (\xmax, \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}{
			\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);
		}

		% AXES
		\draw[->,thick] (0,-\xmax) -- (T) node[left=-1] {$ct$};
		\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
		\draw[->,thick,mydarkred] (90-\ang:-\xmaxp) -- (T')
		node[right=5,above=-1] {you: $ct'$};
		\draw[->,thick,mydarkred] (\ang:-\xmaxp) -- (X') node[right=-1] {$x'$};

		% ANGLES
		\draw pic[->,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = X--O--X'};
		\draw pic[<-,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = T'--O--T};

		% PHOTON
		\draw[photon] (O) --++ (5*\d, 5*\d)
		node[black, above right] {\contour{white}{photon}};
	\end{tikzpicture}
\end{center}

As before, the slanted (red) axes are your reference frame as you're moving. \\
Note that you (and anything moving the same speed as you) are stationary in this reference frame.

\remark{}
In this diagram, we've not only rotated your space axis ($x'$), we've also adjusted the scale of $ct'$ and $x'$ relative to the rest frame. This scaling comes from physical experiments which we will conduct later.

\problem{}
Please verify that in the diagram above, if you shoot a photon behind you, it still moves at speed $c$ in both your reference frame and the ground's reference frame. You can do this by drawing directly on the diagram.

\begin{solution}
	This follows by just extending the boosted axes to the second quadrant and drawing the photon's worldline.
\end{solution}


\vfill
\pagebreak

% \begin{solution}\begin{center}\begin{tikzpicture}[scale=1.8]
%   \message{Lorentz boost^^J}

%   \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 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,-\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);
%   }

%   % 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);
%   }

%   % AXES
%   \draw[->,thick] (0,-\xmax) -- (T) node[left=-1] {$ct$};
%   \draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
%   \draw[->,thick,mydarkred] (90-\ang:-\xmaxp) -- (T')
%     node[right=5,above=-1] {you: $ct'$};
%   \draw[->,thick,mydarkred] (\ang:-\xmaxp) -- (X') node[right=-1] {$x'$};

%   % ANGLES
%   \draw pic[->,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = X--O--X'};
%   \draw pic[<-,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = T'--O--T};

%   % PHOTON
%   \draw[photon] (O) --++ (4*\d,4*\d);

% \end{tikzpicture}\end{center}\end{solution}

\problem{}
A caveat to Lorentz boosts is that we cannot boost to reference frames which are at the speed of light or
faster. Based on the diagram given, why can't we do that? \\
{\em "If my calculations are correct, when this baby hits $c$, you're gonna see some serious stuff."}

\begin{solution}
	If boosted to $c$, our axes would overlap, compressing time and space into one.
	If we boosted past $c$, our axes would flip, making the past the future and the future the past.
\end{solution}
\vfill

\problem{}
This diagram implies some strange things. We'll spend the next sections discussing
some consequences of this, but take a minute to note anything weird that you notice.

In particular, look at a unit of time (or a unit of length) in your frame vs the rest frame.

Which is longer, one unit of time in your reference frame or in the rest frame?

Which is longer, one unit of distance in your reference frame or in the rest frame?

Consider the implications of a slanted space line.
What does it mean if two events both lie on this line?

\vfill
\pagebreak