281 lines
11 KiB
TeX
281 lines
11 KiB
TeX
\section{Special Relativity}
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Galilean relativity is nice until we start going really, {\em really} fast.
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Since most of us are terribly slow, we can use it without any issues.
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However, in reality, things are much weirder. In particular, there is a maximum speed: the speed of light,
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$$c = 299,792,458 \tfrac{m}{s}.$$
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Nothing can move faster than the speed of light and {\bf in every reference frame, light will move at this speed}.
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Let's see if this is consistent with Galilean relativity.
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We are going to making things easier for ourselves now and change units. Instead of measuring time $t$, we are now going to measure $ct$.
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\problem{}<photon diagram>
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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 $ct$.
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\halfdiagramc{You}
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\begin{solution}
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\begin{center}
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\begin{tikzpicture}[scale=2.0]
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\def\ymin{0.2}
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\def\xmin{1.6}
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\def\xmax{2}
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\def\Nlines{4} % number of world lines (at constant x/t)
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\pgfmathsetmacro\d{0.9*\xmax/\Nlines} % grid size
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\coordinate (O) at (0,0);
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\coordinate (T) at (0,\xmax+0.2);
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% WORLD LINES GRID
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\message{ Making world lines...^^J}
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\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
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\draw[world line] ( \x,-\ymin) -- ( \x,\xmax);
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\draw[world line t] (-\xmin, \x) -- (\xmax, \x);
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}
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\draw[world line] (-\d,-\ymin) -- (-\d,\xmax);
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\draw[world line] (-2*\d,-\ymin) -- (-2*\d,\xmax);
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\draw[world line] (-3*\d,-\ymin) -- (-3*\d,\xmax);
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% AXES
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\draw[->,thick] (0,-\ymin) -- (T) node[left=-1] {$ct$};
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\draw[->,thick] (-\xmin,0) -- (\xmax+0.2,0) node[below=0] {$x$};
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% VECTORS
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\draw[vector,myred] (O) -- (0,4*\d)
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node[mydarkred,below left=0] {\contour{white}{you: $x(t)=0$}};
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\draw[photon,shorten >=2] (O) -- (4*\d, 4*\d)
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node[black, above right] {\contour{white}{photon}};
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%\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};
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\end{tikzpicture}
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\end{center}
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\end{solution}
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\problem{}
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Suppose you are now sitting on a train that is moving to the right at $c/2$
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and again send one photon to your right.
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Draw this diagram in the reference frame of the ground.
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Draw your (Galilean) reference frame on top of this diagram
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What is the speed of the photon in your reference frame? Is that a problem?
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\halfdiagramc{Train}
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\begin{solution}
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\begin{center}
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\begin{tikzpicture}[scale=1.5]
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\pgfmathsetmacro\ang{atan(1/2)} % angle
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\def\ymin{0.2}
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\def\xmin{1.6}
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\def\xmax{2}
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\def\Nlines{4} % number of world lines (at constant x/t)
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\pgfmathsetmacro\d{0.9*\xmax/\Nlines} % grid size
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\coordinate (O) at (0,0);
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\coordinate (T) at (0,\xmax+0.2);
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% WORLD LINES GRID
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\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
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\draw[world line] ( \x,-\ymin) -- ( \x,\xmax);
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\draw[world line t] (-\xmin, \x) -- (\xmax, \x);
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}
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\draw[world line] (-\d,-\ymin) -- (-\d,\xmax);
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\draw[world line] (-2*\d,-\ymin) -- (-2*\d,\xmax);
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\draw[world line] (-3*\d,-\ymin) -- (-3*\d,\xmax);
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% AXES
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\draw[->,thick] (0,-\ymin) -- (T) node[left=-1] {$ct$};
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\draw[->,thick] (-\xmin,0) -- (\xmax+0.2,0) node[below=0] {$x=\color{mydarkred}x'$};
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% VECTORS
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\draw[vector,myred] (O) -- (2*\d,4*\d)
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node[mydarkred,below left=0] {\contour{white}{you: $x(t)=ct/2$}};
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\draw[photon,shorten >=2] (O) -- (4*\d, 4*\d)
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node[black, above right] {\contour{white}{photon}};
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%\node[right=8,above,mydarkpurple] at (T) {$x(t)=0$};
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% WORLD LINES GRID - BOOSTED
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\fill[mydarkred,opacity=0.05]
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(O) --++ (90-\ang:\xmax) --++ (\xmax,0) --++ (90-\ang:-\xmax) -- cycle;
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% \fill[mydarkred,opacity=0.05]
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% (O) --++ (90-\ang:-\xmax) --++ (-\xmax,0) --++ (90-\ang:\xmax) -- cycle;
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\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
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\draw[world line'] (\x,0) --++ (90-\ang:\xmax);
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% \draw[world line'] (-\x,0) --++ (90-\ang:-\xmax);
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}
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\end{tikzpicture}
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\end{center}
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In your reference frame, the light is only moving at speed $c/2$. Uh oh.
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\end{solution}
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\pagebreak
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\problem{}
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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.
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Can you brainstorm any ways to fix Galilean relativity to account for this absolute speed of light?
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{\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.
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{\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?
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{\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.}
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\emptydiagramc{Train}
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\pagebreak
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\definition{Lorentz Boost}
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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 $x'$.
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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:
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% SPACETIME DIAGRAM - LORENTZ BOOST
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\begin{center}
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\begin{tikzpicture}[scale=1.8]
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\def\xmax{2}
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\def\xmaxp{2.2} % maximum of rotated axis
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\def\Nlines{5} % number of world lines (at constant x/t)
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\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
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\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
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\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
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\coordinate (O) at (0,0);
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\coordinate (X) at (\xmax+0.2,0);
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\coordinate (T) at (0,\xmax+0.2);
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\coordinate (X') at (\ang:\xmaxp+0.2);
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\coordinate (T') at (90-\ang:\xmaxp+0.2);
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% WORLD LINE GRID
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\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
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\draw[world line] (-\x,-\xmax) -- (-\x,\xmax);
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\draw[world line] ( \x,-\xmax) -- ( \x,\xmax);
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\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
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\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
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}
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% BOOSTED WORLD LINE GRID
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\message{ Making world lines for boosted frame...^^J}
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\fill[mydarkred,opacity=0.05]
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(O) --++ (\ang:\xmaxp) --++ (90-\ang:\xmaxp) --++ (\ang:-\xmaxp) -- cycle;
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\fill[mydarkred,opacity=0.05]
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(O) --++ (\ang:-\xmaxp) --++ (90-\ang:-\xmaxp) --++ (\ang:\xmaxp) -- cycle;
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\foreach \i [evaluate={\x=\i*\D;}] in {1,...,4}{
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\draw[world line'] (\ang:-\x) --++ (90-\ang:-\xmaxp);
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\draw[world line'] (90-\ang:-\x) --++ (\ang:-\xmaxp);
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\draw[world line'] (\ang:\x) --++ (90-\ang:\xmaxp);
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\draw[world line'] (90-\ang:\x) --++ (\ang:\xmaxp);
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}
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% AXES
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\draw[->,thick] (0,-\xmax) -- (T) node[left=-1] {$ct$};
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\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
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\draw[->,thick,mydarkred] (90-\ang:-\xmaxp) -- (T')
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node[right=5,above=-1] {you: $ct'$};
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\draw[->,thick,mydarkred] (\ang:-\xmaxp) -- (X') node[right=-1] {$x'$};
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% ANGLES
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\draw pic[->,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = X--O--X'};
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\draw pic[<-,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = T'--O--T};
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% PHOTON
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\draw[photon] (O) --++ (5*\d, 5*\d)
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node[black, above right] {\contour{white}{photon}};
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\end{tikzpicture}
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\end{center}
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As before, the slanted (red) axes are your reference frame as you're moving. \\
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Note that you (and anything moving the same speed as you) are stationary in this reference frame.
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\remark{}
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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.
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\problem{}
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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.
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\begin{solution}
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This follows by just extending the boosted axes to the second quadrant and drawing the photon's worldline.
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\end{solution}
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\pagebreak
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% \begin{solution}\begin{center}\begin{tikzpicture}[scale=1.8]
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% \message{Lorentz boost^^J}
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% \def\xmax{2}
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% \def\xmaxp{2.2} % maximum of rotated axis
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% \def\Nlines{5} % number of world lines (at constant x/t)
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% \pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
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% \pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
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% \pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
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% \coordinate (O) at (0,0);
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% \coordinate (X) at (\xmax+0.2,0);
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% \coordinate (T) at (0,\xmax+0.2);
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% \coordinate (X') at (\ang:\xmaxp+0.2);
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% \coordinate (T') at (90-\ang:\xmaxp+0.2);
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% % WORLD LINE GRID
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% \message{ Making world lines...^^J}
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% \foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
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% \message{ Running i/N=\i/\Nlines, x=\x...^^J}
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% \draw[world line] (-\x,-\xmax) -- (-\x,\xmax);
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% \draw[world line] ( \x,-\xmax) -- ( \x,\xmax);
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% \draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
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% \draw[world line t] (-\xmax, \x) -- (\xmax, \x);
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% }
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% % BOOSTED WORLD LINE GRID
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% \message{ Making world lines for boosted frame...^^J}
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% \fill[mydarkred,opacity=0.05]
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% (O) --++ (\ang:\xmaxp) --++ (90-\ang:\xmaxp) --++ (\ang:-\xmaxp) -- cycle;
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% \fill[mydarkred,opacity=0.05]
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% (O) --++ (\ang:-\xmaxp) --++ (90-\ang:-\xmaxp) --++ (\ang:\xmaxp) -- cycle;
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% \foreach \i [evaluate={\x=\i*\D;}] in {1,...,4}{
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% \message{ Running i/N=\i/\Nlines, x=\x...^^J}
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% \draw[world line'] (\ang:-\x) --++ (90-\ang:-\xmaxp);
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% \draw[world line'] (90-\ang:-\x) --++ (\ang:-\xmaxp);
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% \draw[world line'] (\ang:\x) --++ (90-\ang:\xmaxp);
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% \draw[world line'] (90-\ang:\x) --++ (\ang:\xmaxp);
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% }
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% % AXES
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% \draw[->,thick] (0,-\xmax) -- (T) node[left=-1] {$ct$};
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% \draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
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% \draw[->,thick,mydarkred] (90-\ang:-\xmaxp) -- (T')
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% node[right=5,above=-1] {you: $ct'$};
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% \draw[->,thick,mydarkred] (\ang:-\xmaxp) -- (X') node[right=-1] {$x'$};
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% % ANGLES
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% \draw pic[->,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = X--O--X'};
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% \draw pic[<-,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = T'--O--T};
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% % PHOTON
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% \draw[photon] (O) --++ (4*\d,4*\d);
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% \end{tikzpicture}\end{center}\end{solution}
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\problem{}
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A caveat to Lorentz boosts is that we cannot boost to reference frames which are at the speed of light or
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faster. Based on the diagram given, why can't we do that? \\
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{\em "If my calculations are correct, when this baby hits $c$, you're gonna see some serious stuff."}
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\begin{solution}
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If boosted to $c$, our axes would overlap, compressing time and space into one.
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If we boosted past $c$, our axes would flip, making the past the future and the future the past.
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\end{solution}
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\vfill
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\problem{}
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This diagram implies some strange things. We'll spend the next sections discussing
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some consequences of this, but take a minute to note anything weird that you notice.
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In particular, look at a unit of time (or a unit of length) in your frame vs the rest frame.
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Which is longer, one unit of time in your reference frame or in the rest frame?
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Which is longer, one unit of distance in your reference frame or in the rest frame?
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Consider the implications of a slanted space line.
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What does it mean if two events both lie on this line?
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\vfill
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\pagebreak |