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\section{Electromagnetism and Relativity}

\subsubsection*{Review of special relativity}

\subsubsection*{Lorentz transformation}

In the special theory of raltivity we use
spacetime coordinates
\[ X^\mu = (ct, x, y, z) = (ct, \bf{x}) \]
(where $c$ is the speed of light) to describe time
and position in an inertial frame $S$.

\myskip
Greek indices such as $\mu$ and $\nu$ run from $0$
to $3$ to cover the dimensions of spacetime. Roman
indices such as $i$ and $j$ run from $1$ to $3$ to
cover the dimensions of space.

\myskip
$X$ is the \emph{position 4-vector}. Under a
\emph{Lorentz transformation} (LT) from $S$ to
another inertial frame $S'$, it transforms
according to
\[ X' = \Lambda X \]
where $\Lambda$ is the $4 \times 4$ \emph{LT
matrix}. In index notation,
\[ X'^\mu = \Lambda^\mu{}_\nu X^\nu \]
where the summation convention implies a sum from
$0$ to $3$ over the repeated index $\nu$. We have
\[ \pfrac{X'^\mu}{X^\nu} = \lambda^\mu{}_\nu .\]
An important example of an LT is a \emph{Lorentz
boost} in standard configuration: $S'$ moves
relative to $S$ with velocity $v$ ($|v| < c$) in
the $x$-direction.
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/5678e84eb74911ed.png}
\end{center}
Then
\begin{align*}
    t'
    &= \gamma \left( t - \frac{vx}{c^2} \right) \\
    x'
    &= \gamma (x - vt) \\
    y'
    &= y \\
    z'
    &= z
\end{align*}
where $\gamma = \frac{1}{\sqrt{1 -
\frac{v^2}{c^2}}}$ is the \emph{Lorentz factor}.
The corresponding matrix is
\[ \Lambda^\mu{}_\nu =
\begin{pmatrix}
    \gamma & -\gamma \beta & 0 & 0 \\
    -\gamma \beta & \gamma & 0 & 0 \\
    0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 1
\end{pmatrix}
\]
with $\beta = \frac{v}{c} \in (-1, 1)$, $\gamma =
\frac{1}{\sqrt{1 - \beta^2}} \ge 1$. A more
general LT may combine a boost with spatial
isometries (rotations and reflections).

\subsubsection*{Four-vectors}

Any \emph{4-vector} $V$ transforms as (1) under a
LT:
\[ V' = \Lambda V \]
The \emph{inner product} of two 4-vectors $V$ and
$W$ is
\[ V \cdot W = \eta_{\mu \nu} V^\nu W^\mu \]
where $\eta_{\mu \nu}$ is the \emph{Minkowski
metric tensor}. This is isotropic, having the same
components
\[ \eta_{\mu \nu}
\begin{pmatrix}
    -1 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0 \\
    0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 1
\end{pmatrix}
= \diag(-1, 1, 1, 1) \]
in any inertial frame. Thus
\[ V \cdot W = -V^0 W^0 + V^1 W^1 + V^2 W^2 + V^3
W^3 .\]
The inner product is \emph{invariant} under a LT:
\begin{align*}
    V' \cdot W'
    &= \eta'_{\mu \nu} V'^\mu W'^\nu \\
    &= \eta_{\mu\nu} \Lambda^\mu{}_\rho V^\rho
    \Lambda^\nu{}_\sigma W^\sigma \\
    &= \eta_{\rho\sigma} V^\rho W^\sigma \\
    &= V \cdot W
\end{align*}
because the LT matrix satisfies the equation
\[ \eta_{\mu \nu} \Lambda^\mu{}_\rho
\Lambda^\nu{}_\sigma = \eta_{\rho\sigma} \]
In matrix notation:
\[ \Lambda^\top \eta \Lambda = \eta \]
which implies $\det(\Lambda) = \pm 1$.

\myskip
The \emph{square} of a 4-vector $V$,
\begin{align*}
    V \cdot V
    &= \eta_{\mu \nu} V^\mu V^\nu \\
    &= -V^0 V^0 + V^i V^i \\
    &= -(V^0)^2 + (V^1)^2 + (V^2)^2 + (V^3)^2
\end{align*}
is also Lorentz-invariant. $V$ is:
\begin{itemize}
    \item \emph{timelike} if $V \cdot V < 0$.
    \item \emph{spacelike} if $V \cdot V > 0$.
    \item \emph{null} (or \emph{lightlike}) if $V
        \cdot V = 0$.
\end{itemize}
An important example is the invariant
\emph{spacetime interval} between two events
\[ \Delta s^2 = \Delta X \cdot \Delta X = -c^2
\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 \]
This is also useful in differential form
\[ \dd s^2 = \dd X \cdot \dd X = -c^2 \dd t^2 +
\dd x^2 + \dd y^2 + \dd z^2 \]

\subsubsection*{Proper time and four-velocity}

A particle traces out a curve in spacetime, called
its \emph{world line}.

\myskip
Let the world line in $S$ be $X^\mu(\lambda)$,
parametrised by the real variable $\lambda$. Then
$V^\mu = \dfrac{X^\mu}{\lambda}$ is a 4-vector.

\myskip
The curve is timelike if $V^\mu$ is timelike,
etc, and this property is independent of how the
curve is parametrised.

\myskip
MAssive particles travel slower than light and
have timelike world lines. We can use the
\emph{proper time} $\tau$ (the time experienced by
the particle) to parametrise the world line.
(Analogous to arclength in Euclidean space).

\myskip
$\tau(\lambda)$ can be defined by
\begin{align*}
    c\dfrac{\tau}{\lambda}
    &= \sqrt{-V \cdot V} \\
    \implies \tau
    &= \frac{1}{c} \int \sqrt{-\eta_{\mu \nu}
    \dfrac{X^\mu}{\lambda} \frac{X^\nu}{\lambda}}
    \dd \lambda
\end{align*}
(with an arbitrary additive constant). By the
chain rule, the RHS, and therefore $\tau$, are
invariant under a reparametrisation $\lambda
\mapsto \tilde{\lambda}(\lambda)$.

\myskip
The differential form is
\[ c^2 \dd \tau^2 = -\dd X \cdot \dd X = -\dd s^2
> 0 \]
The \emph{4-velocity} of a massive particle is
\[ U = \dfrac{X}{\tau} \]
and satisfies $U \cdot U = -c^2$
(Lorentz-invariant, as expected). Since $X^\mu =
(ct, \bf{x})$,
\[ U^\mu = \dfrac{t}{\tau} (c, \bf{u}) \]
where $\bf{u} = \dfrac{\bf{x}}{t}$ is the
3-velocity. Thus
\[ U \cdot U = \left( \dfrac{t}{\tau} \right)^2
(-c^2 + |\bf{u}|^2) = -c^2 ,\]
from which
\[ \dfrac{t}{\tau} = \frac{1}{\sqrt{1 -
\frac{|\bf{u}|^2}{c^2}}} = \gamma_{\bf{u}} .\]
Thus $U^\mu = \gamma_{\bf{u}} (c, \bf{u})$.
