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\begin{titlepage} \setcounter{page}{0} \vspace{2.5cm}
\begin{center}
\Large\bf{Special Relativity and Maxwell Equations}\\
\vspace{2.0cm} \large{by}\\ \vspace{2.0cm}
\large{\bf Bernd A. Berg }\\ \vspace{2.0cm}
{\bf Department of Physics}\\ \vspace{0.4cm}
{\bf Florida State University}\\ \vspace{0.4cm}
{\bf Tallahassee, FL~32306, USA. } \vspace{2.0cm} \vfill

{\bf Copyright \copyright\ by the author.}\\

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\chapter{}\label{Relativity}

% \chapter{Special Relativity and Maxwell Equations
% (by B.A. Berg)} \label{Relativity} ! For PHY4241 (W/S 2004, 2009).

A self-contained summary of the theory of special relativity is given,
which provides the space-time frame for classical electrodynamics.
Historically~\cite{Ei1905} special relativity emerged
out of electromagnetism. Nowadays, is deserves to be emphasized
that special relativity restricts severely the possibilities for
electromagnetic equations. 

\section{Special Relativity}

Let us deal with space and time in vacuum. The conventional time 
unit is one 
\begin{equation} \label{second}
  second~[s].
\end{equation}
Here, and in the following abbreviations for units are placed in 
brackets~$[~]$. For a long time period the second was 
defined in terms of the rotation of the earth as 
${1\over 60}\times {1\over 60}\times {1\over 24}$ of the mean solar day. 
Nowadays most accurate time measurements rely on atomic clocks. They 
work by tuning a electric frequency into resonance with some atomic
transition.
Consequently, the second has been re--defined, so that the frequency
of the light between the two hyperfine levels of the ground state of the
cesium $^{132}Cs$ atom is now exactly 9,192,631,770 cycles per second. 

Special relativity is founded on two basic postulates:

\begin{description}
\item{{\bf 1.}} Galilee invariance: The laws of nature are independent of 
any uniform, translational motion of the reference frame.
\end{description}

This postulate gives rise to a triple infinite set of reference frames 
moving with constant velocities relative to one another. They are called 
{\it inertial frames}. For a freely moving body, i.e. a body which is not 
acted upon by an external force, inertial systems exist. The differential 
equations which describe physical laws take the same form in all inertial 
frames. {\it Galilee invariance} was known long before Einstein. 

\begin{description}
\item{{\bf 2.}} The speed $c$ of light in empty space is independent 
of the motion of its source.
\end{description}

The second Postulate was introduced by Einstein 1905~\cite{Ei1905}. It
implies that $c$ takes the same constant value in all inertial frames.
Transformations between inertial frames follow, which have far reaching 
physical consequences.

The distance unit 
\begin{equation} \label{meter}
 1~meter~ [m] = 100~centimeters~[cm] 
\end{equation}
was originally defined by two scratches on a bar made of 
platinum--iridium alloy kept at the International Bureau of Weights 
and Measures in S\`evres, France. As measurements of the speed of 
light have become increasingly accurate, it has become most 
appropriate to exploit Postulate~2 to define the distance unit. The
standard meter is now defined \cite{Petley}
as the distance traveled by light in
empty space during the time of 1/299,792,458~$[s]$. This makes the 
speed of light exactly 
\begin{equation} c=299,792,458~[m/s] .\label{light}\end{equation}

\subsection{Natural Units}

The units for $second$~(\ref{second} and $meter$~(\ref{meter}) are
not independent, as the speed of light is an universal constant.
This allows to define natural units, which are frequently used in 
nuclear, particle and astro physics. They define 
\begin{equation} \label{natural_units}
c=1
\end{equation}
as a dimensionless constant, and 
$$ 1~[s] = 299,792,458~[m]  $$
holds. The advantage of natural units is that factors of $c$ disappear 
in calculations. The disadvantage is that, for converting back to 
conventional units, the appropriate factors have to be recovered by 
dimensional analysis. For instance, if time is given in seconds $x=t$
in natural units converts to $x=ct$ with $x$ in meters and $c$
given by~(\ref{light}).

\subsection{Definition of distances and synchronization of clocks}

Let us relate the introduced concepts of Galilee invariance and 
of a constant speed of light to physical measurements and reduce 
measurements of spatial distances to time measurements.
Let us consider an inertial frame $K$ with coordinates $(t,\vec{x})$.
We like to place observers at rest at different places $\vec{x}$ in 
$K$. The observers are equipped with clocks of identical making, 
to define the time $t$ at $\vec{x}$. The origin $\vec{x}=0$, in
other notation $(t,\vec{0})$, of $K$ is defined by placing observer
$O_0$ and his clock there. We like to place another observer $O_1$
at $\vec{x}_1$ to define $(t,\vec{x}_1)$. How can $O_1$ know to be
at $\vec{x}_1$? By using a mirror he can reflect light flashed by
observer $O_0$ at him. Observer $O_0$ may measure the polar and
azimuthal angles $(\theta, \phi)$ at which he emits the light
and 
$$ |\vec{x}_1| = c\, \triangle t / 2\, , $$ 
where $\triangle t$ is the time light needs to travel to $O_1$ 
and back. This determines $\vec{x}_1$ and he can signal this information 
to $O_1$. By repeating the measurement, he can make sure that $O_1$ is 
not moving with respect to $K$. For an idealized, force free environment 
the observers will then never start moving with respect to one another. 
$O_1$ synchronizes his clock by setting it to
$$ t_1=t_0+|\vec{x}_1|/c $$
at the instant receiving the signal. If $O_0$ flashes again his instant 
time $t_{01}$ over to $O_1$, the clock of $O_1$ will show time $t_{11}=
t_{01}+|\vec{x}_1|/c$ at the instant of receiving the signal. In the 
same way the time $t$ can be defined at any desired point $\vec{x}$ 
in $K$. 

Now we consider an inertial frame $K'$ with coordinates 
$(t',\vec{x}')$, moving with constant velocity $\vec{v}$ with respect
to $K$. The origin of $K'$ is defined through a third observer $O'_0$.
What does it mean that $O'_0$ moves with constant velocity $\vec{v}$
with respect to $O_0$? At times $t^e_{01}$ and $t^e_{02}$ observer 
$O_0$ may flash light signals (the superscript $e$ stands for
``emit'') at $O'_0$, which are reflected and arrive back after 
time intervals $\triangle t_{01}$ and $\triangle t_{02}$. From 
principle~2 it follows that the reflected light needs the same time 
(measured by $O_0$ in $K$) to travel from $O'_0$ to $O_0$, as it 
needed to travel from $O_0$ to $O'_0$. Hence, $O_0$ concludes that 
$O_0'$ received the signal at
\begin{equation}
  t_{0i}=t^e_{0i}+\triangle t_{0i}/2,\ (i=1,2) 
\end{equation}
in the $O_0$ time. This simple equation becomes quite complicated for 
non-re\-la\-ti\-vistic physics, because the speed on the return path 
would then be distinct from that on the arrival path (consider for 
instance elastic scattering of a very light particle on a heavy 
surface). The constant velocity of light implies that relativistic 
distance measurements are simpler than such non-relativistic 
measurements. For observer $O_0$ the positions $\vec{x}_{01}$ 
and $\vec{x}_{02}$ are now defined through the angles $(\theta_{0i}, 
\phi_{0i})$ and 
\begin{equation}
  |\vec{x}_{0i}| = \triangle t_{0i}\, c/2,\ (i=1,2)\, .
\end{equation}
For the assumed force free environment observer $O_0$ can conclude 
that $O'_0$ moves with respect to him with uniform velocity 
\begin{equation}
\vec{v} = (\vec{x}_{02}-\vec{x}_{01}) / (t_{02}-t_{01})\, .
\end{equation}
Actually, one measurement is sufficient to obtain the velocity when
one employs the relativistic Doppler effect as discussed later in 
section~\ref{sec_Doppler}. 
$O_0$ may repeat the procedure at later times to check that $O_o'$ 
moves indeed with uniform velocity.

Similarly, observer $O'_0$ finds out that $O_0$ moves with velocity 
$\vec{v}{\, '}=-\vec{v}$. According to principle~1, observers in $K'$ 
can now go ahead to define $t'$ for any point $\vec{x}{\, '}$ in $K'$. 
The equation of motion for the origin of $K'$ is as observed by $O_0$ is
\begin{equation} \label{O'_0}
  \vec{x}\, (\vec{x}^{\, '}=0) = \vec{x}_0 + \vec{v}\, t , 
\end{equation}
with $\vec{x}_0=\vec{x}_{01}-\vec{v}\, t_{01}$, expressesing the fact 
that for $t=t_{01}$ observer $O'_0$ is at $\vec{x}_{01}$. Shifting his 
space convention by a constant vector, observer $O_0$ can achieve 
$\vec{x}_0=0$, so that equation (\ref{O'_0}) becomes
$$ \vec{x}\, (\vec{x}^{\, '}=0) = \vec{v}\, t . $$
Similarly, observer $O_0'$ may choose his space convention so that
$$ \vec{x}^{\, '}\, (\vec{x}=0) = - \vec{v}\, t' $$
holds. 

\subsection{Lorentz invariance and Minkowski space}

Having defined time and space operationally, let us focus on a more 
abstract discussion. We consider the two inertial frames with uniform 
relative motion $\vec{v}$: $K$ with coordinates $(t,\vec x)$ and $K'$ 
with coordinates $(t',\vec{x}{\, '})$. We demand that at time $t=t'=0$ 
their two origins coincide. Now, imagine a spherical shell of radiation 
originating at time $t=0$ from $\vec{x}=\vec{x}{\, '}=0$. The 
propagation of the wavefront is described by
\begin{equation}
  c^2 t^2 - x^2 - y^2 - z ^2 = 0 ~~~{\rm in}\ K ,  \label{inK}
\end{equation}
and by
\begin{equation}
  c^2 t'^{\,2}-x'^{\,2}-y'^{\,2}-z'^{\,2}= 0~~~{\rm in}\ K' .\label{inKP} 
\end{equation}
We define 4-vectors $(\alpha = 0,1,2,3)$ by
\begin{equation} \label{4x}
  (x^{\alpha})=\left( \matrix{ct\cr  \vec{x}\cr} \right) 
  ~~{\rm and}~~(x_{\alpha})=\left( ct,\ -\vec{x} \right)\, . 
\end{equation}
Due to a more general notation, which is explained in 
section~\ref{sec_tensor}, the components $x^{\alpha}$ are called 
{\it contravariant} and the components $x_{\alpha}$ {\it covariant}. 
In matrix notation the contravariant 4-vector $(x^{\alpha})$ is 
represented by a column and the covariant 4-vector $(x_{\alpha})$ 
as a row.

The {\it Einstein summation convention} is defined by
\begin{equation} \label{Einsum}
x_{\alpha} x^{\alpha} = \sum_{\alpha=0}^3 x_{\alpha} x^{\alpha} ,
\end{equation}
and will be employed from here on. Equations (\ref{inK}) and 
(\ref{inKP}) read then 
\begin{equation} \label{InvLight}
  x_{\alpha}x^{\alpha} = x'_{\, \alpha} x'^{\, \alpha} = 0\ . 
\end{equation}
{\it (Homogeneous) {Lorentz transformations}} are defined as the group 
of transformations which leave the distance $s^2=x_{\alpha}x^{\alpha}$
invariant:
\begin{equation} \label{HLo}
  x_{\alpha}x^{\alpha} = x'_{\alpha}x'^{\,\alpha} = s^2\ .
\end{equation}
This equation implies (\ref{InvLight}, but the reverse is not true.
An additional transformation, which leave (\ref{InvLight}, but not
(\ref{HLo}), invariant is $x'^{\,\alpha}=\lambda\,x^{\alpha}$, which
is interpreted as a scale transformation.

If the initial condition $t'=0$ and $\vec{x}^{\,'}(\vec{x}=0)
=\vec{x}(\vec{x}^{\,'})=0$ for $t=0$ is replaced by an arbitrary one, 
the equation $(x_{\alpha}-y_{\alpha})(x^{\alpha}-y^{\alpha}) =
 (x'_{\,\alpha}-y'_{\,\alpha})(x'^{\alpha}-y'^{\alpha}) $ still holds.
\index{}{\it Inhomogeneous Lorentz} or 
\index{}{\it Poincar\'e} transformations are
defined as the group of transformations which leave
\begin{equation} \label{ILo}
s^2=(x_{\,\alpha}-y_{\,\alpha})(x^{\alpha}-y^{\alpha})
  ~~~{\rm invariant.} 
\end{equation}
In contrast to the Lorentz transformations the Poincar\'e 
transformations include invariance under 
\index{}{\it translations} 
\begin{equation} \label{translations}
  x^{\alpha} \to x^{\alpha} + a^{\alpha} ~~{\rm and}~~
  y^{\alpha} \to y^{\alpha} + a^{\alpha} 
\end{equation}
where $a^{\alpha}$ is a constant vector. Independently of Einstein, 
Poincar\'e had developed similar ideas, but pursued a more cautious 
approach.

\begin{figure}[tb] \vspace{5pc}
\centerline{\hbox{ \psfig{figure=f01.eps,width=10cm} }}
\caption{Minkowski space: Seen from the spacetime point A at the origin,
the spacetime points in the forward light cone are in the future, those 
in the backward light cone are in the past and the spacelike points are 
``elsewhere'', because their time-ordering depends on the inertial 
frame chosen. Paths of two clocks which separate at the origin (the 
straight line one stays at rest) and merge again at a future space-time 
point B are also indicated. For the paths shown the clock moved along 
the curved path will, at B, show an elapsed time of about 70\% of the 
elapsed time shown by the other clock ({\it i.e.} the one which stays 
at rest).  \label{fig_Minkowski} }
\end{figure}

A fruitful concept is that of a 4-dimensional space-time, 
called {\it Minkowski space}. Equation (\ref{ILo}) gives the 
invariant metric of this space. Compared to the 
norm of 4-dimensional Euclidean space, the crucial difference 
is the relative minus sign between time and space components.
The \index{}{\it light cone} of a 4-vector $x^{\alpha}_0$ is 
defined as the set of vectors $x^{\alpha}$ which satisfy
$$ (x-x_0)^2 =
(x_{\alpha}-x_{0\alpha})\, (x^{\alpha}-x^{\alpha}_0) = 0 .$$
The light cone separates events which are \index{}{\it timelike} 
and \index{}{\it spacelike} with respect to $x^{\alpha}_0$, 
namely
$$ (x-x_0)^2 > 0 ~~{\rm for\ timelike} $$
and
$$ (x-x_0)^2 < 0 ~~{\rm for\ spacelike.} $$
We shall see soon, compare equation~(\ref{2DLT1}), that the time 
ordering of spacelike points is distinct in different inertial 
frames, whereas it is the same for timelike points. For the choice 
$x_0^{\alpha}=0$ this Minkowski space situation is depicted in 
figure~\ref{fig_Minkowski}. On the abscissa we have the 
projection of the three dimensional Euclidean space on 
$r=|\vec{x}|$.  The regions
\index{} {\it future} and \index{} {\it past} of this figure are 
the timelike points of $x_0=0$, whereas {\it elsewhere} are the
spacelike points.

To understand special relativity in some depth, we have to explore 
Lorentz and Poincar\'e transformations in some details. Before we 
come to this, we consider the two-dimensional case and introduce 
some relevant calculus in the next two sections.

\subsection{Two-Dimensional Relativistic Kinematics}

We chose now $\vec{v}$ in $x$-direction and restrict the discussion of 
to the $x$-axis: 
\begin{equation}
 c^2 t^2 - (x^1)^2 = c^2 t'^2 -(x^{'\, 1})^2 \label{2Dinvar}
\end{equation}
It is customary to define $x^0=ct$, $x^{'\, 0}=ct'$ and $\beta=v/c$.
We are looking for a linear transformation
\begin{equation}
  \pmatrix{x^{'\,0}\cr x^{'\,1}\cr} = \pmatrix{a&b\cr d&e\cr}
  \pmatrix{x^0\cr x^1\cr} \label{LTansatz}
\end{equation}
which fulfills (\ref{2Dinvar}) for all $x^0$, $x^1$. Choosing 
$\pmatrix{x^0\cr x^1\cr} = \pmatrix{1\cr 0\cr}$ gives
\begin{equation} \label{a00} 
  a^2-d^2=1\ \Rightarrow\ a=\cosh(\zeta ),\ d=\pm\sinh(\zeta )
\end{equation}
and choosing $\pmatrix{x^0\cr x^1\cr} = \pmatrix{0\cr 1\cr}$ gives
\begin{equation} \label{a11}
  b^2-e^2=-1\ \Rightarrow\ e=\cosh(\eta),\ b=\pm\sinh(\eta)\ . 
\end{equation}
Using now $\pmatrix{x^0\cr x^1\cr}=\pmatrix{1\cr 1\cr}$
yields 
$$ [\cosh (\zeta ) + \sinh (\eta ) ]^2 -
   [\sinh (\zeta ) + \cosh (\eta ) ]^2 = 0\
\Rightarrow\ \zeta = \eta . $$
In equation (\ref{a00}) $d=-\sinh(\zeta)$ is conventionally used. We 
end up with \begin{equation} \label{hyperbolic}
  \pmatrix{a&b\cr d& e\cr} = \pmatrix{\cosh(\zeta )&-\sinh(\zeta)
  \cr -\sinh(\zeta)& \cosh(\zeta)\cr} ,
\end{equation}
where $\zeta$ is called {\it rapidity} or {\it boost}
variable, and has the interpretation of an angle in
a hyperbolic geometry. For our present purposes no 
knowledge of hyperbolic geometries is required. In
components (\ref{hyperbolic}) reads
\begin{eqnarray} \label{2DLT1}
   x'^{\,0} &=& +\cosh(\zeta)\,x^0 -\sinh(\zeta)\,x^1\\
   x'^{\,1} &=& -\sinh(\zeta)\,x^0 +\cosh(\zeta)\,x^1 \label{2DLT2}
\end{eqnarray} 
An interesting feature of equation (\ref{LT1}) is that for spacelike 
points, say $x^1>x^0>0$, a value $\zeta_0$ for the rapidity exists, 
so that 
$$ 0 = +\cosh(\zeta)\,x^0 -\sinh(\zeta)\,x^1 $$
and, therefore,
$$ {\rm sign} (x'^{\, 0}) = - {\rm sign} (x^0) $$ 
for $\zeta > \zeta_0$, {\it i.e.} the time-ordering becomes 
reversed, whereas for timelike points such a reversal of the
time-ordering is impossible as then $|x^0>|>|x^1|$. In 
figure~\ref{fig_Minkowski} this is emphasized by calling the 
spacelike (with respect to $x_0=0$) region {\it elsewhere} in 
contrast to {\it future} and {\it past}.

The physical interpretation is straightforward. Seen from $K$, the 
origin $x'^{\,1}=0$ of $K$ moves with constant velocity $v$. In $K$ 
this corresponds to the equation 
$$ 0 = -\sinh(\zeta)\,x^0 +\cosh(\zeta)\,x^1 $$
and the rapidity is related to the velocity between the frames by
\begin{equation} \label{rapidity}
  \beta = {v\over c} = {x^1\over x^0} = 
  {\sinh(\zeta)\over \cosh(\zeta)} = \tanh(\zeta)\ . 
\end{equation}
Another often used notation is
\begin{equation} \label{gamma}
  \gamma = \cosh (\zeta ) = {1\over\sqrt{1-\beta^2}}~~{\rm and}
  ~~\gamma\,\beta = \sinh (\zeta )\ . 
\end{equation}
Hence, the transformation (\ref{LTansatz}) follows in 
the often stated form
\begin{eqnarray} \label{LT1a}
  x^{'\,0} &=& \gamma\,(x^0-\beta\,x^1)\, ,\\ \label{LT2a}
  x^{'\,1} &=& \gamma\,(x^1-\beta\,x^0)\, .
\end{eqnarray} 
These equation are called {\it Lorentz transformations}. Lorentz 
discovered them first in his studies of electrodynamics, but it 
remained due to Einstein \cite{Ei1905} to fully understand their 
physical meaning. We may perform two subsequent Lorentz 
transformations with rapidity $\zeta_1$ and $\zeta_2$. They 
combine as follows:
$$ \pmatrix{+\cosh (\zeta_2 ) & -\sinh (\zeta_2 )\cr 
           -\sinh (\zeta_2 ) & +\cosh (\zeta_2 )\cr}\
  \pmatrix{+\cosh (\zeta_1 ) & -\sinh (\zeta_1 )\cr 
           -\sinh (\zeta_1 ) & +\cosh (\zeta_1 )\cr}  $$
\begin{equation}
  \pmatrix{+\cosh (\zeta_2+\zeta_1 ) & -\sinh (\zeta_2+\zeta_1 )\cr 
           -\sinh (\zeta_2+\zeta_1 ) & +\cosh (\zeta_2+\zeta_1 )\cr}\ .
\end{equation}
The rapidities add up as
\begin{equation} \label{rapidity_add} 
  \zeta = \zeta_1 + \zeta_2
\end{equation}
in the same way as velocities do under Galilei transformations or 
angles for rotations about the same axis. Note that the inverse to 
the transformation with rapidity $\zeta_1$ is obtained for $\zeta_2
=-\zeta_1$. The relativistic addition of velocities follows from 
(\ref{rapidity_add}). Let $\beta_1=\tanh(\zeta_1)$ and
$\beta_2=\tanh(\zeta_2)$, then
\begin{equation} \label{velocity_add} 
  \beta = \tanh(\zeta_1+\zeta_2) = 
  \frac{\beta_1+\beta_2}{1+\beta_1\beta_2}
\end{equation}
holds. Another immediate consequence of the Lorentz transformations is 
the {\it time dilatation}: A moving clock ticks slower. In $K$ the 
position of the origin of $K'$ is given by 
$$ x^1=v\,x^0/c = \tanh(\zeta)\,x^0 $$
and the Lorentz transformation (\ref{2DLT1}) gives
\begin{equation} \label{Tdil}
  x^{'\, 0} = \cosh(\zeta)\,x^0 - \sinh(\zeta)\,\tanh(\zeta)\,x^0
            = \frac{\cosh^2(\zeta)-\sinh^2(\zeta)}{\cosh(\zeta)}\,x^0
            = \frac{x^0}{\cosh(\zeta)} < x^0\ .
\end{equation} 
This works also the other way round. In $K'$ the position of the 
origin of $K$ is given by 
$$ x^{'\, 1} = -\tanh(\zeta)\, x^{'\, 0} $$
and with this relation between $x^{'\, 1}$ and $x^{'\, 0}$ the
inverse Lorentz transformation gives
$$ x^0=x^{'\, 0} / \cosh(\zeta)\ .$$
There is no paradox, because equal times at separate points in one 
frame are not equal in another (remember that the definition of time 
in one frame relies already on the constant speed of light). In particle 
physics the effect is day by day observed for the lifetimes of unstable 
particles.  To test time dilatation for macrosciopic clocks, we have to 
send a clock on a roundtrip. For this an infinitesimal form of equation 
(\ref{Tdil}) is needed.

Allowing for the fact that at $x^0=x^1=0$ does not have to coincide 
with $x^{'\,0} =x^{'\, 1}=0$, we have Poincar\'e transformations. 
The light radiation may originate in $K$ at $(x^0_0, x^1_0)$ and 
in $K'$ at $(x^{'\, 0}_0,x^{'\, 1}_0)$. This generalizes 
equation (\ref{2Dinvar}) to 
$$ (x^{'\, 0}-x^{'\, 0}_0)^2 - (x^{'\, 1}-x^{'\, 1}_0)^2 = 
   (x^0-x^0_0)^2 - (x^1-x^1_0)^2 , $$
and the Lorentz transformations become
\begin{equation} 
  (x^{'\, 0} -  x^{'\, 0}_0) = \gamma\, 
  [(x^0 -x^0_0) - \beta\, (x^1-x^1_0)]\, , \label{Po1}
\end{equation} 
\begin{equation} 
 (x^{'\, 1} - x^{'\, 1}_0) = 
\gamma\, [(x^1-x^1_0) - \beta\, (x^0-x^0_0)]\, . \label{Po2}
\end{equation} 
Using the rapidity variable and matrix notation:
\begin{equation} \label{Po3}
\pmatrix{ x^{'\, 0}-x^{'\, 0}_0\cr  x^{'\, 1}-x^{'\, 1}_0 \cr} 
=  \pmatrix{ \cosh (\zeta )& -\sinh (\zeta )\cr 
  -\sinh (\zeta )& \cosh (\zeta ) \cr}\
   \pmatrix{ x^0-x^0_0\cr  x^1-x^1_0\cr}
\end{equation} 
In addition we have invariance under translations (\ref{translations}).

Let us explore Minkowski space in more details. It allows to depict 
{\it world lines} of particles. A useful concept for a particle (or 
observer) traveling along its world line is its {\it proper time}
or {\it eigenzeit}. Assume the particle moves with velocity $v(t)$, 
then $dx^1=\beta\,dx^0$ holds, and the infinitesimal invariant along 
its 2D world line is
\begin{equation} \label{2Dworld_line}
  (ds)^2 = (dx^0)^2-(dx^1)^2 = (c\,dt)^2\,(1-\beta^2)\ . 
\end{equation} 
Each instantaneous rest frame of the particle is an inertial frame. 
The increment of time $d\tau$ in such an instantaneous rest frame 
is a Lorentz invariant quantity which takes the form
\begin{equation} \label{dtau}
 d\tau = dt\, \sqrt{1-\beta^2} = dt\, \gamma^{-1}
       = dt/\cosh \zeta \ ,
\end{equation} 
where $\tau$ is called proper time. Clocks click by their proper time.
As $\gamma (\tau) \ge 1$ time dilatation follows
\begin{equation} \label{clocktrip}
  t_2-t_1 = \int_{\tau_1}^{\tau_2} \gamma(\tau)\,d\tau
  = \int_{\tau_1}^{\tau_2} \cosh\zeta(\tau)\,d\tau 
  \ge \tau_2 - \tau_1 . 
\end{equation} 
A moving clock runs more slowly than a stationary clock. Now
equation~(\ref{clocktrip}) applies to general paths of a clock, 
including those with acceleration. Relevant is that the 
entire derivation was done with respect to the inertial system 
in which the time coordinates $t_2$ and $t_1$ are defined.
Two experimental examples for time dilatation are: (i)~Time of flight
of unstable particles in high energy scattering experiments, where 
these particles move at velocities close to the speed of light.
(ii)~Explicit verification through travel with atomic clocks on 
air planes \cite{Hef1,Vessot}. 

Next, let us discuss the relation between velocity and {\it 
acceleration}.  Assume an accelaration $a$ in the instanteneous 
rest frame. To have convenient units we define $\alpha =a/c$ and
\begin{equation} \label{2Dacceleration}
  d\beta = d\zeta = \alpha\,d\tau
\end{equation} 
holds in the instantenous rest frame. The change in another frame
follow from the addition theorem of velocities (\ref{velocity_add})
\begin{equation} \label{dbeta}
  d\beta = \frac{\alpha d\tau+\beta}{1+\alpha d\tau\,\beta}-\beta
         = \alpha\,(1-\beta^2)\,d\tau
\end{equation} 
and simpler for the rapidity
\begin{equation} \label{dzeta}
  d\zeta = (\alpha\,d\tau + \zeta) - \zeta = \alpha\,d\tau\ .
\end{equation} 
Using the proper time, the change of the rapidity is analogue to the 
change of the velocity in non-relativistic mechanics, 
\begin{equation} \label{zeta_tau}
  \zeta - \zeta_0 = \int_{\tau_0}^{\tau} \alpha(\tau)\,d\tau\ .
\end{equation} 
In particular, if $\alpha$ is constant we can integrate and find
\begin{equation} \label{zeta_const}
  \zeta(\tau) = \alpha \tau + \zeta_0~~~{\rm for}~~\alpha~~
  {\rm constant}\ .
\end{equation} 

\subsection{Vector and tensor notation\label{sec_tensor}}

One defines a general transformation $x\to x'$ through
\begin{equation} \label{trans}
x'^{\, \alpha} = x'^{\, \alpha} (x) =
x'^{\, \alpha}\,\left( x^0, x^1, x^2, x^3 \right),\ \alpha = 0,1,2,3. 
\end{equation}
This means, $x'^{\,\alpha}$ is a function of four variables and,
when it is needed, this function is assumed to be sufficiently 
often differentiable with respect to each of its arguments. In the 
following we consider the transformation properties of various
quantities (scalars, vectors and tensors) under $x\to x'$.

A {\it scalar} is a single quantity whose value is not changed
under the transformation (\ref{trans}). The proper time is and
example.

A {\it 4-vector} $A^{\alpha}, (\alpha=0,1,2,3)$ is said 
{\it contravariant} if its components transform according to 
\begin{equation}
  A'^{\alpha} = {\partial x'^{\alpha} \over \partial x^{\beta}} 
  A^{\beta} . \label{con}
\end{equation}
An example is $A^{\alpha} = dx^{\alpha}$, where (\ref{con})
reduces to the well--known rule for the differential of a
function of several variable $(f^{\alpha}(x)=x'^{\alpha}(x))$:
$$ dx'^{\alpha} = {\partial x'^{\, \alpha} \over \partial x^{\beta}} 
   dx^{\beta} . $$
Remark: In this general framework the vector $x^{\alpha}$ is not
always contravariant. When a linear transformation
$$ x'^{\,\alpha} = a^{\alpha}_{\ \beta}\, x^{\, \beta} $$
holds, i.e., with space-time independent coefficients 
$a^{\alpha}_{\ \beta}$, this is the case and one finds
$$ {\partial x'^{\,\alpha}\over\partial x^{\,\beta}} =
   a^{\alpha}_{\ \beta} \ .$$
In the present framework we are only interested in linear 
transformation. Space-time dependent transformations lead 
into general relativity.

A 4-vector is said {\it covariant} when it transforms like
\begin{equation}
  B'_{\alpha} = {\partial x^{\beta} \over \partial x'^{\alpha}} 
  B_{\beta} . \label{cov}
\end{equation}
An example is 
\begin{equation}
  B_{\alpha} = \partial_{\alpha} = 
  {\partial\ \over \partial x^{\alpha}} , \label{partial}
\end{equation}
because of
$$ {\partial\ \over \partial x'^{\alpha}} =
   {\partial x^{\beta} \over \partial x'^{\alpha}}
   {\partial\ \over \partial x^{\beta}} . $$
The {\it inner} or {\it scalar product} of two vectors is defined as
the product of the components of a covariant and a contravariant
vector:
\begin{equation}
 B\cdot A = B_{\alpha} A^{\alpha} . \label{scalarp}
\end{equation}
It follows from (\ref{con}) and (\ref{cov}) that the scalar product is 
an invariant under the transformation (\ref{trans}): 
$$ B' \cdot A' = {\partial x^{\beta} \over \partial x'^{\alpha} }\,
 {\partial x'^{\alpha} \over \partial x^{\gamma} }\, 
 B_{\beta} A^{\gamma} = 
 {\partial x^{\beta} \over \partial x^{\gamma}}\,
 B_{\beta} A^{\gamma} = 
  \delta^{\beta}_{\ \gamma} B_{\beta} A^{\gamma} = B\cdot A .$$
Here the {\it Kronecker delta} is defined by:
\begin{equation} \label{delta}
  \delta^{\alpha}_{\ \beta} = \delta_{\alpha}^{\ \beta} =
  \cases{ 1 ~~{\rm for}~~ \alpha = \beta, \cr
        0 ~~{\rm for}~~ \alpha \ne \beta.} 
\end{equation}
Vectors are {\it rank} one tensors. {\it Tensors} of general rank $k$
are quantities with $k$ indices, like for instance
$$ {T^{\alpha_1\alpha_2 ...}}_{...\alpha_i ... \alpha_k}\ . $$
The convention is that the upper indices transform contravariant
and the lower transform covariant. For instance, a contravariant tensor
of rank two $F^{\alpha\beta}$ consists of 16 quantities that transform
according to
$$ F'^{\alpha\beta} = 
{\partial x'^{\alpha} \over \partial x^{\gamma}} 
{\partial x'^{\beta} \over \partial x^{\delta}} F^{\gamma\delta} .$$
A covariant tensor of rank two $G_{\alpha\beta}$ transforms as 
$$ G'_{\alpha\beta} = 
{\partial x^{\gamma} \over \partial x'^{\alpha}} 
{\partial x^{\delta} \over \partial x'^{\beta}} G_{\gamma\delta} .$$
The {\it inner product} or {\it contraction} with respect to a pair 
of indices, either on the same tensor or between different tensors,
is defined in analogy with (\ref{scalarp}). One index has to be 
contravariant and the other covariant.

A tensor $S^{...\alpha ...\beta ...}$ is said to be symmetric in 
$\alpha$ and $\beta$ when
$$ S^{...\alpha ...\beta ...} = S^{...\beta ...\alpha ...} . $$
A tensor $A^{...\alpha ...\beta ...}$ is said to be antisymmetric in 
$\alpha$ and $\beta$ when
$$ A^{...\alpha ...\beta ...} = - A^{...\beta ...\alpha ...} . $$
Let $S^{...\alpha ...\beta}$ be a symmetric and $A^{...\alpha ...\beta}
$ be an antisymmetric tensor. It holds
\begin{equation} \label{SA} 
S^{...\alpha ...\beta ...} A_{...\alpha ...\beta ...} = 0 . 
\end{equation}
Proof:
$$ S^{...\alpha ...\beta ...} A_{...\alpha ...\beta ...} = 
 - S^{...\beta ...\alpha ...} A_{...\beta ...\alpha ...} =
 - S^{...\alpha ...\beta ...} A_{...\alpha ...\beta ...} , $$
and consequently zero. The first step exploits symmetry and 
antisymmetry, and the second step renames the summation indices.
Every tensor can be written as a sum of its symmetric and
antisymmetric parts in two if its indices
\begin{equation} \label{TSA} 
T^{...\alpha ...\beta ...} = T_S^{...\alpha ...\beta ...} 
 + T_A^{...\alpha ...\beta ...} 
\end{equation}
by simply defining
\begin{equation} 
 T_S^{...\alpha ...\beta ...} = {1\over 2} \left(
T^{...\alpha ...\beta ...} + T^{...\beta ...\alpha ...} 
\right) ~~{\rm and}~~
 T_A^{...\alpha ...\beta ...} = {1\over 2} \left(
T^{...\alpha ...\beta ...} - T^{...\beta ...\alpha ...} 
\right)\ . 
\end{equation}

So far the results and definitions are general. We now specialize to 
Poincar\'e transformations. The specific geometry of the space--time 
of special relativity is defined by the invariant distance $s^2$, see 
equation~(\ref{ILo}). In differential form, the infinitesimal interval 
$ds$ defines the proper time $c\,d\tau =ds$,
\begin{equation}
 (ds)^2 = (dx^0)^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2 .
\label{ILo1} \end{equation}
Here we have used superscripts on the coordinates in accordance to our
insight that $dx^{\alpha}$ is a contravariant vector. Introducing a
{\it metric tensor} $g_{\alpha\beta}$ we re--write equation (\ref{ILo1}) as
\begin{equation}
 (ds)^2 = g_{\alpha\beta}\, dx^{\alpha} dx^{\beta} .
\label{ILo2} \end{equation}
Comparing (\ref{ILo1}) and (\ref{ILo2}) we see that for special relativity
$g_{\alpha\beta}$ is diagonal:
\begin{equation} \label{metric_tensor}
g_{00}=1,\, g_{11}=g_{22}=g_{33}=-1 ~~{\rm and}~~
g_{\alpha\beta}=0 ~{\rm for}~ \alpha\ne \beta . \label{gab}
\end{equation}
Comparing (\ref{ILo2}) with the invariant scalar product 
(\ref{scalarp}), we conclude that
$$ x_{\alpha} = g_{\alpha\beta}\, x^{\beta} . $$
The covariant metric tensor lowers the indices, i.e. transforms
a contravariant into a covariant vector. Correspondingly the 
contravariant metric tensor $g^{\alpha\beta}$ is defined to raise
indices:
$$ x^{\alpha} = g^{\alpha\beta}\, x_{\beta} .$$
The last two equations and the symmetry of $g_{\alpha\beta}$ imply
$$ g_{\alpha\gamma}\, g^{\gamma\beta} = \delta_{\alpha}^{\ \beta} $$
for the contraction of contravariant and covariant metric tensors.
This is solved by $g^{\alpha\beta}$ being the normalized co--factor
of $g_{\alpha\beta}$. For the diagonal matrix (\ref{gab}) the
result is simply
\begin{equation} \label{metric_tensor_up}
 g^{\alpha\beta} = g_{\alpha\beta} . 
\end{equation}
Consequently the equations 
$$ A^{\alpha} = \left( \matrix{A^0\cr \vec{A} \cr} \right), 
   ~~~ A_{\alpha} = (A^0, - \vec{A} ) $$
and, compare (\ref{partial}),
\begin{equation} \label{partial2}
 (\partial_{\alpha}) = 
 \left( {\partial \over c\partial t}\,, \nabla \right), ~~~
 (\partial^{\alpha}) = \left( \matrix{ 
  \frac{{\partial}}{{c}\partial{t}}\cr ~\cr 
   - \nabla\cr} \right)\, . 
\end{equation}
%  \frac{{{\big\partial}}}{{c}\partial{t}}\cr ~\cr 
hold. It follows that the 4-divergence of a 4-vector
$$ \partial^{\alpha} A_{\alpha} = \partial_{\alpha} A^{\alpha} 
= {\partial A^0 \over \partial x^0} + \nabla \cdot \vec{A}$$
and the d'Alembert (4-dimensional Laplace) operator
$$ \square = \partial_{\alpha} \partial^{\alpha} =
{\left(\partial~~\over \partial x^0\right)^2} - \nabla^2 $$
are invariants. Sometimes the notation $\triangle = \nabla^2$ 
is used for the (3-dimensional) Laplace operator.

\subsection{Lorentz transformations}

Let us now construct the Lorentz group. We seek a group of linear 
transformations
\begin{equation}
 x'^{\alpha} = a^{\alpha}_{\ \beta}\, x^{\beta} ,\ \
(\Rightarrow {\partial x'^{\alpha} \over \partial x^{\beta}}
= a^{\alpha}_{\ \beta} ) \label{Ltrans}
\end{equation}
such that the scalar product stays invariant:
$$ x'_{\alpha}\, x'^{\alpha} = a_{\alpha}^{\ \beta} x_{\beta}\,
a^{\alpha}_{\ \gamma} x^{\gamma} = x_{\alpha}\, x^{\alpha}
 = \delta^{\beta}_{\ \gamma} x_{\beta} x^{\gamma} .$$
As the $x_{\beta} x^{\gamma}$ are independent, this yields
$$ 
 a_{\alpha}^{\ \beta}\, a^{\alpha}_{\ \gamma} = 
 \delta^{\beta}_{\ \gamma}\ \Leftrightarrow \
 a_{\alpha\beta}\, a^{\alpha}_{\ \gamma} = 
 g_{\beta\gamma}\ \Leftrightarrow \
 a^{\delta}_{\ \beta}\, g_{\delta\alpha}\, a^{\alpha}_{\ \gamma} = 
 g_{\beta\gamma}\ . $$
In matrix notation
\begin{equation}
\tilde{A} g A = g , \label{Lg1}
\end{equation}
where $g=(g_{\beta\alpha})$ is given by (\ref{metric_tensor}),
\begin{equation} \label{mat_A}
   A = (a^{\beta}_{\ \alpha}) = \left( \matrix{ 
   a^0_{\ 0} & a^0_{\ 1} & a^0_{\ 2} & a^0_{\ 3}\cr 
   a^1_{\ 0} & a^1_{\ 1} & a^1_{\ 2} & a^1_{\ 3}\cr 
   a^2_{\ 0} & a^2_{\ 1} & a^2_{\ 2} & a^2_{\ 3}\cr 
   a^3_{\ 0} & a^3_{\ 1} & a^3_{\ 2} & a^3_{\ 3}\cr } \right)\ , 
\end{equation}
and $\tilde{A}=(\tilde{a}_{\beta}^{\ \alpha})$ with 
$\tilde{a}_{\beta}^{\ \alpha} = a^{\alpha}_{\ \beta}$ 
is the transpose of the matrix $A=(a^{\beta}_{\ \alpha})$, 
explicitly
\begin{equation} \label{mat_A_transpose}
\tilde{A} = (\tilde{a}_{\beta}^{\ \alpha}) = \left( \matrix{ 
\tilde{a}_0^{\ 0}&\tilde{a}_0^{\ 1}&\tilde{a}_0^{\ 2}&\tilde{a}_0^{\ 3}\cr 
\tilde{a}_1^{\ 0}&\tilde{a}_1^{\ 1}&\tilde{a}_1^{\ 2}&\tilde{a}_1^{\ 3}\cr 
\tilde{a}_2^{\ 0}&\tilde{a}_2^{\ 1}&\tilde{a}_2^{\ 2}&\tilde{a}_2^{\ 3}\cr 
\tilde{a}_3^{\ 0}&\tilde{a}_3^{\ 1}&\tilde{a}_3^{\ 2}&\tilde{a}_3^{\ 3}\cr }
 \right) = \left( \matrix{ 
   a^0_{\ 0}& a^1_{\ 0}& a^2_{\ 0}& a^3_{\ 0}\cr 
   a^0_{\ 1}& a^1_{\ 1}& a^2_{\ 1}& a^3_{\ 1}\cr 
   a^0_{\ 2}& a^1_{\ 2}& a^2_{\ 2}& a^3_{\ 2}\cr 
   a^0_{\ 3}& a^1_{\ 3}& a^2_{\ 3}& a^3_{\ 3}\cr } \right)\ .
\end{equation}
For this definition of the transpose matrix the row indices are 
contravariant and the column indices are covariant, vice verse
to the definition (\ref{4x}) for vectors and, similarly, ordinary
matrices. 
Certain properties of the transformation matrix $A$ can be deduced
from (\ref{Lg1}). Taking the determinant of both sides gives us
$ {\rm det} ( \tilde{A} g A ) = {\rm det} (g)\, {\rm det} (A)^2 
= {\rm det} (g) .$ Since ${\rm det} (g) = -1$, we obtain
\begin{equation} \label{detA}
 {\rm det} (A) = \pm 1 . 
\end{equation} 
One distinguishes two classes of transformations. {\it Proper}
Lorentz transformations are continuously connected with the
identity transformation $A = {\bf 1}$. All other Lorentz
transformations are {\it improper}.
Proper transformations have necessarily ${\rm det} (A) = 1$. 
For improper Lorentz transformations it is sufficient, but
not necessary, to have ${\rm det}(A)=-1$. For instance
$A=-{\bf 1}$ (space and time inversion) is an improper
Lorentz transformation with ${\rm det}(A)=+1$. 

Next the number of parameters, needed to specify completely
a transformation in the group, follows from (\ref{Lg1}).
Since $A$ and $g$ are $4\times 4$ matrices, we have 16
equations for $4^2=16$ elements of $A$. But they are not
all independent because of symmetry under transposition. 
The off-diagonal equations are identical in pairs. Therefore,
we have $4+6=10$ linearly independent equations for the 16
elements of $A$. This means that there are {\it six free
parameters}. In other words, the Lorentz group is a 
six--parameter group. 

In the 19th century \index{} {\it Lie} invented the subsequent
procedure to handle these parameters. Let us now consider only 
proper Lorentz transformations. To construct $A$ explicitly, 
Lie makes the ansatz
$$ A = e^L = \sum_{n=0}^{\infty} {L^n\over n!} , $$
where $L$ is a $4\times 4$ matrix. The determinant of $A$ is 
\begin{equation}
{\rm det} (A) = {\rm det} (e^L) = e^{{\rm Tr}(L)} . \label{det}
\end{equation}
Note that ${\rm det}(A)=+1$ implies that $L$ is traceless.
Equation (\ref{Lg1}) can be written
\begin{equation}
g \tilde{A} g = A^{-1} . \label{Lg2}
\end{equation}
From the definition of $L$, $\tilde{L}$ and the fact that 
$g^2 = {\bf 1}$ we have (note $(g\tilde{L}g)^n=g \tilde{L}^n g$ and 
${\bf 1} = 
(\sum_{n=0}^{\infty} L^n/n!)\, (\sum_{n=0}^{\infty} (-L)^n/n!)$) 
$$ \tilde{A}=e^{\tilde{L}},\ \ g\tilde{A}g=e^{g\tilde{L}g}
   ~~{\rm and}~~  A^{-1}=e^{-L}\ . $$
Therefore, (\ref{Lg2}) is equivalent to
$$ g\tilde{L}g=-L ~~{\rm or}~~ \widetilde{(gL)}=-gL . $$
The matrix $gL$ is thus antisymmetric and it is left as an 
exercise to show that the general form of $L$ is:
\begin{equation} \label{matrix_L}
   L = \left( \matrix{ 
    0      &  l^0_{\ 1} &  l^0_{\ 2} &  l^0_{\ 3} \cr
 l^0_{\ 1} &      0     &  l^1_{\ 2} &  l^1_{\ 3} \cr
 l^0_{\ 2} & -l^1_{\ 2} &     0      &  l^2_{\ 3} \cr
 l^0_{\ 3} & -l^1_{\ 3} & -l^2_{\ 3} &     0      \cr} \right).
\end{equation}
It is customary to expand $L$ in terms of six {\it generators}:
\begin{equation}
L=-\sum_{i=1}^3 (\omega_i S_i + \zeta_i K_i) ~~{\rm and}~~
A=e^{-\sum_{i=1}^3(\omega_iS_i+\zeta_i K_i)} . \label{Lie}
\end{equation}
The matrices are defined by
\begin{equation} \label{rotation_matrices}
   S_1 = \left( \matrix{ 0 & 0 & 0 & 0 \cr
                         0 & 0 & 0 & 0 \cr
                         0 & 0 & 0 &-1 \cr
                         0 & 0 & 1 & 0 \cr} \right),\ \
   S_2 = \left( \matrix{ 0 & 0 & 0 & 0 \cr
                         0 & 0 & 0 & 1 \cr
                         0 & 0 & 0 & 0\cr
                         0 &-1 & 0 & 0 \cr} \right),\ \
   S_3 = \left( \matrix{ 0 & 0 & 0 & 0 \cr
                         0 & 0 &-1 & 0 \cr
                         0 & 1 & 0 & 0\cr
                         0 & 0 & 0 & 0 \cr} \right),
\end{equation}
and
\begin{equation} \label{boost_matrices}
  K_1 = \left( \matrix{ 0 & 1 & 0 & 0 \cr
                         1 & 0 & 0 & 0 \cr
                         0 & 0 & 0 & 0 \cr
                         0 & 0 & 0 & 0 \cr} \right),\ \
   K_2 = \left( \matrix{ 0 & 0 & 1 & 0 \cr
                         0 & 0 & 0 & 0 \cr
                         1 & 0 & 0 & 0 \cr
                         0 & 0 & 0 & 0 \cr} \right),\ \
   K_3 = \left( \matrix{ 0 & 0 & 0 & 1 \cr
                         0 & 0 & 0 & 0 \cr
                         0 & 0 & 0 & 0 \cr
                         1 & 0 & 0 & 0 \cr} \right) , 
\end{equation}
They satisfy the following \index{}{\it Lie algebra} commutation 
relations:
$$ [S_i,S_j] = \sum_{k=1}^3 \epsilon^{ijk} S_k,\ \
   [S_i,K_j] = \sum_{k=1}^3 \epsilon^{ijk} K_k,\ \
   [K_i,K_j] =-\sum_{k=1}^3 \epsilon^{ijk} S_k, $$
where the commutator of two matrices is defined by $[A,B]=AB-BA$. 
Further $\epsilon_{ijk}$ is the completely antisymmetric Levi--Cevita 
tensor. Its definition in $n$--dimensions is
\begin{equation} \label{epsilon}
 \epsilon^{i_1 i_2 ... i_n} = \cases{
+1 ~~{\rm for}~~ (i_1, i_2, ..., i_n) 
~~{\rm being\ an\ even\ permutation\ of} ~~(1,2,...,n), \cr
-1 ~~{\rm for}~~ (i_1, i_2, ..., i_n) 
~~{\rm being\ an\ odd\ permutation\ of} ~~(1,2,...,n), \cr
 0 ~~{\rm otherwise}. } 
\end{equation}
To get the physical interpretation of equation (\ref{Lie}) for 
$A$, it is suitable to work out simple examples. First, let
$\vec{\zeta}=\omega_1=\omega_2=0$ and $\omega_3=\omega$. Then
(this is left as exercise)
\begin{equation} \label{S3rot}
A = e^{-\omega\, S_3} =
   \left( \matrix{ 1 &     0      &     0      & 0 \cr
                   0 & \cos\omega & \sin\omega & 0 \cr
                   0 &-\sin\omega & \cos\omega & 0 \cr
                   0 &     0      &     0      & 1 \cr} \right) ,
\end{equation}
which describes a rotation by the {\it angle} $\omega$ (in the 
clockwise sense) around the $\hat{e}_3$ axis. Next, let
$\vec{\omega}=\zeta_2=\zeta_3=0$ and $\zeta_1=\zeta$. Then
\begin{equation} \label{K1bost}
A = e^{-\zeta\, K_1} =
   \left( \matrix{ \cosh\zeta & -\sinh\zeta & 0 & 0 \cr
                  -\sinh\zeta &  \cosh\zeta & 0 & 0 \cr
                        0     &       0     & 1 & 0 \cr
0     &       0     & 0 & 1 \cr} \right) \label{mboost}
\end{equation}
is obtained, where $\zeta$ is known as the {\it boost parameter}
or {\it rapidity}. The structure is reminiscent to a rotation,
but with hyperbolic functions instead of circular, basically 
because of the relative negative sign between the space and 
time terms in eqn.(\ref{ILo1}). ``Rotations'' in the $x^0-x^i$ 
planes are boosts and governed by an hyperbolic gemometry, 
whereas rotations in the $x^i-x^j$ ($i\ne j$) planes are
governed by the ordinary Euclidean geometry.

Finally, note that the parameters
$\omega_i$, $\zeta_i$, $(i=1,2,3)$ turn out to be real, as
equation (\ref{Ltrans}) implies that the elements of $A$ have to
be real. In the next subsection relativistic kinematics is
discussed in more details.

\subsection{Basic relativistic kinematics}

The matrix (\ref{mboost}) gives the Lorentz boost transformation,
which we discussed before in the 2D context (\ref{2DLT1}),
\begin{eqnarray}
 x'^{\, 0} =  x^0 \cosh (\zeta ) - x^1 \sinh (\zeta ) , \label{LT1} \\
 x'^{\, 1} = -x^0 \sinh (\zeta ) + x^1 \cosh (\zeta ) , \label{LT2} \\
 x'^{\, i} = x^i,\ (i=2,3). \label{LT3} 
\end{eqnarray}
% At this point it is an interesting exercise, left to the 
% reader~(\ref{relativistic_consistency},
% to check that the transformations are indeed consistent with the
% time and distance definitions of section~2.1.
To find the transformation law of an arbitrary vector $\vec{A}$ in case 
of a general relative velocity $\vec{v}$, it is convenient to decompose 
$\vec{A}$ into components parallel and perpendicular to $\vec{\beta} = 
\vec{v}/c$. Let $\hat{\beta}$ be the unit vector in $\vec{\beta}$ 
direction, then
$$ \vec{A} = A^{\|} \hat{\beta} + \vec{A}^{\perp} 
   ~~{\rm with}~~ A^{\|} = \hat{\beta} \vec{A} . $$
Then the Lorentz transformation law is simply
\begin{equation}
 A'^{\, 0} = A^0 \cosh (\zeta) - A^{\|} \sinh (\zeta)
      = \gamma (~~A^0 - \beta A^{\|}) , \label{VT1}
\end{equation}
\begin{equation}
 A'^{\, \|} = - A^0 \sinh (\zeta) + A^{\|} \cosh (\zeta)
        = \gamma (- \beta A^0 + A^{\|}) , \label{VT2}
\end{equation}
\begin{equation}
 \vec{A}'^{\, \perp} = \vec{A}^{\perp} . \label{VT3}
\end{equation}
Here I have reserved the subscript notation $A_{\|}$ and 
$\vec{A}_{\perp}$ for use in connection with covariant vectors: 
$A_{\|}=-A^{\|}$ and $\vec{A}_{\perp} = -\vec{A}^{\perp}$.
We proceed deriving the addition theorem of velocities. Assume 
a particle moves with respect to $K'$ with velocity $\vec{u}^{\, '}$:
$$ x'^{\, i} = c^{-1} u'^{\, i} x'{\, ^0} . $$
Equations (\ref{LT1a}), (\ref{LT2a}) imply
$$ \gamma\, (x^1-\beta x^0) = 
   c^{-1} u'^{\, 1} \gamma\, (x^0-\beta x^1) .$$
Sorting with respect to $x^1$ and $x^0$ gives
$$ \gamma\, \left(1+{u'^{\, 1} v\over c^2}\right)\, x^1 = 
   c^{-1}\gamma\, (u'^1 + v)\, x^0\, . $$
Using the definition of the velocity in $K$, 
$\vec{x}=c^{-1} \vec{u}\, x^0$, gives
\begin{equation} \label{u1pv}
 u^1 = c\, {x^1\over x^0} 
     = {u'^1 + v \over 1 + {u'^1 v\over c^2}}\, . 
\end{equation}
Along similar lines,
% exercise~(\ref{ex_uipv}),
we obtain for the two other components
\begin{equation} \label{uipv}
 u^i = {u'^i \over \gamma \left( 1+{u'^1 v\over c^2}\right)},
        ~~(i=2,3). 
\end{equation}
To derive these equations, $\vec{v}$ was chosen along to the
$x^1$-axis. For general $\vec{v}$ one only has to decompose
$\vec{u}$ into its components parallel and perpendicular to
the $\vec{v}$
$$ \vec{u} = u^{\|} \hat{v} + \vec{u}^{\perp}\, , $$
where $\hat{v}$ is the unit vector in $\vec{v}$ direction,
and obtains
\begin{equation}
u^{\|} = {u'^{\, \|}+v \over 1 + {u'^{\, \|} v \over c^2}} 
 ~~~{\rm and}~~~
\vec{u}^{\perp} = {\vec{u}^{\, '\, \perp} \over \gamma\, 
\left( 1 + {u'^{\, \|} v \over c^2} \right) } . \label{addv}
\end{equation}
From this addition theorem of velocities it is obvious that the 
velocity itself is not part of of a 4-vector. The relativistic 
generalization is given in subsection~(\ref{dynamics}). It is
left as an exercise to relate these equations to the addition
theorem for the rapidity~(\ref{rapidity_add}.

The concepts of {\it world lines} in Minkowski space and {\it proper 
time} ({\it eigenzeit} generalized immediately to 4D. Assume the particle 
moves with velocity $\vec{v}(t)$, then $d\vec{x}=\vec{\beta} dx^0$ holds, 
and the infinitesimal invariant along its world line is
\begin{equation} \label{world_line}
(ds)^2 = (dx^0)^2-(d\vec{x})^2 = (c\,dt)^2\, (1-\beta^2)
\end{equation} 
and the relations (\ref{dtau}) and (\ref{clocktrip}) hold as in 2D.

\subsection{Plane waves and the relativistic Doppler effect
            \label{sec_Doppler}}

Let us choose coordinates with respect to an inertial frame $K$.
In complex notation a plane wave is defined by the equation
\begin{equation} \label{wave}
W (x) = W(x^0,\vec{x}) = W_0\,
\exp [\, i\, (k^0\, x^0 - \vec{k}\, \vec{x} )\, ]\, ,
\end{equation}
where $W_0 = U_0 + i\, V_0$ is a complex {\it amplitude}. The 
vector $\vec{k}$ is called {\it wave vector}. It becomes a 4-vector
$(k^{\alpha})$ by identifying 
\begin{equation} \label{wave_k0} 
 k^0 = \omega / c
\end{equation}
as its zero-component,
where $\omega$ is the {\it angular frequency} of the wave. 
Waves of the form (\ref{wave}) may either propagate in a medium
(water, air, shock waves, etc.) or in vacuum (light waves, particle 
waves in quantum mechanics). We are interested in the latter case, 
as the other defines a preferred inertial frame, namely the one
where the medium is at rest. 
The {\it phase} of the wave is defined by
\begin{equation} \label{phase}
\Phi (x) = \Phi (x^0,\vec{x}) = k^0\, x^0 - \vec{k}\, \vec{x} =
\omega\, t - \vec{k}\, \vec{x}\, . 
\end{equation}
When $(k^{\alpha})$ is a 4-vector, it follows that the phase is
a scalar, {\it invariant} under Lorentz transformations 
\begin{equation} \label{phaseinv}
\Phi' (x') = k'_{\, \alpha}\, x'^{\, \alpha}
 = k_{\, \alpha}\, x^{\, \alpha} = \Phi (x)\, .
\end{equation}
That this is correct can be seen as follows:
For an observer at a fixed position $\vec{x}$ (note the term
$\vec{k}\, \vec{x}$ is then constant) the wave performs a periodic 
motion with {\it period} 
\begin{equation} \label{period}
 T = {2 \pi \over \omega} = {1 \over \nu}\, ,
\end{equation}
where $\nu$ is the {\it frequency}. In particular, the phase
(and hence the wave) takes identical values on the two-dimensional
hyperplanes perpendicular to $\vec{k}$. Namely, let $\hat{k}$ be
the unit vector in $\vec{k}$ direction, by decomposing $\vec{x}$ 
into components parallel and perpendicular to $\vec{k}$,
$\vec{x} = x^{\|}\, \hat{k} + \vec{x}^{\, \perp}$, the phase
becomes
\begin{equation} \label{phase2}
 \Phi = \omega\, t - k\, x^{\|}\, ,
\end{equation}
where $k=|\vec{k}|$ is the length of the vector $\vec{k}$. Phases
which differ by multiples of $2\pi$ give the same values for the
wave $W$. For example, when we take $V_0=0$, the real part of
the wave becomes
% \begin{equation} \label{wcos}
$$ W_x = U_0\, \cos (\omega\, t - k\, x^{\|}) $$
% \end{equation}
and $\Phi = 0$, $n\, 2\pi$, $n=\pm 1, \pm 2, ...$ describes the 
wave crests. From (\ref{phase2}) it follows that the crests pass 
by our observer with speed $\vec{u} =u\, \hat{k}$, where
\begin{equation} \label{wspeed}
 u = {\omega \over k} ~~{\rm as\ for}\ \Phi=0\ {\rm we\
 have}\ ~~x^{\|} = {\omega \over k}\, t\, .
\end{equation}
Let our observer count the number of wave crests passing by. How has 
then the wave (\ref{wave}) to be described in another inertial frame 
$K'$?  An observer in $K'$ who counts the number of wave crests, 
passing through the same space-time point at which our first observer 
already counts, must get the same number. After all, the coordinates are 
just labels and the physics is the same in all systems.  When in frame 
$K$ the wave takes its maximum at the space-time point $(x^{\alpha})$ 
it must also be at its maximum in $K'$ at the same space-time point in 
appropriately transformed coordinates $(x'^{\alpha})$. More generally, 
this is holds for every value of the phase, because it is a scalar.

As $(k^{\alpha})$ is a 4-vector 
the transformation law for angular frequency and wave vector is
just a special case of equations (\ref{VT1}), (\ref{VT2}) and
(\ref{VT3})
\begin{equation} \label{Dop1}
 k'^{\, 0} = k^0 \cosh (\zeta) - k^{\|} \sinh (\zeta)
 = \gamma (k^0 - \beta k^{\|})\, ,
\end{equation}
\begin{equation}
 k'^{\, \|} = - k^0 \sinh (\zeta) + k^{\|} \cosh (\zeta)
 = \gamma (k^{\|} - \beta k^0 )\, ,
\end{equation}
\begin{equation}
 \vec{k}'^{\, \perp} = \vec{k}^{\perp}\, ,
\end{equation}
where the notation $k^{\|}$ and $k^{\perp}$ is with respect to 
the relative velocity of the two frames, $\vec{v}$. These
transformation
equations for the frequency and the wave vector describe the
relativistic Doppler effect. To illustrate their meaning, let us 
specialize to the case of a light source, which is emitted in $K$ 
and the observer $K'$ moves in wave vector direction away from the
source, i.e., $\vec{v}~\|~\vec{k}$. The equation for the wave 
speed~(\ref{wspeed}) implies 
$$ c ={\omega \over k}\ \Rightarrow\ k = |\vec{k}| =
      {\omega \over c} = k^0 $$
and chosing directions so that $k'^{\,\|}=k$ holds, (\ref{Dop1}) becomes
$$ k'^{\, 0} = \gamma\, (k^0 - \beta\, k) = \gamma\, 
   (1-\beta)\, k^0 = k^0\, \sqrt{1-\beta \over 1+\beta} $$
or
$$ \omega' = {\nu' \over 2\pi} =  \omega\, \sqrt{1-\beta \over 1+\beta}
 =  {\nu \over 2\pi}\, \sqrt{1-\beta \over 1+\beta}\, . $$
Now, $c = \nu \lambda = \nu' \lambda'$, where $\lambda$ is the
wavelength in $K$ and $\lambda'$ the wavelength in $K'$. 
Consequently, we have
$$ \lambda' = \lambda\, \sqrt{1+\beta \over 1-\beta}\, . $$
For a receeding observer, or source receeding from the observer, 
$\beta>0$ in our conventions for $K$ and $K'$, and the wave length 
$\lambda'$ is larger than it is for a source at rest. This is an 
example of the red-shift, which is, for instance, of major importance 
when one analyzes spectral lines in astrophysics. Using the method of 
section~1, a single light signal suffices now to obtain position and 
speed of a distant mirror.

\subsection{Relativistic dynamics \label{dynamics}}

On a basic level
this section deals with the relativistic generalization of energy, 
momentum and their conservation laws. So far we have introduced two 
units, meter to measure distances and seconds to measure time. Both 
are related through a fundamental constant, the speed of light, so
that there is really only one independent unit up to now. In the
definition of the momentum a new, independent dimensional quantity 
enters, the mass of a particle. This unit is defined through the
gravitational law, which is out of the scope of this article. 
Ideally, one would like to defines mass of a body just as multiples
of the mass of an elementary particle, say an electron or proton.
However, this has remained too inaccurate. The mass unit has
so far resisted modernization and the mass unit
$$ 1~kilogram~[kg]\ =\ 1000~gram~[g] $$
is still defined through a one kilogram standard object a cylinder
of platinum alloy which is kept at the International Bureau of
Weights and Measures at S\'evres, France.

Let us consider a point-like particle in its rest-frame and denote 
its mass there by $m_0$. In any other frame the rest-mass of the 
particle is still $m_0$, which in this way is defined as a scalar.
It may be noted that most books in particle and nuclear physics
simply use $m$ to denote the rest-mass, whereas many books on special 
relativity employ the notation to use $m=\gamma m_0$ for a mass which
is proportional to the energy, {\it i.e.} the zero component of the 
energy-momentum vector introduced below. To avoid confusion, we 
use $m_0$ for the rest mass.

In the non-relativistic limit the momentum is defined by $\vec{p} = 
m_0 \vec{u}$. We want to define $\vec{p}$ as part of a relativistic 
4-vector $(p^{\alpha})$. Consider a particle at rest in frame $K$, 
i.e., $\vec{p}=0$. Assume now that frame $K'$ is moving with a small 
velocity $\vec{v}$ with respect to $K$. Then the non-relativistic limit 
is correct, and $\vec{p}{\, '} = -m_0\vec{v}$ has to hold approximately. 
On the other hand, the transformation laws (\ref{VT1}), (\ref{VT2}) and
(\ref{VT3}) for vectors (note $\vec{p}\,\|\,\vec{\beta}=\vec{v}/c$) 
imply
$$\vec{p}^{\ '} = \gamma\, (\vec{p} - \vec{\beta}\, p^0)\,.$$
For $\vec{p}=0$ we find $\vec{p}^{\ '}=-\gamma\,\vec{\beta}\,p^0)$. As
in the nonrelativstic limit $\gamma\,\beta\to\beta$, consistency 
requires $p^0=c\,m_0$ in the rest frame, so that we get $\vec{p}^{\ '}
=-m_0\,\gamma\,v$. Consequently, for a particle moving with velocity 
$\vec{u}$ in frame $K$ 
\begin{equation}
  \vec{p} = m_0\,\gamma\,\vec{u}
\end{equation}
is the correct relation between relativistic momentum and velocity. 
From the invariance of the scalar product, $p_{\alpha}p^{\alpha}=
(p^0)^2-\vec{p}^{\, 2}=p'_{\alpha}p'^{\alpha}=m_0^2c^2$ holds and
\begin{equation} \label{p0}
 p^{\, 0} = +\sqrt{c^2\,m_0^2 + \vec{p}^{\ 2}} 
\end{equation} 
follows, which is of course consistent with calculating $p^0$ via 
the Lorentz transformation law (\ref{VT1}). It should be noted that 
$c\,p^0$ has the dimension of an energy, i.e. the relativistic energy 
of a particle is
\begin{equation} \label{renergy}
 E = c\, p^0=+\sqrt{c^4\,m_0^2+c^2\,\vec{p}^{\ 2}} =
  c^2\,m_0 + {\vec{p}{\, ^2} \over 2m_0} +\, ...\ ,
\end{equation} 
where the second term is just the non-relativistic kinetic energy 
$T=\vec{p}^{\ 2}/(2m_0)$. 
The first term shows that (rest) mass and energy can be transformed 
into one another~\cite{Ei1906}. In processes where the mass is 
conserved we just do not notice it. Using the mass definition of
special relativity books like \cite{Rind1}, $m=c\, p^0$, together 
with (\ref{renergy}) we obtain at this point the famous equation 
$E=m\, c^2$. Avoiding this definition of $m$, because it is not the
mass found in particle tables, where the mass of a particle is an
invariant scalar, the essence of Einstein's equation is captured by
$$ E_0 = m_0\, c^2\, , $$
where $E_0$ is the energy of a massive body (or particle) in its rest 
frame. The particle and nuclear physics literature does not use a
subscript~$_0$ and denotes the rest mass simply by~$m$.

Non-relativistic momentum conservation $\vec{p}_1+\vec{p}_2 =
\vec{q}_1+\vec{q}_2$, where $\vec{p}_i$, $(i=1,2)$ are the
momenta of two incoming, and $\vec{q}_i$, $(i=1,2)$ are the
momenta of two outgoing particles, becomes relativistic
{\it energy--momentum conservation}:
\begin{equation} \label{emom}
 p^{\alpha}_1 + p^{\alpha}_2 = q^{\alpha}_1 + q^{\alpha}_2 .
\end{equation} 
Useful formulas in relativistic dynamics are
\begin{equation} \label{gambet}
 \gamma = {p_0 \over m_0\, c} = {E \over m_0\, c^2}
  ~~~{\rm and}~~~ \beta = {|\vec{p}| \over p^0} . 
\end{equation} 
Further, the contravariant generalization of the velocity vector is 
given by
\begin{equation} \label{rvelocity}
 U^{\alpha} = {dx^{\alpha} \over d\tau} =
 \gamma u^{\alpha} ~~{\rm with}~~ u^0=c\, ,
\end{equation}
compare the definition of the infinitesimal proper time~(\ref{dtau}).
The relativistic generalization of the force is then the 4-vector
\begin{equation} \label{rforce}
 f^{\alpha} = {d\, p^{\alpha} \over d\, \tau} =
 m_0\, {d U^{\alpha}\over d \tau}\, ,
\end{equation}
where the last equality can only be used for particle with non-zero
rest mass.

% \subsection{Problems}
% \begin{description}
% \item{1.} Show 
% \end{description}

% \vfil\eject

\section{Maxwell Equations}

As before all considerations are in vacuum, as for fields in a
medium a preferred reference system exists. 
Maxwell's equations in their standard form in vacuum are
\begin{equation} \label{IMS0}
\nabla \vec{E} = 4\pi \rho , ~~~\nabla \times \vec{B} -
{1\over c} {\partial \vec{E}\over \partial t}={4\pi\over c} {\vec J} ,
\end{equation}
and
\begin{equation} \label{HMS0}
\nabla \vec{B} = 0 , ~~~\nabla \times \vec{E} +
{1\over c} {\partial \vec{B}\over \partial t} = 0 .
\end{equation}
Here $\nabla$ is the Nabla operator. Note that $\nabla \vec{E} = 
\nabla \cdot \vec{E}$, $\vec{a}\,\vec{b} = \vec{a} \cdot \vec{b}$ 
etc.  throughout the script. $\vec{E}$ is the {\it electric field} 
and $\vec{B}$ the {\it magnetic field} in vacuum. When matter gets 
involved one introduces the {\it applied electric field} $\vec{D}$ and 
the {\it applied magnetic field} $\vec{H}$. Here we follow the convention 
of, for instance, Tipler~\cite{Tipler} and use the notation magnetic 
field for the measured field $\vec{B}$, in precisely the same way as it 
is done for the electric field $\vec{E}$, It should be noted that this 
is at odds with the notation in the book by Jackson~\cite{Jackson}, 
where (historically correct, but quite confusingly) $\vec{H}$ is called 
magnetic field and $\vec{B}$ magnetic flux or magnetic induction.

Equations (\ref{IMS0}) are the {\it inhomogeneous} and equations 
(\ref{HMS0}) are the {\it homogeneous} Maxwell equations in vacuum.

The charge density $\rho$ (charge per unit volume) and the current
density $\vec{J}$ (charge passing through a unit area per time unit)
are obviously given once a charge unit is defined through some 
measurement prescription.  From a theoretical point of view the 
electrical charge unit is best defined by the magnitude of the charge 
of a single electron (fundamental charge unit). In more conventional
units this reads
\begin{equation} \label{qelectron}
 |q_e| = 4.803 204 20 (19) \times 10^{-10}\, [esu] =
         1.602 176 462 (63)\times 10^{-19}\, Coulomb~[C] 
\end{equation}
where the errors are given in parenthesis. Definitions of the through 
measurement prescriptions rely presently on the current unit Amp\'ere
$[A]$ and are given in elementary physics textbooks like~\cite{Tipler}.
The numbers of (\ref{qelectron}) are from 1998~\cite{Co98}. The 
website of the National Institute of Standards and Technology (NIST) 
is given in this reference. Consult it for up to date information.

The choice of constants in the inhomogeneous Maxwell equations
defines units for the electric and magnetic field. The given
conventions $4\pi \rho$ and $(4\pi/c)\vec{J}$ are customarily
used in connection with Gaussian units, where the charge is
defined in electrostatic units $(esu)$.

In the next subsections the concepts of fields and currents are 
discussed in the relativistic context and the electromagnetic field 
equations follow in the last subsection.

\subsection{Fields and currents}

A tensor field is just a tensor function which depends on the
coordinates of Minkowski space: 
$$ T^{...\alpha ...\beta ...} = 
   T^{...\alpha ...\beta ...} (x)\, . $$
It is called {\it static} when there is no time dependence. For
instance $\vec{E} (\vec{x})$ in electrostatics would be a static
vector field in three dimensions. We are here, of course, primarily 
interested in contravariant or covariant fields in four dimensions, 
like vector fields $A^{\alpha} (x)$.

Suppose $n$ electric charge units are contained it a small volume $v$,
such that we can talk about the position $\vec{x}$ of this volume.
The corresponding electrical charge density at the position of that
volume is then just $\rho=n/v$ and the electrical current is defined
as the charge that passes per unit time through a surface element of
such a volume. We demand now that the electric charge density
$\rho$ and the electric current $\vec{J}$ form a 4-vector:
$$ (J^{\alpha}) = \left( \matrix{ c\rho \cr \vec{J}\cr} \right)\, .$$
Here, the factor $c$ is introduced by dimensional reasons and
we have suppressed the space-time dependence, {\it i.e.}
$J^{\alpha} = J^{\alpha} (x)$ forms a vector field. It is left as
a problem to write down the 4-current for a point particle of 
elementary charge $q_e$.

The {\it continuity equation} takes the simple, covariant form
\begin{equation} \label{continuityeqn}
 \partial_{\alpha} J^{\alpha} = 0\, .
\end{equation}
Finally, the charge of a point particle in its rest frame is an 
invariant:
$$ c^2 q_0^2 = J_{\alpha} J^{\alpha}\, .$$

\subsection{The inhomogeneous Maxwell equations}

The inhomogeneous Maxwell equations are obtained by writing down the
simplest covariant equation which yields a 4-vector as first order
derivatives of six fields.
From undergraduate E\&M 
we remember the electric and magnetic fields, $\vec E$ and $\vec B$, as 
the six central fields of electrodynamics. We now like to describe them
in covariant form. A 4-vector is unsuitable as we like to describe
six quantities $E^x, E^y, E^z$ and $B^x, B^y, B^z$. Next, we may try
a rank two tensor $F^{\alpha\beta}$. Then we have $4\times 4 =16$ quantities
at our disposal. These are now too many. But, one may observe that
an symmetric tensor stays symmetric under Lorentz transformation and
an antisymmetric tensor stays antisymmetric.
% , see exercise~(\ref{LT_asymmetry}). 
Hence, instead of looking at the full second rank tensor one has to 
consider its symmetric and antisymmetric parts separately.

By requesting $F^{\alpha\beta}$ to be antisymmetric, 
\begin{equation}
 F^{\alpha\beta} = - F^{\beta\alpha} , \label{Fab}
\end{equation}
this number is reduced to precisely six. The diagonal elements do now
vanish,
$$ F^{00} = F^{11} = F^{22} = F^{33} = 0\, . $$
The other elements follow through (\ref{Fab}) from 
$F^{\alpha\beta}$ with $\alpha < \beta$. As desired, this gives $(16-4)/2=6$ 
independent elements to start with. 

Up to an over--all factor, which is chosen by convention, the only way to 
obtain a 4-vector through differentiation of $F^{\alpha\beta}$ is 
\begin{equation} \label{IMEQ}
\partial_{\alpha}F^{\alpha\beta}={4\pi\over c} J^{\beta}\ .
\end{equation}
This is the {\it inhomogeneous Maxwell equation} in covariant
form. Note that it determines the physical dimensions of the electric 
fields, the factor $c^{-1}4\pi$ on the right-hand side corresponds to
Gaussian units. The continuity equation~(\ref{continuityeqn}) is a 
simple consequence of the inhomogeneous Maxwell equation
$$ {4\pi\over c} \partial_{\beta} J^{\beta} = 
\partial_{\beta} \partial_{\alpha} F^{\alpha\beta} = 0 $$
because the contraction with the symmetric tensor 
$(\partial_{\beta} \partial_{\alpha})$ with the antisymmetric tensor
$F^{\alpha\beta}$ is zero.

Let us choose $\beta=0,1,2,3$ and compare equation~(\ref{IMEQ}) with 
the inhomogeneous Maxwell equations in their standard form~(\ref{IMS0}).
For instance, $\partial_{\alpha} F^{\alpha\,0}=\nabla\vec{E}=4\pi\,\rho$ 
yields the $F^{i\,0}=E^i$, the first column of the $F^{\alpha\beta}$ 
tensor. The final result is
\begin{equation} \label{FEB}
   (F^{\alpha\beta}) = \left(
\matrix{ 0   & -E^x & -E^y & -E^z \cr
         E^x &  0   & -B^z &  B^y \cr
         E^y &  B^z &  0   & -B^x \cr
         E^z & -B^y &  B^x &  0   \cr} \right)\, .
\end{equation}
Or, in components
\begin{equation} \label{EMfields}
 F^{i0}=E^i ~~{\rm and}~~ F^{ij} = -\sum_k \epsilon^{ijk}\,B^k
 \Leftrightarrow B^k = - {1\over 2}\, \sum_i\sum_j 
 \epsilon^{kij} F^{ij}\ .
\end{equation}
Next, $F_{\alpha\beta}=g_{\alpha\gamma}g_{\beta\delta}F^{\gamma\delta}$
implies:
$$ F_{0i}=-F^{0i},\ F_{00}=F^{00}=0,\ F_{ii}=F^{ii}=0,\
 ~~~{\rm and}~~~ F_{ij} = F^{ij} . $$
Consequently,
\begin{equation}
   (F_{\alpha\beta}) = \left(
\matrix{ 0   &  E^x &  E^y &  E^z \cr
        -E^x &  0   & -B^z &  B^y \cr
        -E^y &  B^z &  0   & -B^x \cr
        -E^z & -B^y &  B^x &  0   \cr} \right).
\end{equation}

\subsection{Four-potential and homogeneous Maxwell equations}

We remember that the electromagnetic
fields may be written as derivatives of appropriate potentials. The only
covariant option are terms like $\partial^{\alpha} A^{\beta}$. To make
$F^{\alpha\beta}$ antisymmetric, we have to subtract 
$\partial^{\beta} A^{\alpha}$:
\begin{equation} \label{DaAb}
F^{\alpha\beta}=\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha} .
\end{equation}
It is amazing to note that the homogeneous Maxwell equations follow now
for free. The dual electromagnetic tensor is defined
\begin{equation}
{^*F}^{\alpha\beta} = {1\over 2} \epsilon^{\alpha\beta\gamma\delta}
                          F_{\gamma\delta} , \label{DFab}
\end{equation}
and it holds
\begin{equation}
\partial_{\alpha} {^*F}^{\alpha\beta} = 0 . \label{HMEQ}
\end{equation}
Proof:
$$ \partial_{\alpha} {^*F}^{\alpha\beta} = 
{1\over 2} \left( \epsilon^{\alpha\beta\gamma\delta} 
\partial_{\alpha} \partial_{\gamma} A_{\delta} - 
\epsilon^{\alpha\beta\gamma\delta} 
\partial_{\alpha} \partial_{\delta} A_{\gamma} \right) = 0 . $$

This first term is zero due to (\ref{SA}), because 
$\epsilon^{\alpha\beta\gamma\delta}$ is antisymmetric in $(\alpha,\gamma)$,
whereas the derivative $\partial_{\alpha}\partial_{\gamma}$ is symmetric
in $(\alpha,\gamma)$. Similarly the other term is zero. The homogeneous 
Maxwell equation is related to the fact that the right-hand side of 
equation (\ref{DaAb}) expresses six fields in terms of a single 4-vector.
An equivalent way to write it is the equation
\begin{equation} \label{HMEQa}
\partial^{\alpha} F^{\beta\gamma} + \partial^{\beta} F^{\gamma\alpha} 
+ \partial^{\gamma} F^{\alpha\beta} = 0\ .
\end{equation}
The proof is left as an exercise to the reader.

Let us mention that the homogeneous Maxwell equation (\ref{HMEQ}) or
(\ref{HMEQa}), and hence our demand that the field can be written in 
the form (\ref{DaAb}), excludes magnetic monopoles.

The elements of the dual tensor may be calculated from 
their definition (\ref{DFab}). For example, 
$${^*F}^{02} = \epsilon^{0213} F_{13} = -F_{13} = -B^y , $$
where the first step exploits the anti-symmetries 
$\epsilon^{0231}=-\epsilon^{0213} $ and $F_{31}=-F_{13}$. Calculating
six components, and exploiting antisymmetry of 
${^*F}^{\alpha\beta}$, we arrive at
\begin{equation}
   ({^*F}^{\alpha\beta}) = \left(
\matrix{ 0   & -B^x & -B^y & -B^z \cr
         B^x &  0   &  E^z & -E^y \cr
         B^y & -E^z &  0   &  E^x \cr
         B^z &  E^y & -E^x &  0   \cr} \right).
\end{equation}
The homogeneous Maxwell equations in their form (\ref{HMS0})
provide a non--trivial consistency check for (\ref{HMEQ}),
which is of course passed. It may be noted that, in contrast to the 
inhomogeneous equations, the homogeneous equations determine the 
relations with the $\vec{E}$ and $\vec{B}$ fields only up to an 
over-all $\pm$ sign, because there is no current on the 
right-hand side.

A notable observation is that equation (\ref{DaAb}) does not determine
the potential uniquely. Under the transformation
\begin{equation}
A^{\alpha} \mapsto A'^{\alpha} = 
A^{\alpha} + \partial^{\alpha} \psi , \label{gauge}
\end{equation}
where $\psi = \psi (x)$ is an arbitrary scalar function, the
electromagnetic field tensor is invariant: 
$F'^{\alpha\beta} = F^{\alpha\beta}$, as follows immediately from
$\partial^{\alpha}\partial^{\beta}\psi - 
 \partial^{\beta}\partial^{\alpha}\psi = 0$. The transformations
(\ref{gauge}) are called {\it gauge transformation}$^{1}$. The 
choice of a convenient gauge is at the heart of many application.

\footnotetext[1]{In quantum field theory these are the gauge 
transformations of 2.~kind. Gauge transformations of 1.~kind transform 
fields by a constant phase, whereas for gauge transformation of the 
2.~kind a space--time dependent function is encountered.} 

\subsection{Lorentz transformation for the electric and magnetic fields}

The electric $\vec{E}$ and magnetic $\vec{B}$ fields are {\it not}
components of a Lorentz four-vector, but part of the rank two 
the electromagnetic field $(F^{\alpha\beta})$ given by (\ref{FEB}).
As for any Lorentz tensor, we immediately know its behavior under
Lorentz transformation
\begin{equation}
F'^{\alpha\beta} = a^{\alpha}_{\ \gamma}\,a^{\beta}_{\ \delta}\,
F^{\gamma\delta}\ .
\end{equation}
Using the explicit form (\ref{mboost}) of $A=(a^{\alpha}_{\ \beta})$ 
for boosts in the $x^1$ direction and (\ref{FEB}) for the relation 
to $\vec{E}$ and $\vec{B}$ fields, it is left as an exercise for the 
reader to derive the transformation laws
\begin{equation} \label{LTE}
  \vec{E}^{\,'} = \gamma\,\left( \vec{E} + \vec{\beta} \times
  \vec{B} \right) - {\gamma^2\over \gamma +1}\,\vec{\beta}\,
  \left( \vec{\beta}\,\vec{E} \right)\ ,
\end{equation}
and
\begin{equation} \label{LTB}
  \vec{B}^{\,'} = \gamma\,\left( \vec{B} - \vec{\beta} \times
  \vec{E} \right) - {\gamma^2\over \gamma +1}\,\vec{\beta}\,
  \left( \vec{\beta}\,\vec{B} \right)\ .
\end{equation}


\subsection{Lorentz force}

Relativistic dynamics of a point particle (more generally any mass
distribution) gets related to the theory of electromagnetic fields,
because an electromagnetic field causes a change of the 4-momentum
of a charged particle. On a deeper level this phenomenon is related to
the conservation of energy and momentum and the fact that an
electromagnetic carries energy as well as momentum. Here we are 
content with finding the 
Lorentz covariant form, assuming we know already that such the 
approximate relationship.

We consider a charged point particle in an electromagnetic field
$F^{\alpha\beta}$. Here {\it external} means from sources other
than the point particle itself and that the influence of the point
particle on these other sources (possibly causing a change of the
field $F^{\alpha\beta}$) is neglected. The infinitesimal change of
the 4-momentum of a point point particle is $dp^{\alpha}$ and
assumed to be proportional to (i)~its charge $q$ and (ii)~the
external electromagnetic field $F^{\alpha\beta}$. This means, we
have to contract $F^{\alpha\beta}$ with some infinitesimal covariant 
vector to get $dp^{\alpha}$. The simplest choice is $dx_{\beta}$, what
means that the amount of 4-momentum change is proportional to the 
space-time length at which the particle experiences the electromagnetic 
field. Hence, we have determined $dp^{\alpha}$
up to a proportionality constant, which depends on the choice of
units. Gaussian units are defined by choosing $c^{-1}$ for this
proportionality constant and we have
\begin{equation}   \label{pm_dLF}
 dp^{\alpha} = \pm {q\over c}\, F^{\alpha\beta}\, dx_{\beta} .
\end{equation}
As discussed in the next section, it is a consequence of energy 
conservation, in this context known as Lenz's law, that the force 
between charges of equal sign has to be repulsive. This corresponds 
to the plus sign and we arrive at
\begin{equation}   \label{dLF}
  dp^{\alpha} = {q\over c}\, F^{\alpha\beta}\, dx_{\beta} .
\end{equation}
Experimental measurements are of
course in agreement with this sign. The remarkable  point is that
energy conservation and the general structure of the theory already
imply that the force between charges of equal sign has to be
{\it repulsive}. Therefore, despite the similarity of the Coulomb's
inverse square force law with Newton's law it impossible
to build a theory of gravity along the lines of this chapter,
{\it i.e.} to use the 4-momentum $p^{\alpha}$
as source in the inhomogeneous equation~(\ref{IMEQ}). The
resulting force would necessarily be repulsive. Experiments
show also that positive and negative electric charges exist and
deeper insight about their origin comes from the relativistic Lagrange 
formulation, which ultimately has to include Dirac's equation for 
electrons and leads then to Quantum Electrodynamics.

Taking the derivative with respect to the proper time, we
obtain the 4-force acting on a charged particle, called
{\it Lorentz force},
\begin{equation}  \label{LF}
f^{\alpha} = {dp^{\alpha}\over d\tau} =
{q\over c} F^{\alpha\beta}U_{\beta} .
\end{equation}
As in equation~(\ref{rforce}) $f^{\alpha} = m_0\, d U^{\alpha}/d\tau$
holds for non-zero rest mass and the definition of the
contravariant velocity is given by equation~(\ref{rvelocity}). 

Using the representation~(\ref{FEB}) of the electromagnetic field
the time component of the relativistic Lorentz force, which describes 
the change in energy, is 
\begin{equation}  \label{LFtime}
 f^0 = {dp^0\over d\tau} =
 - {q\over c} \left( \vec{E} \vec{U} \right)\ .
\end{equation}
To get the space component of the Lorentz force we use besides
(\ref{FEB}) equation (\ref{EMfields}) which give the equality
$$ {q\over c}\sum_{j=1}^3F^{ij}U_j = 
-{q\over c}\sum_{j=1}^3\sum_{k=1}^3\epsilon^{ijk}B^kU_j $$
The space components combine into the well-known equation
\begin{equation} \label{LFspace}
\vec{f} = q\,\gamma\,\vec{E} 
	+ {q\over c}\, \vec{U} \times \vec{B}
\end{equation}
where our derivation reveals that the relativistic velocity
(\ref{rvelocity}) of the charge $q$ and not its velocity $\vec{v}$
enters the force equation. This allows, for instance, correct 
force calculations for fast flying electrons in a magnetic field.
The equation (\ref{LFspace}) for $\vec{f}$ may now be used to 
define a measurement prescription for an electric charge unit.
% \hfill\break
% \vfill\eject

\section{Faraday's Law}

From the homogeneous Maxwell equations (\ref{HMEQ}) we have 
\begin{equation}
  \nabla \times \vec{E} + {1\over c}\,
  {\partial\vec{B} \over \partial t} = 0\, . 
\end{equation}
This equation is the \index{Faraday's law}{\it differential form of 
Faraday's law}: A changing magnetic field induces an electric field. 
In the following we derive the integral form, which is needed for 
circuits of macroscopic extensions. We integrate over a simply 
connected surface $S$ and use Stoke's theorem to convert the integral 
over $\nabla\times\vec{E}$ into a closed line integral along
the boundary $C$ of $S$:
$$ \int_S (\,\nabla\times\vec{E}\,)\cdot d\vec{a} =
  \oint_C \vec{E}\cdot d\vec{l} = - {1\over c} \int_S
  {\partial\vec{B} \over \partial t}\cdot d\vec{a}\, .$$
On the right-hand side we eliminate the partial derivative
${\partial/\partial t}$ using
$$ {d\ \over dt} = {\partial\ \over\partial t} +
  \vec{v}\cdot\nabla~~~\left( {\rm note}~~ \vec{v} =
  {\partial\vec{x} \over\partial t} =
  \sum_{i=1}^3 \hat{e}^i\,{\partial x^i\over\partial t}\right) $$
to get 
\begin{equation} \label{F1}
   \oint_C \vec{E}\cdot d\vec{l} =
 - {1\over c} {d\ \over dt} \int_S \vec{B}\cdot d\vec{a}
 + {1\over c} \int_S (\vec{v}\cdot\nabla)\,\vec{B}\cdot d\vec{a}
\end{equation}
Using the other homogeneous Maxwell equation, $\nabla\cdot\vec{B}=0$,
and that the $\partial/\partial x^i$ derivatives of $\vec{v}$ vanish
(e.g., $(\partial/\partial x^1)\,(\partial x^1/\partial t) = (\partial
/\partial t)\,(\partial x^1/\partial x_1) = =0$), the well-known
vector identity
$$[\nabla\times (\,\vec{a}\times \vec{b}\,)] = 
  (\,\nabla\cdot\vec{b}\,)\,\vec{a}+(\,\vec{b}\cdot\nabla\,)\,\vec{a} 
 -(\,\nabla\cdot\vec{a}\,)\,\vec{b}-(\,\vec{a}\cdot\nabla\,)\,\vec{b}\,,$$
gives
$$ \nabla\times \left( \vec{B} \times \vec{v} \right) =
   \vec{v}\,\left( \nabla\cdot\vec{B}\right) +
   \left( \vec{v}\cdot\nabla \right)\,\vec{B} =
   \left( \vec{v}\cdot\nabla \right)\,\vec{B} $$
and we transform the last integral in (\ref{F1}) as follows:
\begin{eqnarray} \nonumber
  {1\over c} \int_S (\vec{v}\cdot\nabla)\, \vec{B}\cdot d\vec{a} &=&
  - {1\over c} \int_S \nabla\times \left(\vec{v}\times\vec{B}\right)
    \cdot d\vec{a}\, = \\ \nonumber
  - {1\over c} \oint_C \left(\vec{v}\times\vec{B}\right)\cdot d\vec{l}
  &=& - \oint_C \left(\vec{\beta}\times\vec{B}\right)\cdot d\vec{l} 
\end{eqnarray}
where Stoke's theorem has been used and $\vec{\beta}=\vec{v}/c$. We 
re-write equation~(\ref{F1}) with both encountered line integral on 
the left-hand side
\begin{equation} \label{F2}
  \oint_C \left(\vec{E}+\vec{\beta}\times\vec{B}\right)
  \cdot d\vec{l} = - {1\over c} {d\ \over dt} \Phi_m
\end{equation}
where $\Phi$ is called \index{Magnetic flux}{\it magnetic flux} and 
defined by
\begin{equation} \label{magflux}
  \Phi_m = \int_S \vec{B}\cdot d\vec{a}\ .
\end{equation}
Equation (\ref{F2}) is the fully relativistic version of Faraday's
law. Let us discuss the approximations which lead to its original,
non-relativistic version. The velocity $\vec{\beta}=\vec{v}/c$ in
equation (\ref{F2}) refers to the velocity of the line element
$d\vec{l}$ with respect to the inertial frame in which the calculation
is done. Let us consider a particular line element $d\vec{l}$ and
transfer the electric field to a frame co-moving with this line
element. Equation (\ref{LTE}) yields 
$$\vec{E}^{\,'}=\gamma\,\left(\vec{E}+\vec{\beta}\times\vec{B}\right)
  -{\gamma^2\over\gamma+1}\,\vec{\beta}\,\left(\vec{\beta}\cdot\vec{B}
  \right) \to \vec{E} + \vec{\beta}\times\vec{B} 
  ~~{\rm for}~~ |\vec{v}|\ll c $$
and in this approximation we have
\begin{equation} \label{Faraday}
  \epsilon_{\rm emf} = \oint_C \vec{E}^{\,'}\cdot d\vec{l}
  = - {1\over c} {d\ \over dt} \Phi_m
\end{equation}
where $\epsilon_{\rm emf}$ is called\index{Electromotive force} 
{\it electromotive force} (emf)\index{emf} and the electric 
field $\vec{E}^{\ '}$ is with respect to the frames co-moving with 
$d\vec{l}$. If the velocity difference between the involved line 
elements are small one may define an appropriate rest frame, normally 
the Lab frame, for the entire current loop and the field 
$\vec{E}_{\rm lab}$ in this frame is a good approximation for 
$\vec{E}^{\, '}$ in~(\ref{Faraday}). In this approximation 
\index{Faraday's law}\index{Induction}{\it Faraday's Law of Induction}
is found in most test books. Due to our initial treatment of special 
relativity we do not face the problem to work out its relativistic 
generalization, but instead obtained (\ref{Faraday}) as the limit of 
the generally correct law (\ref{F2}).

\subsection{Lenz's law\label{Lenz_law}}

With the Lorentz force~(\ref{LF}) given, Energy conservation determines 
the minus sign on the right-hand side of Faraday's law~(\ref{Faraday}). 
This is known as \index{Lenz's law}{\it Lenz's law}. For closed, 
conducting circuits the emf~(\ref{Faraday}) will induce a current, 
whose magnitude depends on the resistance of the circuit. \index{Lenz's 
law}Lenz's law states: {\it The induced emf and induced current are in 
such a direction as to oppose the change that produces them.} 
\cite{Tipler} gives many examples. To illustrate the connection 
with energy conservation, we discuss one of them.

We consider a permanent bar magnet moving towards a closed loop that 
has a resistance $R$. The north pole of the bar magnet is defined so 
that the magnetic field points out of it. We arrange the north--south
axis of the magnet perpendicular to the surface spanned by the loop 
and move the magnet toward the loop. The magnet's magnetic field 
through the loop get stronger when the magnet is approaching and
a current is induced in the loop. The direction of the current is
such that its magnetic field is opposite to that of the magnet,
effectively the loop becomes a magnet with north pole towards the
bar magnet. The result is a {\it repulsive} force between bar
magnet and loop. Work against this force is responsible for the
induced current, and its associated heat, in the loop. Would the
sign of the induced current be different an attractive force would
result and the resulting acceleration of the bar magnet as well
as the heat in the loop would violate energy conservation. Note 
that pulling the bar magnet out of the loop does also produce 
energy.

In our treatment the sign of Faraday's law is already given by
the electromagnetic field equation and energy conservation 
determines the sing in equation (\ref{pm_dLF}) for the Lorenz
force.

\subsection{LCR circuit}

\begin{figure}[tb] \vspace{5pc}
\centerline{\hbox{ \psfig{figure=LCR.eps,width=10cm} }}
\caption{Series LCR circuit with an ac generator.\label{fig_LCR}}
\end{figure}
%
We consider a \index{Resistor}resistor $(R)$, a\index{Capacitor}
capacitor $(C)$ and an \index{Inductor}inductor $(L)$ in series, 
see \index{LCR circuit}\index{Circuit}figure~\ref{fig_LCR}. Charge 
conservation implies
\begin{equation} \label{currents}
 I_R\ =\ I_C\ =\ I_L\ = I
\end{equation}
where $I_R$, $I_C$ and $I_L$ are the currents at the resistor,
capacitor and inductor, respectively. The voltages are related to
the currents by the following equations
\begin{equation} \label{V_RC}
V_R\ =\ R\,I ~~{\rm and}~~ V_C\ =\ {Q\over C} ~~{\rm with}~~
{d Q\over d t}\ =\ I
\end{equation}
where $Q(t)$ is the charge on the capacitor. As inductor we consider an 
idealized infinitely long solenoid. The induced (back) electromotive 
force (voltage over the inductor) is the derivative of the magnetic 
flux~(\ref{magflux})
\begin{equation} \label{V_L}
  \epsilon_{\rm back}\ =\ V_L\ =\ \frac{1}{c}{d\Phi_m\over dt}\ 
  =\ L\,{d I\over d t}
\end{equation}
where the constant $L>0$ is called \index{Self inductance}{\it self 
inductance}\index{Inductance} of the coil. In accordance with Lenz's 
law the induced electromotive 
$\epsilon_{\rm back}$ will work against the electromotive force 
$\epsilon_{\rm emf}$ which drives the circuit.  We use complex 
calculus and the physical quantities are given by the real parts. 
For the total voltage we have
\begin{equation} \label{emf}
  \epsilon_{\rm emf}\ =\ \epsilon_{\max}\, e^{-i\omega t}\
  =\ V\ =\ V_L+V_R+V_C 
\end{equation}
where $\omega$ is the \index{Driving frequency}driving frequency. 
We replace in this relation the voltages by their equations listed 
above and eliminate the current $I$ in favor of the time derivative
of the charge of the capacitor and to arrive at the second order 
differential equation
\begin{equation} \label{LCR_QDeqn}
L\, {d^2 Q\over d t^2} + R\, {d Q\over d t} + {Q\over C}\ 
=\ \epsilon_{\max}\, e^{ - i \omega t}\ . 
\end{equation}
The general solution of this equation is obtained as superposition 
of a special solution of the inhomogenous equation and the general 
solution of the homogeneous equation, $Q=Q_h+Q_i$. A special solution 
of the inhomogeneous equation is
\begin{equation} \label{Q_i}
 Q_i\ =\ Q_0\, e^{ - i \omega t} 
\end{equation}
where
\begin{equation} \label{Q_amplitude}
\left( -\omega^2\,L - i\,\omega\,R + {1\over C} \right)\, Q_0 
= \epsilon_{\max} ~~\Rightarrow~~ Q_0 =  { \epsilon_{\max}
   \over  -\omega^2\,L - i\,\omega\,R + 1/C }\ . 
\end{equation}
The general solution of the homogeneous equation follows from the
ansatz $Q_h\sim e^{i \alpha t}$, which implies the algebraic equation
$$   -\alpha^2\,L + i\,\alpha\,R + C^{-1}\ =\ 0\ . $$
The solution is
\begin{equation} \label{omega_h}
  \alpha\ =\ i\,\frac{R}{2L} \pm \omega_h ~~{\rm with}~~
  \omega_h = \sqrt{ {1\over LC} - {R^2\over 4L^2} }
\end{equation}
and we arrive at the general solution of the homogeneous equation
\begin{equation} \label{Q_h}
 Q_h\ =\ A_1\, e^{-t R / (2L)}\, e^{ +i\omega_h\,t} +
         A_2\, e^{-t R / (2L)}\, e^{- i\omega_h\,t}\ . 
\end{equation}
Due to the exponential damping the homogeneous solution dies out 
for large times and we are left with the inhomogeneous solution 
$Q=Q_i$~(\ref{Q_i}). The homogeneous solution has to be used for
getting the initial values right, e.g.~for turning the circuit
on or off. In the large time limit the essential features of the LCR 
circuit are those of the inhomogeneous solution, which we investigate 
now in more details.

For the inhomogeneous solution (\ref{Q_i}) $Q_0$ is a complex constant.
To separate the amplitude of $Q_0$ from its  phase we write
\begin{equation}
 Q_0\ =\ Q_{\max}\, e^{-i\delta} ~~{\rm with}~~ Q_{\max}\ =\ |Q_0|\ 
 =\ {\epsilon_{\max}\over \sqrt{ \left( 1/ C-\omega^2\,L \right)^2 
 + \omega^2\,R^2 } }\ . 
\end{equation}
Following standard conventions, we rewrite the expression for
$Q_{\max}$ as
\begin{equation} \label{impedance_etc}
 Q_{\max}\ =\ {\epsilon_{\max}\over \omega\, \sqrt{ 
 \left[ 1/(\omega\,C) - \omega\,L \right]^2 + R^2 } }\ =\ 
 {\epsilon_{\max}\over \omega\, \sqrt{(X_C-X_L)^2 + R^2 } }\ =\
 {\epsilon_{\max} \over \omega\, Z}\ .
\end{equation}
Here\index{Capacitive reactance}
\begin{eqnarray} \label{X_C}
  X_C &=& \frac{1}{\omega\,C}~~{\rm is\ called}\ 
                   {\it capacitive\ reactance,} \\ \label{X_L}
  X_L &=& \omega\,L~~{\rm is\ called}\ {\it inductive\ reactance,}
\end{eqnarray}
and\index{Reactance}\index{Inductive reactance} the difference 
$X_L-X_C$ is called \index{Total reactance}{\it total reactance}. 
The quantity\index{Impedance}
\begin{equation} \label{Z_impedance}
 Z = \sqrt{(X_C-X_L)^2+R^2}~~{\rm is\ called}\ {\it impedance}.  
\end{equation} 
Using this notation the original equation (\ref{Q_amplitude}) for 
$Q_0$ becomes 
\begin{equation}
  Q_0\,=\,\frac{\epsilon_{\max}}{\omega\,\left(X_C-X_L-iR\right)}
  \,=\,{\epsilon_{\max}\,(X_C-X_L+i\,R)\over\omega\,Z^2}\ .
\end{equation}
It follows from $Q_0=Q_{\max}\,\cos(\delta)-i\,Q_{\max}\,\sin(\delta)$
that the phase $\delta$ is given by\footnote{Note that in \cite{Tipler}
the phase $\delta$ is defined with respect to the current instead of
the charge, what leads to the equation $\delta=(X_L-X_C)/R$.}
\begin{equation}
  \tan {\delta}\ =\ - {\sin (\delta )\over \cos (\delta )}\ =\
  - {{\rm Im}\,Q_0\over {\rm Re}\,Q_0}\ =\ \frac{R}{X_L-X_C}\ . 
\end{equation}
From equation~(\ref{V_RC}) we see that $V_C$ is in phase with the
charge $Q$ 
\begin{equation}
  V_C = {Q_{\max}\over C}\, e^{-i\delta}\, e^{-i\omega t}
\end{equation}
and the phase of the current $I$ is shifted by $\pi/2$ with respect 
to $V_C$:
\begin{eqnarray} \nonumber
  I\,=\,{d Q\over d t}\,=\,-i\,\omega\,Q_{\max}\,e^{-i\delta}\,
  e^{-i\omega t} &=& \omega\,Q_{\max}\, e^{-i\delta-i\pi/2}\, 
  e^{-i\omega t} \\ \label{current_phase}
  &=& I_{\max}\, e^{-i\delta-i\pi/2}\, e^{-i\omega t}
\end{eqnarray}
where we used $-i=\exp(-i\pi/2)$. Note also that $I_{\max}= \omega\,
Q_{\max}$ together with equation~(\ref{impedance_etc}) implies
\begin{equation} \label{Imax}
  I_{\max}\,=\,\frac{\epsilon_{\max}}{Z}\,=\,{\epsilon_{\max}\over
  \sqrt{\left[ 1/(\omega\,C)-\omega\,L \right]^2 + R^2 } }\ .
\end{equation}
The voltage over the resistor is in phase with the current
\begin{equation}
V_R\ =\ {I_{\max}\over R}\,e^{-i\delta-i\pi/2}\,e^{-i\omega t}
\end{equation}
whereas the voltage over the inductor is shifted by another $\pi/2$ 
phase factor:
\begin{equation}
  V_L\ =\ L\, {d I\over d t}\ =\ \omega\,L\,I_{\max}\,
  e^{-i\delta t-i\pi}\,e^{i\omega t}\ .
\end{equation}
The current amplitude (\ref{Imax}) takes its maximum at the
\index{Resonance frequency}{\it resonance frequency} 
\begin{equation} \label{omega_resonance}
 \omega _0\ =\ {1 \over \sqrt{LC} }\ .
\end{equation}
Note that $Q_{\max}(\omega_0)=\epsilon_{\max}/(\omega_0 R)$ due
to equation~(\ref{impedance_etc}). For $R\ne 0$ the maximum of 
$Q_{\max}$ is shifted by an amount of order $(R/L)^2$ away from 
$\omega_0$. We find it by differentiation of $Q_{\max}$ with 
respect to $\omega^2$:
\begin{equation} \label{omega_max}
 - 2\,L\,( C^{-1}-\omega^2_{\max}\,L)+R^2=0 ~~\Longrightarrow~~
 \omega_{\max} = \sqrt{ {1\over LC} - {R^2\over 2\,L^2} }\ .
\end{equation}
Note, that this $\omega_{\max}$ value is lower than the frequency 
$\omega_h$ of the damped homogeneous solution~(\ref{Q_h}), while
the resonance frequency $\omega_0$ is larger.

\begin{thebibliography}{12}
\addcontentsline{toc}{chapter}{Bibliography}%

\bibitem{Co98} P.J. Mohr and B.N. Taylor, {\it CODATA Recommended 
Values of the Fundamental Physical Constants: 1998}, J. of
Physical and Chemical Reference Data, to appear. See the website
of the National Institute of Standards and Technology (NIST)
at {\tt physics.nist.gov/constants}.

\bibitem{Ei1905} A. Einstein, {\it Zur Elektrodynamik bewegter K\"orper},
Annalen der Physik 17 (1905) 891--921.

\bibitem{Ei1906} A. Einstein, {\it Ist die Tr\"agheit eines K\"orpers 
von seinem Energieinhalt abh\"angig?},
Annalen der Physik 18 (1906) 639--641.

\bibitem{Hef1} C. Hefele and R. Keating, Science 177 (1972) 166, 168.

\bibitem{Jackson} J.D. Jackson, {\it Classical Electrodynamics}, Second 
Edition, John Wiley \& Sons, 1975.

\bibitem{Petley} B.W. Petley, Nature 303 (1983) 373.

\bibitem{Rind1} W. Rindler, {\it Introduction to Special Relativity},
Clarendon Press, Oxford 1982.

\bibitem{Tipler} P.A. Tipler, {\it Physics for Scientists and
Engineers}, Worth Publishers, 1995.

\bibitem{Vessot} R.F.C. Vessot and M.W. Levine, GRG 10 (1979) 181.

\end{thebibliography}

\relax
\end{document}
