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\LARGE ``Proof'' of the constant speed of light \normalsize
% \author{Christopher Neufeld} %
% \date{18 April 1995} %
First off, while it's a bit unconventional to do so, I will be
restricting myself to the MKS units system. That is, measures are taken
in units of metres, kilograms, and seconds.
We will begin this analysis by assuming that the classical Maxwell
equations are valid. These equations were empirically deduced, to quote
my undergraduate E\&M professor, ``by men in drafty castles, wearing
smelly wigs, rubbing bits of cat against bits of glass.'' They are:
\begin{equation}
\begin{array}{ll}
\vec{\nabla} \cdot \vec{D} = \rho &
\vec{\nabla} \cdot \vec{B} = 0 \\
\vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} &
\vec{\nabla} \times \vec{H} = \vec{j} + \frac{\partial \vec{D}}{\partial t}
\end{array}
\label{eq:maxwell}
\end{equation}
Now, for reasons that will become clear we are interested in the
propagation of light in a vacuum. So, it is sufficient to consider these
equations in a vacuum, so called ``free space.'' In free space, the
following relations hold:
\begin{equation}
\begin{array}{ll}
\rho = 0 &
\vec{j} = 0 \\
\vec{D} = \epsilon_{0} \vec{E} &
\vec{H} = \frac{1}{\mu_{0}} \vec{B} \nonumber
\end{array}
\end{equation}
Where $\epsilon_{0}$ and $\mu_{0}$ are fundamental physical constants
describing properties of the vacuum in MKS units.
Substituting these values in to equations~\ref{eq:maxwell} yields:
\begin{equation}
\begin{array}{ll}
\epsilon_{0} \vec{\nabla} \cdot \vec{E} = 0 &
\vec{\nabla} \cdot \vec{B} = 0 \\
\vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} &
\vec{\nabla} \times \vec{B} = \epsilon_{0} \mu_{0}
\frac{\partial \vec{E}}{\partial t}
\end{array}
\label{eq:vacuummaxwell}
\end{equation}
We can combine the bottom two equations to produce:
\begin{equation}
\vec{\nabla} \times ( \vec{\nabla} \times \vec{E} ) = - \epsilon_{0}
\mu_{0} \frac{\partial^{2} \vec{E}}{\partial t^{2}} \nonumber
\end{equation}
Exploiting the fact that $\vec{\nabla} \cdot \vec{E} = 0$ allows us to
write this as:
\begin{equation}
\vec{\nabla}^{2} \vec{E} = \epsilon_{0} \mu_{0}
\frac{\partial^{2} \vec{E}}{\partial t^{2}} \nonumber
\end{equation}
This is just a wave equation for the electric field, and represents a
wave whose speed of propagation is
$\frac{1}{\sqrt{\epsilon_{0} \mu_{0}}}$.
All this was just to arrive at the following observation: if the we are
to believe Maxwell's equations, we are forced to conclude that the speed
of light is a function solely of the two parameters $\epsilon_{0}$ and
$\mu_{0}$. These two parameters are physical constants describing the
properties of the vacuum.
If we now assume that the laws of physics are constant for observers
moving with fixed but different velocities, then we have to conclude that
the speed of light itself is a constant for these different observers.
This conclusion has been verified experimentally in the negative results
of searches for the ``luminiferous ether.''
\end{document}