The Mathematical Theory of RelativityEddington, Arthur Stanley, Sir
Science
The Mathematical Theory of Relativity
Eddington, Arthur Stanley, Sir
Relativity (Physics)
In empty space this becomes
\[
\Wave F_{\mu\nu} = 2B_{\mu\nu\alpha\epsilon} F^{\epsilon\alpha}
\Tag{(74.42)}
\]
for an infinite world. For a curved world undisturbed by attracting matter,
in which $G_{\mu}^{\epsilon} = \lambda g_{\mu}^{\epsilon}$, $B_{\mu\nu\alpha\epsilon} = \frac{1}{3}\lambda(g_{\mu\nu} g_{\alpha\epsilon} - g_{\mu\alpha} g_{\nu\epsilon})$, the result is
\[
(\Wave + \tfrac{4}{3}\lambda)F_{\mu\nu} = 0.
\Tag{(74.43)}
\]
It need not surprise us that the velocity of propagation of electromagnetic
potential and of electromagnetic force is not the same (cf.~\Eq{(74.33)} and~\Eq{(74.43)}).
The former is not physically important since it involves the arbitrary convention
$\kappa_{\alpha}^{\alpha} = 0$.
But the result~\Eq{(74.42)} is, I think, unexpected. It shows that the equations
of propagation of electromagnetic force involve the Riemann-Christoffel tensor;
and therefore this is not one of the phenomena for which the ordinary Galilean
equations can be immediately generalised by the principle of equivalence.
\PageSep{177}
This naturally makes us uneasy as to whether we have done right in adopting
the invariant equations of propagation of light ($ds = 0$, $\delta\int ds = 0$) as true in
all circumstances; but the investigation which follows is reassuring.
\Subsection[(c)]{Propagation of a wave-front.}
The conception of a ``ray'' of light in physical optics is by no means
elementary. Unless the wave-front is of infinite extent, the ray is an abstraction,
and to appreciate its meaning a full discussion of the phenomena of
interference fringes is necessary. We do not wish to enter on such a general
discussion here; and accordingly we shall not attempt to obtain the formulae
for the tracks of rays of light for the case of general coordinates \Foreign{ab initio}.
Our course will be to reduce the general formulae to such a form, that the
subsequent work will follow the ordinary treatment given in works on physical
optics.
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