The Mathematical Theory of RelativityEddington, Arthur Stanley, Sir
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The Mathematical Theory of Relativity
Eddington, Arthur Stanley, Sir
Relativity (Physics)
The vanishing of~\Eq{(90.52)} gives the remarkable equation
\[
\kappa^{\mu} = -\tfrac{1}{3}\beta J^{\mu}.
\Tag{(90.71)}
\]
And since $J_{\mu}^{\mu} = 0$ \Eq{(73.77)}, we must have
\[
\kappa_{\mu}^{\mu} = 0,
\Tag{(90.72)}
\]
agreeing with the original limitation of~$\kappa_{\mu}$ in~\Eq{(74.1)}.
We see that the formula for~$\rho_{0}$ \Eq{(90.62)} agrees with that previously found~\Eq{(89.3)}
having regard to the limitation $\kappa_{\mu}^{\mu} = 0$.
The result~\Eq{(90.62)} becomes by~\Eq{(90.71)}
\[
\rho_{0} = -\frac{\beta^{2}}{12\pi}\, J_{\mu} J^{\mu}.
\]
This shows that matter cannot be constituted without electric charge and
current. But since the density of matter is always positive, the electric charge-and-current
inside an electron must be a \emph{\Chg{space-like}{spacelike}} vector, the square of its
\index{Electron!magnetic constitution of}%
\index{Magnetic constitution of electron}%
length being negative. It would seem to follow that the electron cannot be
built up of elementary electrostatic charges but resolves itself into something
more akin to magnetic charges.
It will be noticed that the result~\Eq{(90.72)} is inconsistent with the formula
$\kappa_{\alpha} \kappa^{\alpha} = \kappa_{\alpha}^{\alpha}$ which we have found for empty space~\Eq{(89.4)}. The explanation is afforded
by~\Eq{(90.71)} which requires that a charge-and-current vector must exist wherever
$\kappa_{\mu}$~exists, so that no space is really empty. On Weyl's hypothesis $\kappa_{\alpha}^{\alpha} = 0$ is the
condition which holds in all circumstances; whilst the additional condition
$\kappa_{\alpha}^{\alpha} = \kappa_{\alpha} \kappa^{\alpha}$ holding in empty space reduces to the condition expressed by $J^{\alpha} = 0$.
It is supposed that outside what is ordinarily considered to be the boundary
of the electron there is a small charge and current $\dfrac{3}{\beta}\, \kappa^{\alpha}$ extending as far as the
electromagnetic potential extends.
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