The Mathematical Theory of RelativityEddington, Arthur Stanley, Sir
Science
The Mathematical Theory of Relativity
Eddington, Arthur Stanley, Sir
Relativity (Physics)
There is another way of specialising coordinates which may be mentioned
here for completeness. It is always possible to choose coordinates such that
the determinant $g = -1$ everywhere (as in Galilean coordinates). This is
explained in \SecRef{49}.
We may also consider another class of specialised coordinates---those
which are permissible in special problems. There are certain (non-Euclidean)
coordinates found to be most convenient in dealing with the gravitational
field of the sun, Einstein's or de~Sitter's curved world, and so on. It must be
remembered, however, that these refer to idealised problems, and coordinate-systems
\index{Coordinate-systems!natural}%
\index{Coordinate-systems!proper}%
with simple properties can only be approximately realised in nature.
\PageSep{81}
\index{Coordinate-systems!statical}%
\index{Static coordinates}%
If possible a \emph{static system} of coordinates is selected, the condition for this
being that all the~$g_{\mu\nu}$ are independent of one of the coordinates~$x_{4}$ (which
must be of timelike character\footnotemark).\footnotetext
{$dx_{4}$ will be timelike if $g_{44}$ is always positive.}
In that case the interval corresponding to
any displacement~$dx_{\mu}$ is independent of the ``time''~$x_{4}$. Such a system can,
of course, only be found if the relative configuration of the attracting masses
is maintained unaltered. If in addition it is possible to make $g_{14}$, $g_{24}$, $g_{34} = 0$
the time will be reversible, and in particular the forward velocity of light
along any track will be equal to the backward velocity; this renders the
application of the name ``time'' to~$x_{4}$ more just, since one of the alternative
conventions of \SecRef{11} is satisfied. We shall if possible employ systems which
are static and reversible in dealing with large regions of the world; problems
in which this simplification is not permissible must generally be left aside as
insoluble--e.g.\ the problem of two attracting bodies. For small regions of the
world the greatest simplification is obtained by using natural coordinates.
\Section{37.}{Einstein's law of gravitation}
\index{Einstein's law of gravitation}%
\index{G@$G_{\mu\nu}$ (Einstein tensor)}%
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