But of course a certain dynamical value is also attached to the
g's, because by the transfer of our axes from the Galilean type
we have made a change which is equivalent to the introduction of
a gravitational field; and the g's must specify the field. That
is, these g's are the expressions of our experiences, and hence
their values can not depend upon the use of any special axes; the
values must be the same for all selections. In other words, whatever
function of the coordinates any one g is for one set of axes, if other
axes are chosen, this g must still be the same function of the new
coordinates. There are ten g's defined by differential equations;
so we have ten covariant equations. Einstein showed how these g's
could be regarded as generalized potentials of the field. Our own
experiments and observations upon gravitation have given us a certain
knowledge concerning its potential; that is, we know a value for it
which must be so near the truth that we can properly call it at least
a first approximation. Or, stated differently, if Einstein succeeds in
deducing the rigid value for the gravitational potential in any field,
it must degenerate to the Newtonian value for the great majority of
cases with which we have actual experience. Einstein's method, then,
was to investigate the functions (or equations) which would satisfy
the mathematical conditions just described. A transformation from
the axes used by the observer in the following box may be made so as
to introduce into the equations the gravitational field recognized
by an observer on the earth near the box; but this, obviously, would
not be the general gravitational field, because the field changes as
one moves over the surface of the earth. A solution found, therefore,
as just indicated, would not be the one sought for the general field;
and another must be found which is less stringent than the former
but reduces to it as a special case. He found himself at liberty to
make a selection from among several possibilities, and for several
reasons chose the simplest solution. He then tested this decision
by seeing if his formulæ would degenerate to Newton's law for the
limiting case of velocities small when compared with that of light,
because this condition is satisfied in those cases to which Newton's
law applies. His formulæ satisfied this test, and he therefore was
able to announce a "law of gravitation," of which Newton's was a
special form for a simple case.
Public-domain text, read in full here on John Shaqi.
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