The question of course must be raised as to what can be said in regard
to phenomena which so far as we know do not have an electromagnetic
origin. In particular what can be done with respect to gravitational
phenomena? Before, however, showing how this problem was attacked by
Einstein; and the fact that the subject of my address is Einstein's
work on gravitation shows that ultimately I shall explain this, I
must emphasize another feature of Minkowski's geometry. To describe
the space-time characteristics of any event a point, defined by its
four coordinates, is sufficient; so, if one observes the life-history
of any entity, e.g., a particle of matter, a light-wave, etc., he
observes a sequence of points in the space-time continuum; that is,
the life-history of any entity is described fully by a line in this
space. Such a line was called by Minkowski a "world-line." Further,
from a different point of view, all of our observations of nature
are in reality observations of coincidences, e.g., if one reads
a thermometer, what he does is to note the coincidence of the
end of the column of mercury with a certain scale division on
the thermometer tube. In other words, thinking of the world-line
of the end of the mercury column and the world-line of the scale
division, what we have observed was the intersection or crossing of
these lines. In a similar manner any observation may be analyzed;
and remembering that light rays, a point on the retina of the eye,
etc., all have their world-lines, it will be recognized that it is
a perfectly accurate statement to say that every observation is the
perception of the intersection of world-lines. Further, since all we
know of a world-line is the result of observations, it is evident
that we do not know a world-line as a continuous series of points,
but simply as a series of discontinuous points, each point being where
the particular world-line in question is crossed by another world-line.
It is clear, moreover, that for the description of a world-line
we are not limited to the particular set of four orthogonal axes
adopted by Minkowski. We can choose any set of four-dimensional
axes we wish. It is further evident that the mathematical expression
for the coincidence of two points is absolutely independent of our
selection of reference axes. If we change our axes, we will change
the coordinates of both points simultaneously, so that the question
of axes ceases to be of interest. But our so-called laws of nature are
nothing but descriptions in mathematical language of our observations;
we observe only coincidences; a sequence of coincidences when put in
mathematical terms takes a form which is independent of the selection
of reference axes; therefore the mathematical expression of our laws
of nature, of every character, must be such that their form does
not change if we make a transformation of axes. This is a simple but
far-reaching deduction.
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