Einstein's Theories of Relativity and Gravitation: A selection of material from the essays submitted in the competition for the Eugene Higgins prize of $5,000Bird, J. Malcolm (James Malcolm)
Philosophy
Einstein's Theories of Relativity and Gravitation: A selection of material from the essays submitted in the competition for the Eugene Higgins prize of $5,000
Bird, J. Malcolm (James Malcolm)
Relativity (Physics)
Thus Einstein, interpreting the equality of gravitational and inertial
mass, showed that the same quality manifests itself according to
circumstances as "weight" or as inertia, and that all force is purely
relative and may be treated as one phenomenon (an interruption in
energy flow). This "Principle of Equivalence" shows that small portions
of the World-Fabric, observed from a freely moving particle (free of
force), could be treated as small portions of the World-Frame. [9]
If such observations were practicable, we could determine the Fabric
curvature by referring point-event measurements to equation (1). We
cannot observe from unique tracks but we can observe them from our
restrained situation. Their importance is now apparent, because, by
tracing them over a region, we are tracing something absolute in the
Fabric--its geometrical character. We study this curvature by exploring
separation-intervals on the tracks of freely moving bodies, relating
these separation-intervals to actual measurements in terms of space
and time components depending on the observer's reference system. The
law of curvature must be the law of gravitation. To illustrate the
lines on which Einstein proceeded to survey the World-Fabric from
the earth we will consider a similar but more simple problem--the
survey of the sea-surface curvature from an airship. We study this
curvature by exploring small distances on the tracks of ships (which we
must suppose can only move uniformly on unique tracks--arcs of great
circles), relating such distances to actual measurements in terms of
length and breadth components depending on the observer's reference
system. This two-dimensional surface problem can be extended to the
four-dimensional Fabric one.
We consider the surface to be covered by two arbitrarily drawn
intersecting series of curves: curves in one series not intersecting
each other, vide figure. This Gaussian system of coordinates is
appropriate only when the smaller the surface considered, the
more nearly it approximates to Euclidean conditions. It admits of
defining any point on the surface by two numbers indicating the curves
intersecting at that point. $P$ is defined by $x_1$, $x_2$. $P_1$
(very near $P$) is defined by $x_1 + dx_1$, $x_2 + dx_2$. The equation
for the minute distance $s$ between two adjacent points in such a
system is given by the general formula
$$s^2 = g_{11}d{x_1}^2 + g_{12}dx_1dx_2 + g_{22}d{x_2}^2\,.$$
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