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)
The $g$'s may be constants or functions of $x_1$, $x_2$. Their
value is dependent on the observer's reference system and on the
geometrical character of the surface observed. The curves being
arbitrary, the formula is appropriate for any reference system,
or even if the observer does not know exactly what his reference
system is. (The Fabric observer does not know what his space and time
partitioning actually is because he is in a gravitational field). It
is the $g$'s which disclose the geometry of an observer's partitions,
and their values also contain a reflection of the character of the
region observed.
We find $s$ by direct exploration with a moving ship ($\Omega$ is found
by direct exploration with a freely moving particle); $dx_1$, $dx_2$
are the observed length and breadth measurement differences which we
have to relate to $s$. By making sufficient observations in a small
area and referring them to the general formula we can find the values
of the $g$'s for the observer's particular reference system. Different
values for $g$'s will be found if the observer changes his reference
system, but there is a limitation to the values so obtainable owing to
the part played by the surface itself, which is diffidently expressing
its intrinsic geometrical character in the $g$'s in each observation.
EINSTEIN'S RESULTS
Thus we approach the absolute character of the surface through
the relative nature of the observer's reference system. There is a
relationship common to all values of the $g$'s that belong to the
same curvature. This relationship is expressed by a differential
equation. It is this equation of curvature that the airship's observer
must find. Einstein's problem was similar, but he was concerned with
four dimensions, which entailed a general formula with ten $g$'s,
and he had to find a set of differential equations of the second
order to determine the law of Fabric curvature. He divided the Fabric
into regions: I. World-Frame--beyond influence of energy. II. Empty
region--free of energy, but under its influence. III. Region containing
free energy only. Each region has a characteristic curvature. By
means of an absolute differential calculus--a wonderful mathematical
scaffolding erected by Riemann, Christoffel and others--involving the
theory of tensors, he succeeded in finding such a set of equations. He
kept the following points in view: (1) The equations must not only
give the character of region II, but must satisfy the special case
of region I; (2) They must be independent of any partitioning system,
because the General Theory of Relativity demands that a law of nature
be in a form appropriate for all observers whatever their position
and motion; (3) They must be concerned with energy which is conserved,
not mass which the Special Theory showed dependent on velocity. This
set of differential equations which shows how the curvature of the
Fabric at any point links to the curvature at neighboring points is
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