146 The Poetry of Physics and The Physics of Poetry
perpendicular to the x and t- axes so that a length, which is purely spatial
in the primed co-ordinate system of the moving observer, becomes both
spatial and temporal in the unprimed co-ordinates system of the
stationary observer.
Another example of this rotation in the space-time continuum
was encountered when we considered the time dilation of a moving
clock. In the frame moving with the clock, the two events of the clock
reading 8:00 a.m. and 9:00 a.m. are separated by a purely temporal
length. In the stationary frame, however, the two events are separated by
both a spatial and temporal duration, since the clock moves with respect
to the stationary observer. Einstein utilized the Minkowski four-
dimensional space-time continuum to formulate his theory of gravity. He
acknowledged his debt to Murkowski referring to his contribution as
follows: “Without it, the General Theory of Relativity would perhaps get
no farther than its long clothes.”
Although the space-time continuum is four-dimensional in the
Special Theory of Relativity, its geometry is still basically Euclidean.
Space is not warped or curved. The shortest distance between two
points is a straight line, as is the trajectory of a light beam. In the
General Theory of Relativity, however, the geometry of the space-time
continuum is no longer Euclidean; space is curved. The curvature of
space was demonstrated, theoretically, using the equivalence principle
and, subsequently, verified experimentally, with the detection of the
curvature of a beam of light passing close to the Sun during a solar
eclipse. The shortest distance between two points in the vicinity of the
Sun is not a straight line. The most basic axiom of Euclidean geometry,
which Kant had elevated to an a priori truth, is ironically and empirically
untrue in the case of light passing close to the Sun. One must describe
the four-dimensional space-time continuum in terms of non-Euclidean
geometry, which, fortunately for Einstein, had been worked out by
Lobachevsky, Gauss and Riemann in the nineteenth century.
In non-Euclidean geometry, the ratio of the circumference of a circle
to its diameter is no longer equal to π as it is in Euclidean geometry. We
can show that the ratio of the circumference to the diameter of a circle in
a rotating frame of reference is less than π and hence, invoking the
equivalence principle demonstrates the need to describe the space-time
continuum embedded in a gravitational field in terms of non-Euclidean
geometry. Let us consider a disc, which is a perfect circle at rest with the
circumference equal to π times its diameter (C = πD). Let us consider