The General Theory of Relativity 147
the same disc rotating such that its outer edge has the velocity v with
respect to an observer sitting at rest at the centre of the disc. To this
observer, a length on the edge of the disc will appear contracted by the
Lorentz–Fitzgerald factor 1 −v^2 /c^2 because of the motion of the outer
edge. This observer will measure the circumference to be πD 1 −v^2 /c^2.
The diameter of the circumference will not appear contracted, however,
because it lies perpendicular to the motion of the disc. The ratio of
the circumference to the diameter of a circle in this rotating frame
of reference is not π but π 1 −v^2 /c^2. The geometry in this frame of
reference is non-Euclidean, and since a frame in non-uniform motion is
equivalent to a frame in which there is a gravitation field, we see that the
geometry of the space-time continuum is also non-Euclidean. Space is
curved in a gravitational field. The slowing down of a clock is also easily
demonstrated in our rotating frame. A clock at the edge of the disc will
appear to slow down to our observer at the centre by virtue of its
velocity.
The curvature of space is difficult to comprehend in three dimensions,
let alone four dimensions, as required in relativity theory. Let us consider
a two-dimensional example to give ourselves a feeling for non-Euclidean
geometry. We shall compare a flat, two-dimensional plane, described by
Euclidean geometry, with the surface of a sphere. On the flat plane, the
shortest distance between two points is a straight line and the ratio of the
circumference to the diameter of a circle is π. The surface of a sphere is a
curved, two-dimensional space, described by non-Euclidean geometry.
The shortest distance between two points is not a straight line but a
segment of a circle. This circle is called a great circle and the path
between the two points is called the geodesic. Let us now consider a
circle inscribed on the surface of the sphere. This circle is defined as the
locus of all points equidistant from the centre of the circle. If we were to
measure the circumference and the diameter of this circle, we would
discover that the ratio of these quantities is less than π.
In formulating the gravitational interaction between masses, Einstein
does not use the concept of one mass exerting a force upon another.
Instead, he calculates the curvature of the space-time continuum due to
the presence of matter. He then assumes that a mass will travel along the
shortest possible path in this non-Euclidean, four-dimensional space.
The path, geodesic or world line, along which the particle travels, is
determined by the curvature of the space and the initial position and
velocity of the particle. The world line of the Earth is not due to the force