196 Measurements in physically relevant space-times
• an increase in the thrust if the observer is in region I; then the thrust points
to the center of symmetry;
• a decrease in the thrust if the observer is in region II; then the thrust points
away from the center of symmetry;
• an increase in the thrust if the observer is in region III; then the thrust points
away from the center of symmetry.
We see clearly that monitoring the thrust alone would not permit the observer
to distinguish between regions I and III, although he could recognize at once
that he was in region II and in that case deduce the local inward direction.
One must have an independent way to recognize which of the above regions the
space-ship occupies. As implied by the previous discussion, the observer must be
able to change the angular velocity of revolution without changing the radius of
the orbit, at least not in an appreciable way. To this end we shall describe the
following experimental device. The main thrust fixes the radial direction from
within the space-ship while the direction of the harmonic oscillations inside the
pipe fixes the plane of the orbit as the one orthogonal to it. Then let a set of
rails go across the space-ship orthogonal to the radial direction and parallel to the
orbital plane. The rails will be approximately parallel to the orbit, at least within
the space-ship. Let us have a small box, of negligible mass, which slides freely on
these rails. Inside the box we have a test mass μ
0
linked by a spring to one side
of the box which is perpendicular to the direction of the main thrust. The spring
is rigidly fastened to that side of the box, so it can stretch or compress but not
bend transversely. The acceleration produced by the main thrust is transferred
to the mass μ
0
by the spring, which is then stretched or compressed by a given
amount and is regulated so that the test mass remains at the center of the box.
We assume that this position of the test mass corresponds to the average radius
of the orbit, although its actual coordinate value is not known. The whole set
orbits rigidly with proper angular velocity |
p
ζ|. Let us now give an impulse to
the small box parallel to the rails; the small box will then slide on the rails and
consequently the mass μ
0
will acquire a new angular velocity ζ with respect to
infinity or, equivalently, will be acted upon by a Coriolis force with respect to the
orbiting frame. In this case the initial acceleration exerted by the spring on the
mass will no longer be sufficient to keep it in equilibrium at the center of the box.
The spring, in fact, will either stretch or compress with respect to its initial state.
If it is stretched, then it means that in order to remain at the center of the box the
particle needs a bigger acceleration if it was initially stretched or a smaller one
if it was initially compressed. If on the contrary the spring is compressed, then
it means that the mass needs either a bigger or a smaller acceleration according
to whether it was initially compressed or stretched, respectively.
In all cases, by readjusting the elastic properties of the spring, one can measure
the extra acceleration needed to keep the mass μ
0
at the center of the box. This
test case can be exploited to vary the angular velocity of revolution of the entire