Extracts from “The Philosophy of Space and Time”

by Hans Reichenbach

1The relativity theory of dynamics is not a purely academic matter, for it upsets the Copernican worldview. It is meaningless to speak of a difference in truth claims of the theories of Copernicus and Ptolemy; the two conceptions are equivalent descriptions. What had been considered the greatest discovery of western science compared to antiquity, is now denied its claim to truth. However much this fact may caution us in the formulation and evaluation of scientific results, it does not signify a regression in the historical development of science. The theory of relativity does not say that the conception of Ptolemy is correct; rather it contests the absolute significance of either theory. It can defend this statement only because the historical development passed through both of them, and because the conquest of the Ptolemaic cosmology by Copernicus gave rise to the new mechanics, which in turn gave us the means to recognize also the one-sidedness of the Copernican worldview. The road to truth has followed in this case the purest form of the dialectic which Hegel considered essential in every historical development.

Of course, it would be an overestimation to interpret Mach’s ideas as the completion of the synthesis. When Mach replied to Newton that centrifugal forces must be explained by means of relative motion alone, he did not formulate a physical theory, but only the beginning of a program for a physical theory which must eventually deal with all mechanical phenomena, not only centrifugal force. Above all it must explain relativistically the phenomena of motion in a gravitational field, e.g., the motion of the planets. The greatest achievement of Newtonian mechanics was that it gave a dynamic foundation to the Copernican worldview. While from the point of view of kinematics there existed no difference between the universes of Ptolemy and Copernicus, Newton decided in favor of Copernicus from the point of view of dynamics. It was only for this particular world description that his theory of gravitation offered a mechanical explanation. The complicated planetary orbits of Ptolemy, on the other hand, did not fit into any explanation. If we wish to establish the equivalence of both world conceptions, we must find a theory of gravitation sufficiently general to explain the Copernican and also the Ptolemaic planetary motion as a gravitational phenomenon. Herein lies the great achievement of Einstein, compared to which the ideas of Mach appear only as preliminary suggestions: Einstein has indeed found such a comprehensive theory of gravitation; and it is with this discovery, which places him on the same plane as Copernicus and Newton, that the problem of the relativity of motion has been brought to a conclusion…

…It is for this reason that the idea of simplicity cannot be used to decide between the Ptolemaic and Copernican conceptions. The Copernican conception is indeed simpler, but this does not make it any "truer," since this simplicity is descriptive. The simplicity is due to the fact that one of the conceptions employs more expedient definitions. But the objective state of affairs is independent of the choice of definitions; this choice can result in a simpler description, but it cannot yield a "truer" picture of the world. That these definitions, e.g., the definition of rest according to Copernicus, lead to a simpler description, of course expresses a feature of reality and is therefore an objective statement. The choice of the simplest description is thus possible only with the advance of knowledge and can in general be carried through only within certain limits. One description may be simplest for some phenomena while a different description may be simplest for others; but no simplest description is distinguished from other descriptions with regard to truth. The concept of truth does not apply here, since we are dealing with definitions…

2We now turn to the consequences of dynamic relativity, which goes beyond the epistemological relativity. For this purpose we must analyze Einstein's theory of gravitation, since Einstein adopted Mach's idea of dynamic relativity and developed it further. Whereas Mach restricted his investigations to rotation, Einstein applied the principle to all kinds of motions; consequently his formulation is superior. He was able to give this general formulation by transforming the ideas of Mach into a differential principle.

Einstein expressed his principle of equivalence in the form of a thought experiment. Let a mass m be suspended by a spring in a closed compartment such as an elevator (see figure below).

Figure showing equivalence of acceleration and gravitation.

A physicist in this compartment observes suddenly that the spring expands. He can easily verify this expansion by using a measuring rod. The increase in the tension of the spring indicates a stronger pull of the mass m. How can the physicist find the cause of this pull? He could give two explanations.

Explanation I. The compartment has received an upward acceleration (in the direction of arrow b) from some external force. The effect of the inertia of the mass m is therefore a downward pull opposite to the direction of the acceleration.

Explanation II. The compartment has remained at rest, but a downward directed gravitational field g (arrow g) has arisen and therefore exerts a stronger pull on the mass m.

It is impossible to decide experimentally between these two explanations inside the compartment. This is still true if we permit the physicist to look out of a window, since he will observe only kinematic phenomena, and these do not enable him to decide between the two explanations…

3We shall now investigate the problem of rotation, on the basis of Einstein’s idea that a special gravitational field has to be assumed for every coordinate system.

If we consider, on the one hand, the earth as rotating, this motion must be relative to one of the Newtonian inertial systems which can be introduced as an approximation. Relative to this coordinate system there exists no gravitational field, only an inertial field. If we consider, on the other hand, the earth to be at rest, there must be a tensorial gravitational field relative to a system of axes to which the earth is rigidly connected. This gravitational field manifests itself as a rotational field, which might be compared with the rotational field of a three-phase electric current whose three phases run through a correspondingly divided coil. Just as such a field whirls iron filings in its interior, so the tensorial gravitational field moves the fixed stars around in a circle. The stars will thus move with the same velocities around the axis of the field. At the same time, light rays, which are deflected in the gravitational field like heavy bodies, are rotated. Only the introduction of such a rotational field leads to the dynamic equivalent of the rotating coordinate system. A number of objections4 that have been raised against this conception will be discussed in the following paragraphs.

The equivalence of rotating coordinate systems introduces velocities above the velocities of light. Points of the coordinate system that lie farther out will have increasingly higher peripheral velocities and points which lie outside of a circle with radius r = c / omega (omega is the angular velocity) will therefore have a peripheral velocity omega r greater than c relative to the coordinate system. The planet Neptune will already have a velocity greater than that of light if we consider the earth at rest; the fixed stars will have even greater velocities. This consequence appears to contradict a requirement of the theory of relativity.

The conclusion is erroneous, however, since we are here dealing with a problem in the general and not in the special theory of relativity. In the special theory we can maintain the velocity 3\super. 100 cm/sec. as the limiting value because we allow only certain coordinate systems relative to which all velocities are to be measured. If we admit arbitrary coordinate systems, the number 3\super.00 cm/sec. can be exceeded. The limiting character of the velocity of light, however, can be maintained even in the general theory of relativity, yet the assertion is formulated differently. Given any two mass points, light signals will be the fastest connection between them. Light is the fastest messenger; it moves faster than any other means of communication from the same place and at the same time. We formulated this property above with the aid of the concept first-signal. This principle is satisfied even if we assume that the coordinate system is rigidly connected to the earth. A light signal sent from the planet Neptune along a tangent to the planet’s orbit moves faster than the planet itself and will run away from it. Here light has an even greater velocity, although the planet itself already exceeds the velocity 3\super.00 cm/sec. This value has therefore no significance if completely arbitrary space-time measurements are admitted.

In addition there results another restriction from the limiting character of the velocity of light. The system of axes rigidly connected to the earth can ideally be extended indefinitely, but outside the circle r = c/\'f9 it is impossible to realize the axes materially. Outside this circle there can be no material points which are at rest relative to the coordinate system; in this case, the coordinate system is no longer a real system. Neptune has its high peripheral velocity therefore only relative to ideal rest points, but not relative to points which can be realized materially. This restriction is due to the fact that in the immediate neighborhood of Neptune, as at every point, the special theory of relativity holds in infinitesimal regions. This requirement leads to an important distinction. Not all kinematically possible systems can be realized by material structures at rest relative to them. We must therefore distinguish between real and unreal systems.

According to the general relativity of rotation, we can consider not only the earth but also any given rotating system, e.g., a merry-go-round, as the rest system. This conception, however, has absurd consequences. The horse, which in the usual interpretation pulls the merry-go-round, must in the second interpretation be able to put the earth, even the universe, in motion by means of treading, since now the merry-go-round remains at rest. How can the horse have the strength to do so?

This objection overlooks the fact that, in the relativistic conception, the rotation of the stars is due to a gravitational rotational field, and not to the horse. The latter has an entirely different task; it prevents the merry-go-round from following the rotational field and taking part in the general rotation. We see that even according to the relativistic interpretation, the horse has to perform a task determined by the mass of the merry-go-round and not by the mass of the stars. If an elevator glides down slowly and a fly inside crawls upward so that it remains at the same level relative to the building, it has to transport only its own mass – it does not have to “push down” the elevator.