# Einstein:GR1916 chapter 02 - Physical Meaning of Geometrical Propositions

Die Grundlage der allgemeinen Relativitätstheorie / The Foundation of the General Theory of Relativity, *Annalen der Physik* **49** (1950)

## 2: Physical Meaning of Geometrical Propositions

**IN** classical mechanics, and no less in the special theory of relativity, there is an inherent epistemological defect which was, perhaps for the first time, clearly pointed out by Ernst Mach. We will elucidate it by the following example:– two fluid bodies of the same size and nature hover freely in space at so great a distance from each other and from all other masses that only those gravitational forces need be taken into account which arise from the interaction of different parts of the same body. Let the distance between the two bodies be invariable, and in neither of the bodies let there be any relative movements of the parts with respect to one another.
But let either mass, as judged by an observer at rest relatively to the other mass, rotate with constant angular velocity about the line joining the masses. This is a verifiable relative motion of the two bodies. Now let us imagine that each of the bodies has been surveyed by means of measuring instruments at rest relatively to itself, and let the surface of [math]S_1[/math] prove to be a sphere, and that of [math]S_2[/math] an ellipsoid of revolution. Thereupon we put the question – What is the reason for this difference in the two bodies? No answer can be admitted as epistemologically satisfactory,^{1} unless the reason given is an *observable fact of experience*. The law of causality has not the significance of a statement as to the world of experience, except when *observable facts* ultimately appear as causes and effects.

Newtonian mechanics does not give a satisfactory answer to this question. It pronounces as follows:– The laws of mechanics apply to the space [math]R_1[/math], in respect to which the body [math]S_1[/math] is at rest, but not to the space [math]R_2[/math], in respect to which the body [math]S_2[/math] is at rest. But the privileged space [math]R_1[/math] of Galileo, thus introduced, is a merely *fictitious* case, and not a thing that can be observed. It is therefore clear that Newton's mechanics does not really satisfy the requirement of causality in the case under consideration , but only apparently does so, since it makes the fictitious cause [math]R_1[/math] responsible for the observable difference in the bodies [math]S_1[/math] and [math]S_2[/math].

The only satisfactory answer must be that the physical system consisting of [math]S_1[/math] and [math]S_2[/math] reveals within itself no imaginable cause to which the differing behaviour of [math]S_1[/math] and [math]S_2[/math] can be referred. The cause must therefor else *outside* this system. We have to take it that the general laws of motion, which in particular determine the shapes of [math]S_1[/math] and [math]S_2[/math], must be such that the mechanical behaviour of [math]S_1[/math] and [math]S_2[/math] is partly conditioned, in quite essential respects, by distant masses which we have not included in the system under consideration. These distant masses and their motion relative to [math]S_1[/math] and [math]S_2[/math]must then be regarded as the seat of the causes (which must be susceptible to observation) of the different behaviour of our two bodies [math]S_1[/math] and [math]S_2[/math]. They take over the role of the fictitious cause [math]R_1[/math]. of all imaginable spaces [math]R_1[/math], [math]R_2[/math], etc., in any kind of motion relatively to one another, there is none which we may look upon as privileged *a priori* without reviving the above-mentioned epistemological objection. * The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion.* Along this road we arrive at an extension of the postulate of relativity.

In addition to this weighty argument from the theory of knowledge, there is a well-known physical fact which favours an extension of the theory of relativity. Let [math]K[/math] be a Galilean system of reference, i.e. a system relatively to which (at least in the four-dimensional region under consideration), a mass, sufficiently distant from all other masses, is moving with uniform motion in a straight line. Let [math]K'[/math] be a second system of reference which is moving relatively to [math]K[/math] in *uniformly accelerated* translation.Then, relatively to [math]K'/math\gt, a mass sufficiently distant from other masses would have an accelerated motion such that its acceleration and direction of acceleration are independent of the material composition and physical state of the mass.
Does this permit an observer at rest relatively to \ltmath\gtK'/math\gt to infer that he is on a "really" accelerated frame of reference? The answer is in the negative:; for the above-mentioned relation of freely movable masses to \ltmath\gtK'[/math]may be interpreted equally well in the following way. The system of reference [math]K'[/math] is unaccelerated, but the space-time territory in question is under the sway of a gravitational field, which generates the accelerated motion of the bodies relatively to [math]K'[/math].

This view is made possible for us by the teaching of experience as to the existence of a field of force, namely, the gravitational field, which possesses the remarkable property of imparting the same acceleration to all bodies. ^{2} The mechanical behaviour of bodies relatively to [math]K'[/math] is the same as presents itself to experience in the case of systems which we are wont to regard as "stationary" or as "privileged". Therefore, from the physical standpoint, the assumption readily suggests itself that the systems [math]K[/math] and [math]K'[/math] may both with equal right be looked upon as "stationary," that is to say, they have an equal title as systems of reference for the physical description of phenomena.

It will be seen from these reflexions that in pursuing the general theory of relativity we shall be led to a theory of gravitation, since we are able to "produce" a gravitational field merely by changing the system of coordinates. It will also be obvious that the principle of the constancy of the velocity of light *in vacuo* must be modified, since we easily recognize that the path of a ray of light with respect to [math]K'[/math] must be in general curvilinear, if with respect to [math]K[/math] light is propagated in a straight line with a definite constant velocity.

- Of course an answer may be satisfactory from the point of view of epistemology, and yet be unsound physically, if it i in conflict with other experiences.
- Eötvös has proved experimentally that the gravitational field has this property in great accuracy.

- I've changed the original translator's use of the word "facticious" to "fictitious" throughout.
**EB 2016**