Can we ascribe gravity to a certain state of the aether?

(Communicated in the meeting of February 26, 1916).

§ 1.

The form of Einstein’s equations is not changed by an arbitrarily chosen change of the system of coordinates[1].

Hilbert[2] then showed the use that may be made of a variation law that may be regarded as Hamilton’s principle in a generalized form.

These create a definitive form to the “general theory of relativity”, though much remains still to be done to develop it and in apply it to special problems.

A four-dimensional geometric representation may be of much use for this latter purpose; by means of it we shall be able to indicate for a system containing a number of material points and an electromagnetic field (or eventually only one of these) the quantity

{\displaystyle H}, which occurs in the variation theorem, and which we may call the principal function.

This quantity consists of 3 parts, of which the first relates to the material points, the second to the electromagnetic field and the third to the gravitation field itself.

As to the material points, it will be assumed that the only connexion between them is that which results from their mutual gravitational attraction.

§ 2. We shall be concerned with a four-dimensional extension � 4 {\displaystyle R_{4}}, in which “space” and “time” are combined, so that each point � {\displaystyle P} in it indicates a definite place � {\displaystyle A} and at the same time a definite moment of time � {\displaystyle t}. If we say that � {\displaystyle P} refers to a material point we mean that at the time � {\displaystyle t} this point is found at the place � {\displaystyle A}.

In the course of time the material point is represented every moment by a new point � {\displaystyle P}; all these points lie on the “world-line”, which represents the state of motion (or eventually the state of rest) of the material point[3].

In the same sense we may speak of the world-line of a propagated light-vibration.

An intersection of two world-lines means that the two objects to which they belong meet at a certain moment, that a “coincidence” takes place[4].

Einstein has made the striking remark[5] that the only thing we can learn from our observations and with which our theories are essentially concerned, is the existence of these coincidences.

Let us suppose e.g. that we have observed an occultation of a star by the moon or rather the reappearance of a star at the moon’s border.

Then the world-line of a certain light-vibration starting from a point on the world-line of the star has in its further course intersected the world-line of a ​point of the border of the moon and finally that of the observer’s eye. A similar remark may be made when the moment of reappearance is read on a clock.

Let us suppose that the light-vibration itself lights the dial-plate, reaching it when the hand is at the point � {\displaystyle a}; then we may say that three world-lines, viz. that of the light-vibration, that of the hand and that of the point � {\displaystyle a} intersect.

§ 3. We may imagine that, in order to investigate a gravitation field as e.g. that of the sun, a great number of material points, moving in all directions and with different velocities, are thrown into it, that light-beams are also made to traverse the field and that all coincidences are noted[6]. It would be possible to represent the results of these observations by world-lines in a four-dimensional figure — let us say in a “field-figure” — the lines being drawn in such a way that each observed coincidence is represented by an intersection of two lines and that the points of intersection of one line with a number of the others succeed each other in the right order.

Now, as we have to attend only to the intersections, we have a great degree of liberty in the construction of the “field-figure”.

If, independently of each other, two persons were to describe the same observations, their figures would probably look quite different and if these figures were deformed in an arbitrary way, without break of continuity, they would not cease to serve the purpose.

After having constructed a field-figure � {\displaystyle F} we may introduce “coordinates”, by which we mean that to each point � {\displaystyle P} we ascribe four numbers � 4 {\displaystyle x_{1},x_{2},x_{3},x_{4}}, in such a way that along any line in the field-figure these numbers change continuously and that never two different points get the same four numbers. Having done this we may for each point � {\displaystyle P} seek a point � ′ {\displaystyle P’} in a four-dimensional extension

′ {\displaystyle R’{4}}, in which the numbers � 4 {\displaystyle x{1},\dots x_{4}} ascribed to � {\displaystyle P} are the Cartesian coordinates of the point � ′ {\displaystyle P’}. In this way we obtain in � ′ {\displaystyle R’{4}} a figure � ′ {\displaystyle F’}, which just as well as � {\displaystyle F} can serve as field-figure and which of course may be quite different according to the choice of the numbers � 1 , {\displaystyle x{1},\dots x_{4}}, that have been ascribed to the points of � {\displaystyle F}.

If now it is true that the coincidences only are of importance it must be possible to express the fundamental laws of the phenomena by geometric considerations referring to the field-figure, in such a way that this mode of expression is the same for all possible field-figures; from our point of view all these figures can be considered as being the same. In such a geometric treatment the introduction of ​coordinates will be of secondary importance; with a single exception (§ 13) it only serves for short calculations which we have to intercalate (for the proof of certain geometric propositions) and for establishing the final equations, which have to be used for the solution of special problems.

In the discussion of the general principles coordinates play no part; and it is thus seen that the formulation of these principles can take place in the same way whatever be our choice of coordinates. So we are sure beforehand of the general covariancy of the equations that was postulated by Einstein.