1

A. Einstein, "On the Foundations of the Generalized Theory of Relativity and the Theory of Gravitation," Physikalische Zeitschrift 15(1914), pp. 176-180.

2

A. Einstein, "Comments" on "Outline of a Generalized Theory of Relativity and of a Theory of Gravitation," Zeitschrift fuer Mathematik und Physik, 62(1914), pp. 260-61.

3

A. Einstein and M. Grossmann, "Covariance Properties of the Field Equations of the Theory of Gravitation Based on the General Theory of Relativity," Zeitschrift fuer Mathematik und Physik, 62(1914), pp. 215-225.

4

A. Einstein, "The Formal Foundation of the General Theory of Relativity," Koeniglich Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte. (1914), pp. 1030-1085.

Einstein's Point-Coincidence Argument

The Published Version

A. Einstein, "The Foundation of the General Theory of Relativity," Annalen der Physik, 49 (1916).

It is clear that a physical theory that satisfies this postulate [of general covariance] will also be suitable for the general postulate of relativity. For the sum of all substitutions in any case includes thoses which correspond to all relative motions of three-dimensional systems of co-ordinates. That this requirement of general co-variance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflection. All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points. Moreover, the results of our measurements are nothing but verifications of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock-dial, and observed point-events happening at the same place at the same time.

The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences. We allot to the universe four space-time variables x₁, x₂, x₃,x₄, in such a way that for every point-event there is a corresponding system of values of the variables x₁ . . . x₄. To two coincident point-events there corresponds one system of values of the variables x₁ . . . x₄, i.e., coincidence is characterized by the identity of the co-ordinates. If, in place of the variables x₁, x₂, x₃,x₄, we introduce functions of them, x'₁, x'₂, x'₃,x'₄, as a new system of co-ordinates, so that the systems of values are made to correspond to one another without ambiguity, the equality of all four co-ordinates in the new system will also serve as an expression for the space-time coincidence of the two point-events. As all our physical experience can be ultimately reduced to such coincidences, there is no immediate reason for preferring certain systems of coordinates to others, that is to say, we arrive at the requirement of general covariance.

In Correspondence

Einstein to Michele Besso, 3 January 1916 (translation from John Stachel, "The Hole Argument...")

Everything in the hole argument was correct up to the final conclusion.It has no physical content if, with respect to the same coordinate system K, two different solutions G (x) and ′ G (x ) exist, To imagine two solutions simultaneously on the same manifold has no meaning, and indeed the system K has no physical reality. The hole argument is replaced by the following consideration. Nothing is physically real but the totality of space-time point coincidences. If, for example, all physical events were to be built up from the motions of material points alone, then the meetings of these points, i.e., the points of intersection of their world lines, would be the only real things, i.e., observable in principle. These points of intersection naturally are preserved during alltransformations (and no new ones occur) if only certain uniqueness conditions are observed. It is therefore most natural to demand of the laws that they determine no more than the totality of space-time coincidences. From what has been said, this is already attained through the use of generally covariant equations.

Einstein to Paul Ehrenfest, 26 December 1915 (My translation).

In §12 of my work of last year, everything is correct (in the first three paragraphs) up to the italics at the end of the third paragraph. One can deduce no contradiction at all with the uniqueness of occurrences from the fact that both systems G(x) and G'(x), related to the same reference system, satisfy the conditions of the grav. field. The apparent force of this consideration is lost immediately if one considers that

(1) the reference system signifies nothing real
(2) that the (simultaneous) realization of two different g-systems (betters said, two different grav. fields) in the same region of the continuum is impossible according to the nature of the theory.

In the place of §12 steps the following consideration. The reality of the world-occurrence (in opposition to that dependent on the choice of reference system) subsists in spacetime coincidences. [Footnote: and in nothing else!] For example the intersection points of two different world lines are real as is the assertion that they do not intersect one another. Those assertions, which refer to physical reality, are not lost then through any (unambiguous) coordinate transformation. If two systems of g_μ_ν (or [more] gen.[erally], variables used for describing the world) are so constituted, that one can obtrain the second from the first merely through a space-time transformation, then they refer to exactly the same thing. For they have all timespace coincidences in common, i.e. all that is observable. This consideration shows immediately how natural is the requirement of general covariance.

Background reading
Marco Giovanelli, “Erich Kretschmann as a proto-logical-empiricist: Adventures and misadventures of the point-coincidence argument,” Studies in History and Philosophy of Modern Physics, 44: 115–134. Preprint at http://philsci-archive.pitt.edu/10158.