Hole Argument - Einstein's Resolution

Einstein's Resolution

In 1915, Einstein realized that the hole argument makes an assumption about the nature of spacetime: it presumes that the gravitational field as a function of the coordinate labels is physically meaningful by itself. By dropping this assumption, general covariance became compatible with determinism, but now the gravitational field is only physically meaningful to the extent that it alters the trajectories of material particles. While two fields that differ by a coordinate transformation look different mathematically, after the trajectories of all the particles are relabeled in the new coordinates, their interactions are manifestly unchanged. This was the first clear statement of the principle of gauge invariance in physical law.

Einstein believed that the hole argument implies that the only meaningful definition of location and time is through matter. A point in spacetime is meaningless in itself, because the label which one gives to such a point is undetermined. Spacetime points only acquire their physical significance because matter is moving through them. In his words:

"All our spacetime verifications invariably amount to a determination of spacetime coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meeting of two or more of these points." (Einstein, 1916, p.117)

He considered this the deepest insight of general relativity. When asked by reporters to summarize his theory, he said:

"People before me believed that if all the matter in the universe were removed, only space and time would exist. My theory proves that space and time would disappear along with matter."