"World Structure" and the Physical Meaning of Spacetime Geometry

Robert DiSalle

The work of Hermann Minkowski on special relativity, and that of Hermann Weyl on general relativity, are generally acknowledged as crucial steps in the evolution of our understanding of those theories. In each case what was thought of primarily as a "relativity theory" - a theory chiefly concerning the behaviour of fundamental physical quantities under coordinate transformations-was revealed to be an account of what Weyl came to call "world-structure"; Weyl's notion, in fact, was the extension to the context of curved spacetime of Minkowski's notion of "the absolute world." In this way the theories' implications concerning spacetime ontology came to be seen much more clearly than they had been by Einstein, and by the logical empiricist philosophers who were most concerned to assimilate his philosophical lessons. In this paper I will argue that Minkowski and Weyl offered something more than accounts of the spacetime structure underlying Einstein's theories: they also offered a methodological and epistemological account of our knowledge of spacetime structure, and in particular of the physical interpretation of spacetime geometry, that far surpassed the more familiar philosophical accounts (such as those of the logical empiricists) in depth, clarity, and plausibility.

Before the later 19th century, the most plausible account of the interpretation of geometry was the Kantian one, on which the content of geometrical postulates was assumed to have an immediate source in intuition. The work of Helmholtz and Poincaré made this account untenable by revealing the role of physical assumptions in what had been assumed to be pure intuition, but it left intact the idea that the interpretation of geometry rested on the same principles of intuitive construction (now understood as based on the motion of rigid bodies) that Kant had appealed to. With the introduction of special and general relativity, however, and the change of focus from space to spacetime, the empirical interpretation of geometry obviously could no longer be based on intuitive assumptions about spatial displacements. It seemed obvious that spacetime geometry had to be regarded as a purely formal structure, whose connections with experience would have to be fixed by the adoption of interpretive rules-rules which must by their very nature be conventional, since they define the empirical meanings of spatio-temporal concepts and so cannot be regarded as empirical principles in themselves. Even after the general decline of logical empiricism, and the widespread adoption of a model-theoretic approach to theories, the general notion of spacetime structure as a formalism requiring interpretation persisted, even if the question of the nature of interpretation no longer received the kind of attention that the logical empiricists had given it. On none of these views is it possible to give a clear account of the idea that we have empirical knowledge of the structure of spacetime.

Such an account was the contribution of Minkowski's analysis of special relativity. It is not merely the representation of special relativity in a four-dimensional form. Nor is it the "explanation" of special relativity by means of the hypothesis that there exists a certain underlying spacetime structure. Rather, it is Minkowski's attempt to show that our knowledge of the invariance group of electrodynamics is, in virtue of Einstein's analysis of time, knowledge of the structure of spacetime. In other words, the claim at the heart of Minkowski's analysis is, at the same time, extremely far- reaching and extremely modest: it is the claim that a world in which special relativity is true simply is a world with a particular spacetime structure. Properly understood, then, spacetime structure cannot be reasonably regarded as a formalism awaiting an interpretation. For its postulates themselves constitute the interpretation of characteristic physical principles-in this case the invariance of the velocity of light-in spatio-temporal terms. Thus the empirical meaning of spacetime geometry is determined in as direct a way as the meaning of spatial geometry had been, through the principle of free mobility; what had made this difficult to see, before Minkowski, is the fact that spatio-temporal principles have no direct intuitive significance, since their content arises not from the simple experience of motion but from a more or less complicated structure of dynamical laws. Minkowski revealed, in short, how the liberation of physical geometry from spatial intuition was not a separation of formalism from empirical content, but the simple result of shifting our attention from purely spatial principles (such as free mobility) to dynamical principles involving time.

The fact that Minkowski spacetime is a fixed structure with global symmetries, made it impossible, of course, that it could be an appropriate global structure for general relativity. The work of Weyl was necessary to show that, in spite of the radical change in the nature of spacetime, from a fixed and homogeneous background to a dynamically varying field, the basic insight of Minkowski into the relation between spacetime structure and physical assumptions could be maintained-that one could regard Einstein's arguments concerning the equivalence principle in the way that Minkowski regarded the arguments about time, as revealing the geometrical implications of fundamental physical laws. Despite its lack of philosophical prominence during most of the 20th century, Weyl's interpretation of general relativity's geometrical content must be seen as an important part of the theory's assimilation by mathematical physicists.

I conclude that the predominant logical empiricist, and post-logical- empiricist, way of considering spacetime by way of the distinction between structure and interpretation has led us to overlook an alternative way of thinking that is not only more illuminating, but also historically more crucial to the evolution of general relativity as an empirical physical theory. Reconsidering this way of thinking should have interesting implications for current problems facing general relativity.

Robert DiSalle
Department of Philosophy
University of Western Ontario
London, Ontario N6A 3K7