Algebraic Representation of Space-Time as Esoteric?

If the philosophical analysis of the singular feature of space-time is able to shed some new light on the possible nature of space-time, one should not lose sight of the fact that, although connected to fundamental issues in cosmology, like the ‘initial’ state of our universe, space-time singularities involve unphysical behaviour (like, for instance, the very geodesic incompleteness implied by the singularity theorems or some possible infinite value for physical quantities) and constitute therefore a physical problem that should be overcome. We now consider some recent theoretical developments that directly address this problem by drawing some possible physical (and mathematical) consequences of the above considerations.


Indeed, according to the algebraic approaches to space-time, the singular feature of space-time is an indicator for the fundamental non-local character of space-time: it is conceived actually as a very important part of General Relativity that reveals the fundamental pointless structure of space-time. This latter cannot be described by the usual mathematical tools like standard differential geometry, since, as we have seen above, it presupposes some “amount of locality” and is inherently point-like. The mathematical roots of such considerations are to be found in the full equivalence of, on the one hand, the usual (geometric) definition of a differentiable manifold M in terms of a set of points with a topology and a differential structure (compatible atlases) with, on the other hand, the definition using only the algebraic structure of the (commutative) ring C(M) of the smooth real functions on M (under pointwise addition and multiplication; indeed C(M) is a (concrete) algebra). For instance, the existence of points of M is equivalent to the existence of maximal ideals of C(M). Indeed, all the differential geometric properties of the space-time Lorentz manifold (M,g) are encoded in the (concrete) algebra C(M). Moreover, the Einstein field equations and their solutions (which represent the various space-times) can be constructed only in terms of the algebra C(M). Now, the algebraic structure of C(M) can be considered as primary (in exactly the same way in which space-time points or regions, represented by manifold points or sets of manifold points, may be considered as primary) and the manifold M as derived from this algebraic structure. Indeed, one can define the Einstein field equations from the very beginning in abstract algebraic terms without any reference to the manifold M as well as the abstract algebras, called the ‘Einstein algebras’, satisfying these equations. The standard geometric description of space-time in terms of a Lorentz manifold (M,g) can then be considered as inducing a mathematical representation of an Einstein algebra. Without entering into too many technical details, the important point for our discussion is that Einstein algebras and sheaf-theoretic generalizations thereof reveal the above discussed non-local feature of (essential) space-time singularities from a different point of view. In the framework of the b-boundary construction M = M ∪ ∂M, the (generalized) algebraic structure C corresponding to M can be prolonged to the (generalized) algebraic structure C corresponding to the b-completed M such that CM = C, where CM is the restriction of C to M; then in the singular cases, only constant functions (and therefore only zero vector fields) can be prolonged. This underlines the non-local feature of the singular behaviour of space-time, since constant functions are non-local in the sense that they do not distinguish points. This fundamental non-local feature suggests non-commutative generalizations of the Einstein algebras formulation of General Relativity, since non-commutative spaces are highly non-local. In general, non-commutative algebras have no maximal ideals, so that the very concept of a point has no counterpart within this non-commutative framework. Therefore, according to this line of thought, space-time, at the fundamental level, is completely non-local. Then it seems that the very distinction between singular and non-singular is not meaningful anymore at the fundamental level; within this framework, space-time singularities are ‘produced’ at a less fundamental level together with standard physics and its standard differential (commutative) geometric representation of space-time.

Although these theoretical developments are rather speculative, it must be emphasized that the algebraic representation of space-time itself is “by no means esoteric”. Starting from an algebraic formulation of the theory, which is completely equivalent to the standard geometric one, it provides another point of view on space-time and its singular behaviour that should not be dismissed too quickly. At least it underlines the fact that our interpretative framework for space-time should not be dependent on the traditional atomistic and local (point-like) conception of space-time (induced by the standard differential geometric formulation). Indeed, this misleading dependence on the standard differential geometric formulation seems to be at work in some reference arguments in contemporary philosophy of space-time, like in the field argument. According to the field argument, field properties occur at space-time points or regions, which must therefore be presupposed. Such an argument seems to fall prey to the standard differential geometric representation of space-time and fields, since within the algebraic formalism of General Relativity, (scalar) fields – elements of the algebra C – can be interpreted as primary and the manifold (points) as a secondary derived notion.


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