Matrix Models for Spacetime Topodynamics

The machinery is suggested to describe the varying spacetime topology on the level of its substitutes by finite topological spaces. Introduction The approximations of (or substitutes for) continuous spacetime by finitary structures are studied in this paper. The results presented furnish a framework in which one might express such ideas as variable spacetime topology or, for instance, the topological fluctuations on small scales. The paper is organized as follows. In Section 1 the coarse-graining procedure is described. Being applied to a continuous manifold, it yields the so-called pattern space [12] which, being finite or at most countable set, may be thought of as topological space or, equivalently, as a directed graph. The kinematics of the spacetime topology is then addressed to that of the pattern space which substitutes its continuous predecessor. In Section 2 the finitary counterpart of the supespace (in Wheeler’s sense) is introduced providing the arena for the variation of the topology of pattern spaces. To construe it we use the remarkable isomorphism between pattern spaces and finite quasiorders. The latter, being subject of combinatorial studies, are associated with certain finite-dimensional algebras [9]. Thus, the study of the variety of finite topological spaces (being, loosely speaking, discrete by its nature) is replaced by dealing with finitedimensional algebras whose matrix representation is treated. In Section 3 the main topological features of pattern spaces are formulated in algebraic terms. In Section 4 the spatialization procedure is suggested restoring points of the pattern spaces by given finite-dimensional algebra. The ideas used in this procedure are Stanley’s techniques [11] in algebraic combinatorics. Now, possessing the algeraic means to capture the topological features, we are interested in introducing finitary substitutes for differential structures, to which the Section 5 is devoted. The elements of the tensor calculus needed to introduce the basic constituents of general relativity turn to be successfully transplanted to pattern spaces and their matrix representations. The Einstein-Hilbert variational principle is then rewritten in terms of matrix equations. 1 The coarse-graining procedure