Combinatorial Spacetimes

Abstract

I investigate a class of dynamical systems in which finite pieces of spacetime contain finite amounts of information. Most of the guiding principles for designing these systems are drawn from general relativity: the systems are deterministic; spacetime may be foliated into Cauchy surfaces; the law of evolution is local (there is a light-cone structure); and the geometry evolves locally (curvature may be present; big bangs are possible). However, the systems differ from general relativity in that spacetime is a combinatorial object, constructed by piecing together copies of finitely many types of allowed neighborhoods in a prescribed manner. Hence at least initially there is no metric. The role of diffeomorphism is played by a combinatorial equivalence map which is local and preserves information content. Most of my results come in the 1+1-dimensional oriented case. There sets of spaces may be described equivalently by matrices of nonnegative integers, directed graphs, or symmetric tensors; local equivalences between space sets are generated by simple matrix transformations. These equivalence maps turn out to be closely related to the flow equivalence maps between subshifts of finite type studied in symbolic dynamics. Also, the symmetric tensor algebra generated by equivalence transformations turns out to be isomorphic to the abstract tensor algebra generated by commutative cocommutative bialgebras. In higher dimensions I study the case where space is a special type of colored graph (discovered by Pezzana) which may be interpreted as an n-dimensional pseudomanifold. Finally, I show how one may study the behavior of combinatorial spacetimes by searching for constants of motion, which typically are associated with local flows and often may be interpreted in terms of particles.