Continuous spatial automata

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Continuous spatial automata, unlike cellular automata, have a continuum of locations. The state of a location is a finite number of real numbers. Time is also continuous, and the state evolves according to differential equations. One important example is reaction-diffusion textures, differential equations proposed by Alan Turing to explain how chemical reactions could create the stripes on zebras and spots on leopards. When these are approximated by CA, such CAs often yield similar patterns. MacLennan [1] considers continuous spatial automata as a model of computation.

There are known examples of continuous spatial automata which exhibit propagating phenomena analogous to gliders in Conway's Game of Life. For example, take a 2-sphere, and attach a handle between two nearby points on the equator; because this manifold has Euler characteristic zero, we may choose a continuous nonvanishing vector field pointing through the handle, which in turns implies the existence of a Lorentz metric such that the equator is a closed timelike geodesic. An observer free falling along this geodesic falls toward and through the handle; in the observer's frame of reference, the handle propagates toward the observer. This example generalizes to any Lorentzian manifold containing a closed timelike geodesic which passes through relatively flat region before passing through a relatively curved region. Because no closed timelike curve on a Lorentzian manifold is timelike homotopic to a point (where the manifold would not be locally causally well behaved), there is some timelike topological feature which prevents the curve from being deformed to a point. Because it has been conjectured that these might serve as a model of a photon, these are sometimes also called pseudo-photons.

It is an important open question whether pseudo-photons can be created in an Einstein vacuum space-time, in the same way that a glider gun in Conway's Game of Life fires off a series of gliders. If so, it is argued that pseudo-photons can be created and destroyed only in multiples of two, as a result of energy-momentum conservation.