An Analog Cellular Automaton Model of Gravitation: Planck-Scale Black Holes Randall C. O’Reilly ∗ University of Colorado Boulder 345 UCB Boulder, CO 80309 Randy.OReilly@colorado.edu Copyright c 2005 by Randall C. O’Reilly, initially published online at URL listed below on 12/28/05 (Dated: December 28, 2005) Many features of general relativity can be captured in an analog cellular automaton model operat- ing within a three dimensional regular face-centered cubic lattice. The gravitational field is modeled as a real-valued state variable that propagates according to a standard second-order wave equation. The value of the field at each point determines the effective distance between cells and the effec- tive “mass” of any other waves in the system (e.g., electromagnetic waves). The effects of this gravitational field were explored in the context of a second-order wave field representing the electro- magnetic wave field. The energy of this wave field provides the driving source of the gravitational field, and sufficiently high-energy waves drive a gravitational well that traps the wave oscillations (i.e., a miniature “black hole”), producing a stable dynamic state. These miniature black holes ex- hibit chaotic, random brownian motion, and are appropriately accelerated by a constant gravitational gradient. I. INTRODUCTION This paper presents an analog (continuous-valued) cellular automaton (CA) model of gravitation as described by general relativity. An analog CA uses real-valued state variables in a regular face-centered cubic lattice, updated synchronously by simple local neighborhood computations. In other work, a complete model of the coupled Maxwell-Dirac equations for electrodynamics has been developed [1]. Here, we apply this same approach to gravitation. The resulting model, coupled to a simple second-order electromagnetic wave fi eld, produces a stable black-hole at the Planck scale. These miniature black holes exhibit chaotic, random brownian motion, and are appropriately accelerated by a constant gravitational gradient CA models are appealing because they represent arguably the simplest way of implementing physical processes: space is carved into a lattice of identical small cubes (cells), each cell contains one or more state variables, and physics emerges through the local interactions between these cells (Figure 1). A number of different CA models of various physical and other phenomena have been developed, and their potential virtues as physical models discussed [2–14]. Most of these CA models involve discrete (binary) state vari- ables (i.e., a digital CA), and despite all the promising efforts, nobody has yet come up with a binary-state CA system that produces something like fundamental physics (e.g., quantum electrodynamics). However, by introducing continuous-valued state variables (i.e., an analog CA), several researchers have been able to model the evolution of fundamental quantum wave functions [13, 14]. The present model also adopts continuous-valued state variables. As discussed at greater length in [1], the use of continuous-valued states may seem problematic from the perspective of a digital computer, but they are more natural in terms of analog computers. Furthermore, ∗ URL: http://psych.colorado.edu/ ∼ oreilly/realt.html