Relativistic computers and the Turing barrier q Istva ´n Ne ´meti a, * , Gyula Da ´vid b a Renyi Institute, Budapest, Hungary b Department of Atomic Physics, Eo ¨ tvo ¨ s University, Budapest, Hungary Abstract We examine the current status of the physical version of the Church-Turing Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with [D. Deutsch, A. Ekert, R. Lupacchini, Machines, logic and quantum physics, Bulletin of Symbolic Logic 6 (3) (2000) 265–283] that PhCT is not only a conjecture of mathematics but rather a conjecture of a combination of theoretical physics, mathematics and, in some sense, cosmology. Since the idea of computability is inti- mately connected with the nature of time, relevance of spacetime theory seems to be unquestionable. We will see that recent developments in spacetime theory show that temporal developments may exhibit features that traditionally seemed impos- sible or absurd. We will see that recent results point in the direction that the possibility of artificial systems computing non- Turing computable functions may be consistent with spacetime theory. All these trigger new open questions and new research directions for spacetime theory, cosmology, and computability. Ó 2005 Elsevier Inc. All rights reserved. Of all the entities I have encountered in my life in physics, none approaches the black hole in fascination. And none, I think, is a more important constituent of this universe we call home. The black hole epit- omizes the revolution wrought by general relativity. It pushes to an extreme—and therefore tests to the limit—the features of general relativity (the dynamics of curved spacetime) that set it apart from special relativity (the physics of static, ‘‘flat’’ spacetime) and the earlier mechanics of Newton. Spacetime cur- vature. Geometry as part of physics. Gravitational radiation. All of these things become, with black holes, not tiny corrections to older physics, but the essence of newer physics. —John Archibald Wheeler (2000). 1. Aims, perspective We discuss the perspectives and scope of applicability of the Physical Church-Turing Thesis (PhCT). Roughly, PhCT is the conjecture that whatever physical computing device (in the broader sense) or physical 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.075 q With contributions from Attila Gohe ´r and Hajnal Andre ´ka. * Corresponding author. E-mail address: inemeti@axelero.hu (I. Ne ´meti). Applied Mathematics and Computation 178 (2006) 118–142 www.elsevier.com/locate/amc