Causally Simple Spacetimes and Domain Theory

Abstract

Globally, hyperbolic spacetimes are the simplest kind of spacetimes which are studied in General Relativity. It is shown by Martin and Panangaden that it is possible to reconstruct globally hyperbolic spacetimes in a purely order theoretic manner using the causal relation J +. Indeed these spacetimes belong to a category that is equivalent to a special category of domains known as interval domains [8]. In this paper, it is shown that this result is true for a larger superclass of spacetimes.

Keywords

• Domain theory,causality,spacetime,causally simple,Alexandrov topology

References

1. [1] S. Abramsky, A. Jung, Domain theory, Handbook of Logic in Computer Science, S. Abramsky, D. M. Gabbay, T. S. E. Maibaum (Editors), Volume III, Oxford University Press (1994).
2. [2] J. K. Beem, P. E. Ehrlich, K. L. Easley, Global Lorentzian Geometry, Marcel Dekker, New York, (1996).
3. [3] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge University Press, (2002).
4. [4] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lowson, M. Mislove, D. S. Scott, Continuous Lattices and Domains, Encyclopedia Math. Appl. 93, Cambridge University Press, (2003).
5. [5] S. W. Hawking and G. F. R. Ellis, The large scale structure of spacetime, Cambridge Monographs on Mathematical Physics, Cambridge University Press, (1972).
6. [6] K. Keimel, Bicontinuous Domains and some old problems in Domain theory, Electron. Notes Theor. Comput. Sci., 257, (2009), 35–54.
7. [7] K. Martin, P. Panangaden, Spacetime topology from causality, Arxiv gr-qc/0407093v1, (2004).
8. [8] K. Martin, P. Panangaden, ‘A domain of spacetime intervals in General Relativity”, Commun. Math. Phys. 267, (2006), 563–586.

9. [9] R. Penrose, Techniques of differential topology in Relativity, AMS Colloquium Publications, SIAM Philadelphia, (1972).
10. [10] D. Scott, Outline of a mathematical theory of computation, Technical Monograph PRG-2, Oxford University Computing Laboratory, (1970).
11. [11] Alex Simoson, Part III: Topological Spaces from a Computational Perspective, Mathematical Structures for Semantics, (2001-2002).
12. [12] R. M. Wald, General Relativity, University of Chicago Press, (1984).