Related approaches & their problems

For Space-Times a seminal contribution of S. W. Hawking[1] introduced a unique combinatorial structure –a partial-order– attached to Lorentzian Manifolds with some additional restrictions, that up to conformal mappings defines the manifold (Alfonso García-Parrado and José M. M. Senovilla review[2] on Causal Space-times). David Malament[3] showed how this combinatorial structure alone, under suitable conditions, is sufficient to reconstruct Space-time up to a conformal factor.

Rafael Sorkin[4] and his school used the above results to establish the concept of a Causal set, an interval-finite, combinatorial model. Yet there is no direct structural link to the originating structure and they try to complete in one step the model presenting volume as the only missing concept, similar as John Stachel[5] proposes on a continuous background to combine projective and conformal structure. We decided to do one step at a time that is first combine both concepts before jumping into metrics. And –quite different to the Causal Set approach- we insist that there must be a structural connection between the discrete and the continuous model.

There has  been another approach somewhat close to ours. Hans-Jürgen Borchers and Rathindra Nath Sen reconstruct the complete Einstein-Weyl Causality[6] starting from the total order on light rays. Light rays in a certain sense connect the conformal and the projective structure by their inherent order and being locally geodesic. Yet the authors still assume a global partial order and that light rays are order-dense, which precludes finite structures i.e. Petri-nets.

For Petri-Nets since 1973 there as been some systematic effort; Olaf Kummer and Mark-Oliver Stehr present some more recent results[7] to detect the underlying combinatorial structures, specifically in the theory of Concurrency or causal structures defined by event-occurrence systems.

The problems

1. the mentioned Space-Time models in their definition make a heavy use of concepts typical for the continuous world, like Hausdorff-spaces as basic model-domain or using properties borrowed from Linear Algebra, all which as such can not be transported into the finite/countable domain.
2. the mentioned Petri-Net models -namely concurrency-theory- require countable models to work and therefore are as such a not suited to express all the technical concepts as used in continuous models. On structural level, there is no Linear Algebra, hence appears on first sight impossible to express concepts like convex let alone tensors or more complicated constructs.
3. Both models depend on Global Partial Orders even when expressing purely local concepts, a slight contradiction with the basic idea of General Relativity as something locally defined.
4. Geometry without additional constraints can not be derived from order alone. It must be introduced as an additional concept. It's long known that line-geometry -i.e. Geometry based on Points, Lines and Incidences- has finite, countable and continuous models. Yet the concept of a geodesic line is neither present in Causal Structures nor -as far as I know- in Petri Net-Theory.

The ideas for solution

1. Both model domains use Paths respectively Curves as a basic building block, where Curves in both domains model trajectories -world-lines- of particles (more precisely potential trajectories see Malament[8]). All expressed relations and properties can be re-written using only curves and the relations among points as defined by curves.
2. As Carl Adam Petri[9] pointed out quite early, on partial orders there exists a generalization for the concept of Dedekind-continuity and -completeness that allows for countable models, yet if applied to full-orders produces the known results. Crucial are two types of points, closed and open, while retaining the idea that the emerging topologies should be connected.
3. A little bit later Petri proposed the separation relation {{a,b},{c,d}} -an unordered pair of unordered pairs- as the basic order-producing relation. This relation expresses the separation of 4 points on a line, and is well defined on any Jordan-Curve, open or closed, i.e. there is no difference between a line, may be with suitable compactification, and a circle.
4. A careful analysis of the original article from Hawking, specifically analyzing the relation between local time-like cones, which form the base for the topology, the definition of regular paths in that topology and their relation to time-like curves, allowed to eliminate the reference to linear concepts like convex and to define local time-like cones and their properties using only combinatorial concepts.
5. This revision in turn demanded a revision of concepts in Petri-Nets. While in the original model the open elements are conditions and the closed elements are the events, and likewise sets border by conditions considered open while sets bordered by events closed, we need exactly the dual: events and event-bordered sets are open, conditions and condition-bordered sets are closed. It should be noted that for countable structures -Petri-nets are normally assumed to be countable- both sets -open and closed- define the same dual Alexandrov[10] Topology. However already the comparison of Dedekind continuity between total orders and half-orders alas Occurrence-Nets shows that the common type of elements in both -the non-branching conditions- must be closed.
6. In a Hawking-space all points are closed. Therefore it was necessary to overcome the initial interpretation of Einstein as if world points would correspond to physical events. They do not! If a Hawking-space models the loci -the geometry-, then a physical event can not have an exact place as Quantum-Mechanics tells us. A similar observation made decades ago Pauli[11]. Curiously enough, in this interpretation nothing ever happens in Hawking-Space as there are no events. To have events we must coarse grain first.
7. Likewise a too naive interpretation by Net-Theory of GRT had to be abandoned, as if each world-point branches into infinite many world-lines. Actually a world-point summarizes the whole time-like pre- respectively post-cones as such and not individual lines. This is the essence of the construction of regular paths by Hawking and the distinguishing conditions from Malament.
8. W.r.t. Geometry, we will start at the most elementary level: locally a line shall be uniquely defined by 2 points, locally any 2 points shall be connected by a line, finally the geometry shall be non-trivial i.e the local space shall be connected by lines with at least 3 points.

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[1] S. W. Hawking A.R. King and P. J. McCarthy, A new topology for curved space-time which incorporates the causal, differential and conformal structures Journal of Mathematical Physics Vol. 17, No 2, February 1976
[2] Alfonso García-Parrado, José M. M. Senovilla, Causal structures and causal boundaries, arXiv:gr-qc/0501069v2
[3] David  Malament, The class of continuous timelike curves determines the topology of spacetime Journal of Mathematical Physics, July 1977, Volume 18, Issue 7, pp. 1399-1404
[4] Rafael Sorkin, Causal Sets: Discrete Gravity, Notes for the Valdivia Summer School, Jan. 2002, arXiv:gr-qc/0309009v1 1 Sep 2003
[5] John Stachel, Projective and Conformal Structures in General Relativity, Loops ’07, Morelia June 25-30, 2007,
[6] Hans-Jürgen Borchers, Rathindra Nath Sen, Mathematical Implications of Einstein-Weyl Causality, Lect. Notes Phys. 709 (Springer, Berlin Heidelberg 2006)
[7] Olaf Kummer, Mark-Oliver Stehr: Petri's Axioms of Concurrency - A Selection of Recent Results , Proceedings of the 18th International Conference on Application and Theory of Petri Nets, Toulouse, June 23-27, 1997, Lecture Notes in Computer Science 1248, © Springer-Verlag , 1997
[8] David B. Malament, Classical Relativity Theory, arxiv.org/abs/gr-qc/0506065v2
[9] Carl Adam Petri, Concurrency. Lecture Notes in Computer Science Vol. 84: Net Theory and Applications, Proc. of the Advanced Course on General Net Theory of Processes and Systems, Hamburg, 1979 / Brauer, W. (ed.) --- Berlin, Heidelberg, New York: Springer-Verlag, 1980, Pages: 251-260
[10] Not to be confused with the Alexandrov Topology as used by Hawking
[11] Pauli, Vorlesungen in Turin über nichtlokale Feldtheorien in Google-Books http://books.google.com/books?id=NU9OUj-f8cYC&hl=es Page 34 ff.