**Introduction**

We are looking for a combinatorial framework that, in an *essential* way, includes the structure of *Space-Time** **as a continuous model* on one side and the structure of of *Petri-Nets as a finite (countable) model* on the other.

*Essential* means that physically different Space-Times and logically different Petri-Nets shall have different models and that different models produce different Space-Times and different Petri-Nets respectively.

**A wrong way** to get *time & space *would be simply to *assume* them, as Einstein showed convincingly more but a century ago, refuting thereby Immanuel Kant, who another century before had declared *time & space *as logical “

*a priori”*beyond material experience.

{Here and later on we will use deliberately time-space instead of the usual space-time when referring to our model}.

**There is a second *** caveat *already raised by Einstein –see The Challenge-:

At the very end, all measurement (and hence all Physics) boils down to have/to observe the

*coincidence*of

*something*at the

*same*

*time-space point*, while the description of

*time-space points*using

*coordinates*is just a convenient means to describe these coincidences.

*Hence it should make no difference in the description at least, if another such system of coordinates is used, if only there is a one-to-one correspondence between the former and the latter.*

Though a specific *time & space *may be extremely practical for a description and hence a necessary heuristic tool -among other to detect *symmetries* which reflect physical *invariants*-, the *Physics* described should not depend on the specifics of the used *space & time:* **Any time & space should do**, as long it produces the **same*** pattern of coincidence*.

The principle of General Covariance, even more its expression as diffeomorphism** covariance,** is for Einstein a **sequitur** of the idea of background independence. Actually he uses –and others since- nonetheless only these very specific coordinate-systems, that are appropriate for the description of Differential Manifolds. Yet he never stated anywhere that the realm of Differential Manifolds would be for him the *only* domain for admissible Coordinate-Systems.

Without getting here into more of the heuristic details, why it might be convenient to use less *sophisticated* structures for the sake of the spirit of GRT itself, in this article we will introduce an axiom-set of 5 groups of axioms, which uses just very elementary concepts from set-topology without *any *metric, yet by the end provides a *Category of Topological Spaces* powerful enough to include in a non-trivial way Differential Manifolds with Lorentzian metric but also other finite and countable models. Finite and countable models turn out to be Petri-Nets with additional interpretation.

{Non-trivial means that any diffeomorphism of the manifold implies a corresponding homeomorphism in this category, including the required updates of time & geometry.}

*Related approaches*

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 *

- 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.
- 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. - Both models depend on G
*lobal Partial Orders*even when expressing purely local concepts, a slight*contradiction*with the basic idea of*General Relativity*as something*locally*defined. -
*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*

- 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.* - 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.* - 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.* - 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. - 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***c**It should be noted that for countable structures -Petri-nets are normally assumed to be countable- both sets -*onditions and condition-bordered sets are closed.**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*. - 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 aas Quantum-Mechanics tells us. A similar observation made decades ago Pauli[11]. Curiously enough, in this interpretation*physical event can not have an exact place**nothing ever happens in Hawking-Space as there are no events.*To have*events*we must coarse grain first. - 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. - 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*.

[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/0506065**v2**

[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.

**The Axioms for Einstein-Spaces**

Based on the above I obtained **Axioms for Einstein-Spaces** as presented below.

Some models for **E-Spaces** are:

(1) Occurrence-Nets (with the above change and some additional requirements) as subclass of Petri-Nets

(2) The Real Numbers (but not Rationales nor Integers) (Q-order is derived from classical order)

(3) The Unit-Circle S1 (and the Circle Group) (but not n-cyclic Groups) and the Real Line (Q-order is derived from the relation among four points)

(4) The Minkowski-Space and the Quaternion (Q-order is derived from Q-Topology)

(5) The Causal structure of a Lorentzian manifold as defined by Hawking and others (Q-order is derived from the relation among four points on a time-like curve)

{For more see E-Space Examples, though it’s not yet updated to the most recent findings of this version for the axioms.}

**Axiom 1 A–Space **Atomic Topological Space

**Axiom-set 1** presents an almost standard definition for a Topology (1,6) and its homeomorphisms (2).

(2) will serve us as test-instrument: what not remains invariant will not be acceptable.

(3) introduces the notion of closure and interior operation. (4) is a may be not-so-usual but equivalent way to define *connected* for a set.

(5) singles out the subsets of *closed* respectively *open* elements. Be aware that we do not ask all elements to be closed, just either *closed* or* open* (9). The name *atomic* is our invention. **As shorthand we will call point only the closed elements and add open where required. **

(7) defines the topology as connected, (8) eliminates topologically equivalent elements and finally (10) requires the existence of a countable dense subset, the usual definition for separable.

{Only natural numbers were made by God Himself, anything else is human invention, as Kronecker always said.}

{As well known results from set-topology, the properties (7), (8), (9) (10) are topological invariants}

{By definition a manifold complies with

**Axiom 1**}

As here, in the sequel we will use the letter **D** to introduce a definition-line, **A **for a line that claims a property as axiom. As a hint: you may open a larger picture of each axiom in a second window just by clicking .. and then switch between image-window and text … to avoid loosing the reference context.

Before starting the presentation of **Axiom 2,** some heuristics about what we would like to achieve.

- We are looking for a substitute for the usual definition of
*geodesics*yet without using*any*metric concept, where*any*shall be understood*literally*that is we refuse even to rely on an underlying Euclidean metric space as the usual definition of a manifold does. Hence we can not use concepts from*differential calculus*either, as they do require at least a normed linear vector-space. Yet in combinatorial geometry –where we may perfectly define affine, projective or simpler linear spaces- there is also no differential calculus necessary. - The most simple combinatorial structure is the
*Linear Space*with*Lines*and*Points*such that any two points are on at least one line, each line has at least two points, two different lines share at most one point and finally two different points are on at least two different lines. This structure can be extended by a canonical procedure into a projective plane, preserving the initial lines. This purely combinatorial structure seems as a good candidate, moreover as –if desired- we may add additional properties to require right from the outset a projective or affine geometry without touching metrics. - The definition of
*geodesics*in GRT is strictly local i.e. applies only in local context. We will need some means to define this context, yet assure consistency of the definitions, similar as it’s done in*sheaf-theory*. - The definition has to be
*background-independent*or -what is the same in our limited world of Topology- a topological invariant w.r.t. homeomorphisms. - The final idea had parents: Albert Einstein, with his famous equivalence principle –there is no difference if someone moves on a
*geodesic*or stays put yet the world moves around him- and Ruth Moufang, who introduced*Lines*as fix-points for translation-symmetries in combinatorial geometry. As a*child idea*, we will try to define a*geodesic*as the local fix-point for those global homeomorphisms (obviously a subgroup) that move us along the*geodesic*. If successful, we are done, as homeomorphisms map subgroups.

**Axiom 2 G–Space **Geometric Space

**Axiom-set 2 **adds a property to A-Space (1). *Blocks* (2) –our local context to be used- are *closed* subspaces, for which all *open neighborhoods* contain another *open neighborhood *within which they are *connected* (Beware: as we may have open elements, the intersection of all *open neighborhoods* of a point/a closed set may contain still other elements besides the point/the set itself).

For the same reason (4) we consider initially only *points *as elements of *Pre-Geodesics*, applying some technicalities (5) later to add may be missing elements.

{Yet –without getting into details now, but important for the later work- this allows some at the first sight *strange *geometries with *strange* geodesics (geodesics of points all with a rational ratio of their intervals on the Real line.). If all elements are closed, then (5) is void, i.e. nothing is added.}

(3) Is the *cornerstone *of the axiom-set: It defines a predicate γ that combines *blocks* and subsets of *points*, and delivers *true* when they match the conditions.

- (3.1) is more technical: any local subset of a
*pre-geodesic*is a pre-geodesic. - (3.2) and (3.3) express partially the requirements for
*Linear Spaces*: any point is on some*pre-geodesic*with at least 2 points. If two*pre-geodesic*share more than 2 points, they are part of another*pre-geodesic.*Take a chain of these, then the maximal element is the*one and only one*on which all the*points*are. - (3.4) Tries to implement the
*child-idea*: for any 2 local points on a*pre-geodesic*, there exists a*global homeomorphism*that carries one onto the other (*the world moves,*3.4 first half*),*yet this homeomorphism carries also the whole*pre-geodesic*in a way that the original point joined to the local part of the image form again a*pre-geodesic*(3.4 second half). It shares with the original*pre-geodesic*2 points. Hence their join is part of the same*maximal pre-geodesic*(3.3). - The
*moving*homeomorphisms form a*subgroup*of the homeomorphism-group of the A-topology, which leaves invariant the maximal element (the*one and only one, voilá our fix-point).*This subgroup is an invariant of the homeomorphism-group itself, that is when mapping the points, the sub-group is mapped accordingly. Hence as final result (3.1), (3.2)and (3.3) remain likewise intact.

(4) Constructs the class of all possible *pre-geometries, *admitting only those whose permitted *blocks *form a *cover *for the set of *points*.

(5) Adds elements in case that not all elements of the set are *points*. (Beware: we do not ask that a *pre-geodesic* to be topologically connected. Neither *pre-geodesics* nor *geodesics *are necessarily *topological paths *i.e. images of a *continuous* mapping of [0,1]. They rather will serve to measure (or count) not to define topology.

{As a hint: a *physical light-ray* considered as *geodesic* can not be connected topologically, due to the quantum-nature of light. Yet it follows a *topologically connected path* with distance measure 0, at least as long as we don’t get into QED.}

(6) Contains another part of the *Linear Space* requirements: once completed, (6.1) connects a *block *using now *geodesics *and (6.2) assures that all elements can be *told apart *using *geodesics*. I named this set of requirements *G-Definite *.

(7) Appears as if it were a repeat of (4), now for *geometries. *Well, it’s not!

The class contains *geometries *as completed in (5), yet the predicate γ is only applied to the original* pre-geometry, *while *definiteness* is tested for the completed *geometry*.

{Beware: a completed *geodesic *may contain *open* elements, which would cripple γ right away: you can not move an open onto a closed element and vice-versa. Likewise we do not require that the topology is *homogeneous* in all points. As will be needed later elsewhere, some *points* will correspond to *observables, *others are *unobservable details*. All obviously only in the case that the topology has *open elements*.}

(8) Is finally the axiom itself. The class of *geometries* is not empty i.e. there is at least one. And if there is one … there are many as we can move around using the homeomorphism-group. That’s the content of the Theorem (9).

{The crafting or la carpintería wasn’t done yet. However all the definitions above are based only on the A-topology or (3) firmly tied to it. So it should/might be tedious but appears *true *and feasible to compose if minor details are still wrong.}

{Now take a Lorentzian Manifold, use as one initial *geometry *the *geodesics *of the corresponding Lorentzian metric, take as *blocks *the closure of some *normal **neighborhoods *for each point, such that their join forms a *cover. *Then it appears as if this *geometry *satisfies at least (3.1), (3.2), (3.3), (6.1), (6.2).

The missing part: find the move-around diffeomorphism. Well, I’m not very rapid/clever/trained in Lorentzian Manifolds, but again it appears to me that they do exist. It might take some time –more time for me- but appears to me as a feasible approach.

Done this, (5) is void as the underlying Topology is Hausdorff, hence there is no difference between (4) and (7). Finally a even larger class results from applying the diffeomorphism to the initial geometry according to (9).}

We will leave G-Spaces and turn our attention to another way to add properties, based on curves, to an A-Topology. Basically these combinatorial structures –some times called Space-Time, sometimes *Causal Structure* are fairly well known since David Malament proved his famous equivalence-theorem about time-like curves and causal structure of a Lorentzian Manifold.

Yet all these approaches –or at least many of them- introduce a very basic asymmetry between *Time* and *Space*: while the former is assumed to be some type of *partial order* with hence no closed *time-like* (*causal*) curves, the authors do not put the same type of restriction on *Space*, where a *closed *(*spatially) *universe is still an option.

In other entries of my BLOG I explain why I don’t share this approach, which more over heavily relies on arguments outside GRT itself (like the famous Grandfather paradox) and –in my humble opinion- enters into open contradiction with very basic assumptions of GRT. We need something to replace *Partial Order* as a building block while retaining *orientability*. This is the central attempt of Axiom 3 and Axiom 4.

**Axiom-set 3 **starts (1) with some A-Space.

(2) defines as *pre-path* sets that *fall apart* if –except may be end-elements Z- a single element is removed. It’s an attempted replacement for the classical definition of [the image] of a Jordan-curve, yet without using the whole baggage of *Real*-Topology and intrinsically substituting the concept of *injective *by *monotone, *needed as the A-Topology is not required to be Hausdorff.

(3) extends the idea to *closed curves*. Both together form *paths*. Please note we are talking about *images*, so there is no parameter nor parameterization, which again would introduce *metric* concepts at a far to early stage.

(4) defines which sets we will consider *path-connected, *such that (5) may claim that for every *element *every *open neighborhood* contains a *path-connected* *open neighborhood. *Elements neither *closed *nor *open *would damage these definitions. Finally *path-connected* extends to the whole set, as it’s itself *connected* (A 1.7) {well known result from set-topology}.

(6) Eliminates *loose ends* i.e.* *any element has at least two neighbors.

{As well known result from set-topology, paths themselves and the above definitions based on paths are topological invariants}

{As well known result any *manifold* complies with **Axiom 3**}

Paths define a symmetry-relation among their elements: the order by which they are arranged on the path. This relation is known since ancient times. The next theorem explores this relation.

**Theorem 1 Q–Relation ***4 elements on a path*

(1) sets the domain: we will talk about the *paths* of a P-Space.

(2) defines the Q-Relation: 4 elements may be grouped into two pairs such that each pair separates the other (5).

(3), (4), (5) explore the relation (details in the referred entries).

(6) shows that the Q-Relation is *persistent* i.e. once defined it does not change in broader settings, a property important later on for instance for approximations.

(7) shows that the Q-Relation is a topological invariant, almost obvious by looking at (2) and recalling that *paths *themselves are invariant.

The next step consists in introducing a *combinatorial* *concept* for *time*. To put it very bluntly, we will do as mankind already has done: simply extended/extrapolate to the whole *universe*, what we know already for sure from one *path -*by the way, due to Theorem 1, any *path*-, that is we assume that there is a structure that behaves almost like a *path. *This structure* *we call a *time*.

(1) sets the domain: we will talk about the *paths* of a P-Space.

(2) introduces the structure *time* we are looking for, in which the Q-Relation shall hold if it holds for any member-path.

(3) is just a simplification of the Q-Relation: We forget the order and only remember that the 4 elements are on some *path. *(4) defines a predicate θ,

*true*if the structure behaves like we assume it does (i.e. like a

*path)*. (The entry Going around in Circles I explores its meaning).

(5) Defines a

*rule of interference*to combine

*different paths of a time.*

(5.1) whenever all pairs of 3 points can be found on some

*path*–regardless their configurations there-, there should be a 4th element to give a complete path, where all 4 may be positioned.

(5.2) whenever all triplets of 4 points

*are on some path*, the 4 points themselves may be positioned on a path.. (The entry Going around in Circles II provides some heuristics for this rule and shows its consequences).

(6) states the already familiar concept we used for topology and

*geometries*that elements are connected yet may be separated using only

*time*

*.*

Done? Well not yet. A

*path*is a topologically connected structure. Therefore the

*interval*between two

*points*is always

*open*. The predicate (7) carries this on to

*time.*

{In GRT

*time-like cones*and

*time-like double-cones*are always

*open.*}

(8) Defines the class of possible

*time(s)*while (9) as axiom claims that this class is not empty.

(10) shows that

*time(s)*are a topological invariants, almost obvious as we used only

*paths,*the Q-Relation and the topology itself to define

*time(s).*

In the next step, we will combine *time(s)* and *geometries* into a single framework.

**Axiom 5 E–Space **Spaces with Time and Geometry

(1) sets the domain: we will talk about *spaces* that have both *time *and *geometry. *There is a

*compatibility*condition, well known from GRT: a

*geodesic once time-like remains time-like.*This condition is expressed by (2).

(3) as axiom claims that there is at least one

*compatible*pair of (

*time,geometry*).

{The

*compatibility condition*rules out

*effects at distance,*

*spooky effects*as Einstein calls them in the

*Einstein-Rosen paradox*. Yet it does not exclude

*symmetries at distance*of the

*geometry,*that’s Bell or not Bell is not a question, at least in our

*Einstein Spaces*.}

{Take as a

*Time*the

*time-like curves*of a Lorentzian Manifold, as

*Geometry*the

*geodesics*of a

*cover*by

*normal neighborhoods*, then the manifold becomes an

*Einstein-Space.*}

#### Our principal theorem:

**Theorem 2 Einstein Spaces** E – Spaces form a Topological Category

(1) defines just the *class of homeomorphisms *between two topological spaces.

(2) states that if they are isomorphic w.r.t to their topology, then one of them is an *E-Space *iff the other is also an *E-Space, *that is *E-Spaces *form a *Category of topological spaces, *which we call *Einstein-Spaces. *{The crafting or la carpintería wasn’t done yet. However from earlier remarks, it seems quite obvious that

*G-Spaces*and

*T-Spaces*each are

*topological categories*. The

*compatibility condition*as such is

*topologically invariant*.}

We add two pages with the bare-bone axiom-sets and theorems.

{If Theorem 2 __is__ true, then it might have far reaching consequences.

At least to me it would explain a lot about the intrinsic, tricky relation between the *Einstein Field-Equation* –it appears as if it combines *Geometry* and *Physics*- on one side and *Causal Structure* *-*in the sense of Hawking and Malament- on the other, which may be modeled quite naturally as a *Time. *The solution to this *puzzle* would come close to solve the second Einstein Challenge*,* still an *arduous task *for many researchers and thinkers far better equipped with mathematical and physical background than I am,* * but unsolved now for almost a 100 years).}

**We’re done! Cornelius Hopmann, December 2010**