Thursday, October 11, 2012

More –this time mathematical- heuristics

Let me try to explain what I’m trying to do. [Especially not trying to re-invent the wheel when there are so many proven, useful models at hand].

I’m aiming at Combinatorial General Relativity, as Einstein mentioned at the end of some of his writings as an unsolved puzzle.

Yet “my way” is not the usual top-down –i.e. trying to find quantization-rules on the very top- but “bottom-up” that is investigating in a more or less systematic fashion the implicit and explicit combinatorial elements and structures on each level. Here “causal structure”, based on the work of Stephen Hawking, David Malament and others, is already an almost closed chapter (Q-Orders).

The current “target”: differential manifolds.

Usually a manifold is defined using local homeomorphism between some local neighbor-hoods in the manifold-topology and some local part of a Euclidean space. For the latter normally one uses its algebraic definition (module over some field), gets from there to the usual Euclidean metric and then obtains the topology.

Now there is a completely equivalent set of Axioms for Euclidean Geometries: the famous Hilbert Axioms. Why would one like to use this set? Well, it’s known that there are finite, discrete models that satisfy the first set of Hilbert-Axioms (incidence Axioms for points, lines and planes), yet do not satisfy the second set (Order-Axioms) nor the third (Congruence-Axioms) and where the forth (Parallel-Axiom) is optional.

Not only this, but as v. Staudte, Moufang, Veblen, v. Neumann and others have shown, already the first set allows the construction of coordinates and hence metric, i.e. coordinates and metric arise using incidence relations and nothing else. [And that’s what I understand Einstein had in mind with his remarks].

Incidence relations as “founding ground” have a second great advantage: topological homeomorphisms carry them over automatically, i.e. suitably defined the manifold inherits these relations automatically, while congruence-relations or algebraic properties do not.

The “missing link”: Hilbert takes “lines” and “planes” as primitives (as does synthetic Geometry). So in my next –failed- intent I was looking where to find “lines” and “planes” in the context of Riemann and Pseudo-Riemann manifolds, and voilá there they were: geodesics and totally normal neighborhoods, yet with a tremendous drawback: their definition already requires concepts of the very, very top like the metric-tensor and the exponential map.

So I did a step backward, asking myself: is it possible to introduce lines using ONLY topological concepts? The model at hand: balls and spheres as defined by the usual Euclidean metric.

Next step: is it possible to define balls and spheres without using metric? Well it is. One might use simply connected neighborhoods and simply connected surfaces and, as recently shown, we get nice spheres in all dimensions.

Problem: “simply connected” as concept requires Hausdorff-spaces and imports via Homotopy the whole “World of Reals”, incompatible with my aim to allow also discrete, finite models not talk about my suspicion that it is a far to restrictive assumption for the real World (there are holes almost everywhere). Therefore I replaced “simply connected” by just “locally path connected”.

Next step: given sets of surfaces of locally path connected neighborhoods, what would be a minimal set of axioms based only on incidence and topology that would convert this set locally into something similar to the Euclidean Balls and –Spheres we started with? There a point lies on a straight line between two points iff there are Spheres for each of them that touch each other exactly in that point. More over the Spheres made using Euclidean Distance form a complete order around each point. There are additional compatibility rules among the spheres of different points on the same straight line. Finally “completeness” respectively “coherence” is to be considered i.e. in Euclidean Geometry there is always a straight line that connects them. This again would provoke “continuous models only” already in three dimensions. So I weakened this requirement to: “it should be possible to go from anywhere to anywhere in straight steps” and “if there are intersecting spheres for two points, then there are touching spheres (and hence a line that connects them)”.

Done this, we get a set of definitions invariant under any topological homeomorphism. That means two things: they carry over “automatically” to the manifold (locally of course), but second they produce a natural gauge. (Not such a surprise: there quite many metrics already for the real line, which produce the same topology. For instance the max-norm reproduces the same topology yet has cubes as balls).

Once I’ve got lines, the remainder is almost straight forward, using well known properties from synthetic geometry for triangles and quadrilaterals as axioms required to be able to “construct” coordinates and metrics. Yet (a) they all use nothing but incidence relations (b) are satisfied by usual Euclidean Geometries and therefore (c) inherited –locally- by any manifold.

Adding the additional structure of lines, adds to the compatibility-conditions for an Atlas of “charts with lines”. While the topological part remains –coincidence of topology on intersections-, I’m not yet finished of what is required in my case. [Again not such a surprise, taking into account that a Differential Manifold imposes additional compatibility conditions and that –exotic spheres from 4 dimensions onward- there might be more but only one Differential Structure].

Compatibility-Conditions and polish-up may still take some weeks or months, among other because I’m not sure whether going from “simply connected” to “locally path connected” is to weak, that is at least something similar to “star-connected” might be required (yet this as concept already uses “straight lines” so … ).

Tuesday, September 11, 2012

Topology and Geometry: Heuristics

This is yet another attempt to define and deduce the two very basic ingredients of the type of structures we would like to use as common underpinning for General Relativity Theory (normally assumed to be continuous) and General Net Theory (for which the so called Petri-Nets are just one example). This time we will concentrate on Topology and Geometry.
Be aware: this is work in progress!

III Geometry

I.0 Initial Heuristics

0.1 Einstein himself

EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM
The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a "neighboring" one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing “jumps". I am sure the reader will appreciate with sufficient clearness what I mean here by “neighboring" and by “jumps" (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.

Let us now imagine that a large number of little rods of equal length have been made, their lengths being small compared with the dimensions of the marble slab. When I say they are of equal length, I mean that one can be laid on any other without the ends overlapping. We next lay four of these little rods on the marble slab so that they constitute a quadrilateral figure (a square), the diagonals of which are equally long. To ensure the  equality of the diagonals, we make use of a little testing-rod. To this square we add similar ones, each of which has one rod in common with the first. We proceed in like manner with each of these squares until finally the whole marble slab is laid out with squares. The arrangement is such, that each side of a square belongs to two squares and each corner to four squares.

It is a veritable wonder that we can carry out this business without getting into the greatest difficulties. We only need to think of the following. If at any moment three squares meet at a corner, then two sides of the fourth square are already laid, and, as a consequence, the arrangement of the remaining two sides of the square is already completely determined. But I am now no longer able to adjust the quadrilateral so that its diagonals may be equal. If they are equal of their own accord,  then this is an especial favor of the marble slab and of the little rods, about which I can only be thankfully surprised. We must needs experience many such surprises if the construction is to be successful.

If everything has really gone smoothly, then I say that the points of the marble slab constitute a Euclidean continuum with respect to the little rod, which has been used as a “distance" (line-interval). By choosing one corner of a square as “origin", I can characterize every other corner of a square with reference to this origin by means of two numbers. I only need state how many rods I must pass over when, starting from the origin, I proceed towards the “right” and then “upwards”, in order to arrive at the corner of the square under consideration. These two numbers are then the “Cartesian coordinates" of this corner with reference to the “Cartesian coordinate system" which is determined by the arrangement of little rods.

By making use of the following modification of this abstract experiment, we recognize that there must also be cases in which the experiment would be unsuccessful. We shall suppose that the rods “expand" by an amount proportional to the increase of temperature. We heat the central part of the marble slab, but not the periphery, in which case two of our little rods can still be brought into coincidence at every position on the table. But our construction of squares must necessarily come into disorder during the heating, because the little rods on the central region of the table expand, whereas those on the outer part do not.

With reference to our little rods –defined as unit lengths- the marble slab is no longer a Euclidean continuum, and we are also no longer in the position of defining Cartesian coordinates directly with their aid, since the above construction can no longer be carried out. But since there are other things which are not in influenced in a similar manner to the little rods (or perhaps not at all) by the temperature of the table, it is possible quite naturally to maintain the point of view that the marble slab is a ”Euclidean continuum". This can be done in a satisfactory manner by making a more subtle stipulation about the measurement or the comparison of lengths.

But if rods of every kind (i.e. of every material) were to behave in the same way as regards the influence of temperature when they are on the variably heated marble slab, and if we had no other means of detecting the effect of temperature than the geometrical behavior of our rods in experiments analogous to the one described above, then our best plan would be to assign the distance one to two points on the slab, provided that the ends of one of our rods could be made to coincide with these
two points; for how else should we define the distance without our proceeding being in the highest measure grossly arbitrary?

The method of Cartesian coordinates must then be discarded, and replaced by another which does not assume the validity of Euclidean geometry for rigid bodies. The reader will notice that the situation depicted here corresponds to the one brought about by the general postulate of relativity.

Taken from:  Relativity: The Special and the General Theory, Pages 77-80, A Popular Exposition, 3rd ed. Author: Albert Einstein, Translator: Robert W. Lawson, retrieved from http://www.gutenberg.org/ebooks/36114)

0.2 Appreciation and Findings

Carefully reading the above, there are no numbers involved except counting and actually the whole configuration stays the same when applying heat somewhere, even if rods may need bending not only expansion and angels are no longer right ones. So at the very bottom Einstein talks about Topology and not Geometry, since the whole Geometric content reduces to certain configuration-setsbended squares-that stay invariant under homeomorphisms.  Please note, that in Einstein's considerations there is explicit geometric content –the construction itself- and there is implicit geometric content, for instance the surprise that thinks workout so nicely or the assumption that rods represent somehow shortest connections.

The question hence is: can we express the above using only means of topology, that is converting the implicit assumptions into to properties of purely topologically defined configurations i.e. without referring to metrics, including angles? The answer is: Yes, we can. Before doing so within the framework of our Q-Topology, some additional Gedanken-experiments.

Einstein starts off with straight rods and squares constructed using the rods. Now straight is itself a concept deeply founded in Euclidian thinking as the shortest connection between to points. We now know, just thanks to General Relativity, that the straightest connection itself is defined by the curvature of space-time, hence not a very good starting point to define space-time. What about square? Well, actually Einstein does not use the whole square but rather only its border. And border is a topological concept. So let’s try what happens if we start with circles, assuming that we’ve got for each point a set of circles surrounding it.

EinsteinKreise1

Here we got some points scatter in a plane. Actually they were produced plotting for some points sets of circles surrounding these points, or more precisely some points of these circles.

 

 EinsteinKreise 3

 

 

 

Here we have the same plane, yet for 3 points we added 4 circles for each and a straight line that runs through these 3 points. Did we just invent this line? No! If you look closer, wherever two different circles share just a single point, there goes the straight line. Or to put it differently, a third point is on a straight line between to other, if there is a circle for each of the 2 and these circle share exactly this third point. By tweaking Mathematica a little bit more, these 3 would have been part of the original set of scattered points.

EinsteinKreise4

Here the same picture without the clutter. There are more relations to discover: If a point is at the 1 point-intersection of two circles, then this point itself has two circles, which intersect exactly with one circle of one point and the other point. This means the relation for three points to be on one straight line is permutation invariant.

EinsteinKreise5

 

 

Here we added just a forth circle. As it appears, if two different points are each on a straight line with two other points, then all four are on the same line.
And: a point is just in the middle of two other points on the same line, if it has exactly 1 circle running through these 2 points, met a each by a circle from the opposite point.

EinsteinKreise6

And here we got a set of circles that is definitively off the line.
In the case of classical Euclidian Geometry, all the observed relations are easily proven. i.e. they are Theorems. But that’s not our interest here. The point is, that once established, they remain valid at least locally under continuous, bijective transformations, i.e. –topologically speaking homeomorphisms- as they preserve sets, intersections and borders, the only concepts we’ve used so far.

 

 

EinsteinKreise7

EinsteinKreise8

 

 

 

 

 

These are just two examples of different projections of the same configuration: tossing and stretching has no effect, even if our circles aren’t circles anymore.

EinsteinKreise9

 

However, we may not transform circles individually. Here the ellipsis  is the result of trying to transform individually the last circle and the result spells disaster. The ellipsis has still 1-point intersections with other circles yet somewhere else. Transformations therefore affect at least locally complete neighborhoods and we can not transform things nicely, i.e. if some borders are circles and others not, there is no transformation that would turn all at once into circles. 

 

 

EinsteinKreise 10a EinsteinKreise 10

A final, drastic example: this transformation leaves the circle-sets as sets intact – each depicted in a different color-, yet suddenly our straight line isn’t straight anymore; well it depends: If we take Einstein and our above observations seriously –i.e. that a straight line is defined by its intersecting circle-sets, whatever these circles or equivalent descriptions may be, then the line is now as straight as it was before. We are just using a different but equivalent set of coordinates to describe the configuration or –what finally becomes the same- it’s the same configuration, only in a different gravitational field.

Now let’s see what happens if instead of circles or ellipses we use something else, for instances, as Einstein did, squares respectively quadrilaterals. Topologically speaking, that is without metrics, they are equivalent anyway.

EinsteinKreise12

EinsteinKreise 12a

Apparently in this example and it’s transformation, everything works fine. Yet the next example shows that we are in deep, deep trouble.

EinsteinKreise 13

EinsteinKreise 13a 

The straight line runs thru the lower right corner, the center and the upper left corner, yet the central quadrilateral though it touches upper and lower quadrilaterals, it does so not in a single point but in a whole segment.

Gauss and Einstein would say: don’t worry! Just adapt and refine your grid and at the limit everything will be fine.  The next picture tries to illustrate this idea.

EinsteinKreise 14

EinsteinKreise 14a 

The smaller the quadrilaterals get, the smaller gets their intersection, which appears as if closing in on the line. Yet Heisenberg now would say: be aware, you can’t always refine to the limit, while Feynman would add: well any point of the segment of intersection is part of a possible shortest path, they just differ in their probabilities. The point with the highest probability will be the point closest to a straight line.  

The first approach assumes that somehow, at least in a small neighborhood, we would already know what straight line means, and then starting from there construct an appropriate grid. This would be in discordance with our whole approach, where we would like to use a given grid to define what straight line means.

The second approach requires tools ways beyond pure topology, and even then remains somewhat circular, as its requires already a metric to define the respective wave-functions, which in turn yield the probabilities, where we would like to deduce the metric using the grid and not use a metric to deduce the grid as peaks and lows of the probability function.

The critical condition, that could not be met easily nor always, was that two circles meet just in one single point. What if relaxing this condition just a little bit? To get a better idea, let’s have again a look on the offline circle, now an offline quadrilateral.

EinsteinKreise 15

EinsteinKreise 15a

There are appearing differences: the green quadrilateral cuts segments out of blue and red, as they do with respect to green, likewise black cuts green and blue and they in turn cut black. Yet while the green-red and red-green respectively green-blue and blue-green segments are the same, the black-blue and blue-black respectively black-green and green-black segments differ; and that is a pure topological difference, which allows us the distinguish the configuration of black wrt.  blue and green  from the configuration of green wrt. red and blue.

For further considerations we will hence relax “share just one point” to “share just one segment”. As to end the heuristics part, we’ll have a look without further discussion on touching and intersecting spheres and cubes in 3 dimensions.

EinsteinKreise16

EinsteinKreise16a

EinsteinKreise 17 EinsteinKreise 17a EinsteinKreise 17b

Monday, December 5, 2011

Linear Q-Space - Third Intent

We had again to modify the definition in two aspects: (1) Taking into account,as we did before when defining the topology  that not all elements are places, and (2) introduce some means to connect the locally defined linear structures.

While the Light-cones are the underlying combinatorial structure from General Relativity to model the Causal (or Conformal) Structure, it seems that totally normal neighborhoods serving as charts and the atlas formed by a subset of the class of totally normal neighborhoods  are the proper candidates upon which to model the combinatorial equivalent for the Projective Structure.

In my last version for the Q-Space Axioms, I committed a terrible conceptual mistake, which for its detection and clean-up took some time. Mislead by the appearance of the concept of geodesics as if something global yet based on only locally defined properties, I introduced a similar global relation G with local properties. In the second intent, I limited the definition to a pure local G yet this way I lost the connection from local context to local context, which for manifolds is naturally provided by their atlas of charts.

Axiom Set 3 Linear Q-SpacesEqn4

(1) As before, take Topological Q-Spaces as departing point. (2) As before introduces the topological closure and the closed hull of Alexandrov-Sets (=Double Cones) as base-set.
(3) Defines as strictly local relation G within a base-set.
(4) Define 3 properties required to make G a useful Local Linear Structure:
Before continuing, remember that in our topology not necessarily all elements are closed. We singled out one set, the set of Places as not-open, while all others are open.
(4.1) may be called the line property: 2 places define a line i.e. if two different elements c and d are collinear to the same places a and b, then they are on the same line.
(4.2) may be called the separation property: different elements differ in at least one line and any line contains at least two places.
(4.3) may be called the connectivity property: it’s possible to get from anywhere to anywhere in a final number of linear steps.
The combination of (4.1) and (4.2) imply that locally G satisfies the axioms for Incidence geometry: all lines have at least 2 places, two places define exactly one line and finally any 2 different elements are on at least 2 different lines, in our case even stronger: 2 different lines with each at least 3 elements.
(5) defines linear sets i.e. those that with any 2 places contain all locally collinear elements. The <linear set generated by a set of elements> is the smallest linear set that contains them all. The concept of a generated linear set allows later to introduce a combinatorial concept of linear independence.
(6) introduces the set of 1-dimensional linear sets or lines.
The predicate (7) connects lines to the previously topologically defined paths: all lines are paths. This turns lines and hence the local linear structure into a topological invariant. The predicate more over requires that the whole linear structure may be densely generated at each element that is that there is a countable set of lines all containing this element, which generates the whole neighborhood.
(8) defines Normal Blocks as equivalent for totally normal neighborhoods i.e. a block and a local linear structure, right as the definition of a totally normal neighborhood requires a specific affine connection .
(9) introduces the base set for global G, which –as it appears in (10)- is nothing else but the another expression of forming an atlas out of totally normal neighborhoods.
(10) defines the class of global G, formed taking subsets of Normal Blocks. Such a subset has to meet 3 conditions: (10.1) its blocks have to cover the whole set, where the global structure is just the join of the local linear structures. (10.2) The local structures have to be compatible, i.e. agree on shared subsets and finally (10.3) the whole subset of normal blocks has to be connected by means of local linear structures with shared subsets. Actually (10) was modeled following the idea of an atlas of compatible charts for a manifold.
(11) Define the class of Q-orders that allow such a global G and the class of these.
(12) Simply asks that this class is not empty.

Going back to Totally Normal Neighborhoods of Lorentz-Manifolds. All points in a Totally Normal Neighborhood are connected by at most one geodesic, whence different points differ at least in one geodesic. A geodesic relies in its definition on the affine connection. Obviously lines are paths in the underlying topology. In case of standard Lorentz-Manifolds, just 4 lines are sufficient to generate the whole neighborhood. Totally Normal Neighborhoods define charts, which in turn for a fixed affine connection define jointly an atlas, which obviously is connected for connected manifolds.

Voilá exactly what the axiom-set 3 claims for Linear Q-orders.

Monday, November 28, 2011

Linear Q-Spaces – Second Intent

While the Light-cones are the underlying combinatorial structure from General Relativity to model the Causal (or Conformal) Structure, it seems that Normal Neighborhoods and Normal Coordinate-Systems (alas the Inertial-Frames or the heuristic base for the Einstein Equivalence Principle) are the proper candidates upon which to model the combinatorial equivalent for the Projective Structure.

In my last version for the Q-Space Axioms, I committed a terrible conceptual mistake, which for its detection and clean-up took some time. Mislead by the appearance of the concept of geodesics as if something global yet based on only locally defined properties, I introduced a similar global relation G with local properties.
Now G will be a strictly local relation with only local properties.
Here is the new Axiom Set 3 titled Linear Q-Spaces.

Axiom Set 3 Linear Q-Spaces Linear Spaces

(1) As before, take Topological Q-Spaces as departing point. (2) As before introduces the topological closure and the closed hull of Alexandrov-Sets (=Double Cones) as base-set.
(3) However defines G as a strictly local relations within a base-set.
(4) Define 3 properties required to make G a useful Local Linear Structure:
(4.1) may be called the line property: 2 points define a line i.e. if two different points c and d are collinear to the same points a and b, then they are on the same line.
(4.2) may be called the separation property: different points differ in at least one line.
(4.3) may be called the connectivity property: it’s possible to get from anywhere to anywhere in a final number of linear steps.
The combination of (4.1) and (4.2) imply that locally G satisfies the axioms for Incidence geometry: all lines have at least 2 points, two points define exactly one line and finally any 2 different points are on at least 2 different lines, in our case even stronger: 2 different lines with each at least 3 points.
(5) defines linear sets i.e. those that with any 2 points contain all locally collinear points. The <linear set generated by a set of points> is the smallest linear set that contains them all. The concept of a generated linear set allows to introduce a combinatorial concept of linear independence.
(6) introduces the set of 1-dimensional linear sets or lines.
(7) defines a set of Frames i.e. minimal subsets of mutually linear independent lines (7.2), which generate the whole set (7.1) and are fixed at some point (7.3).
(8) Is a later required technicality: a Frame should be countable at least.
(9) singles out the anchor-points for Frames.
(10) defines for a Q-Order a Locally Linear Structure (though without metric part yet): all those pairs of a base-set B and a relation G defined on B such that it defines a linear structure on B (10.1), every line is at the same time a path in the underlying topology (10.2), all frames are at least countable (10.3) and finally there is at least one anchor point (10.4). Please note that there might be more but only one G i.e. the same base-set may carry more but one linear structure. Finally a Locally Linear Structure is a topological invariant, as a local homeomorphism takes lines to lines (consequence of 10.2).
(11) Selects all those Q-Orders for which there is a corresponding LLS such that there is a countable subset, whose anchor-points cover the whole set. The final LLS is just the union of LLS for all Q-orders.
(12) Just claims that there are Q-Orders with a Locally Linear Structure

Going back to Normal Neighborhoods of Lorentz-Manifolds. These are defined by selecting first one anchor-point (in the Einstein-concept the famous free-falling-observer). All points in a Normal Neighborhood of this anchor-point are connected by at most one geodesic, whence different points differ at least in one geodesic. All points connect by means of a geodesic to the anchor-point, and finally there is at least one frame –the normal coordinate system at the anchor-point- that generates the whole set. Obviously lines are paths in the underlying topology. Finally there is a Normal Neighborhood for any point. The dimension (8) for Frames is just 4. And the second-countability for Lorentz-Manifolds implies the countability restriction in (11)
Voilá exactly what the axiom-set 3 claims for Linear Q-orders.

Next step will be to explore the combinatorial elements of connections when expressed as mappings of frame-sets.

Saturday, November 19, 2011

Combinatorial Pre-Weyl-Spaces

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, those which 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. 

Phrased more general by Fotini Markopoulou[1] : Background independence I (BI-I): A theory is background independent if its basic quantities and concepts do not presuppose the existence of a background space-time metric. Hence one might  argue that using Differential Manifolds as basic framework as such violates the background-independence postulate: the concept Manifold as such uses by its very definition at least locally always the Euclidian metric, not to mentioned that Topological Manifolds with 4 or more dimensions may carry more but only one Differential Structure, such that the very construction of the Riemann Curvature Tensor depends on a choice made before.

Following ideas already touched by Hermann Weyl[2] and resumed later by John Stachel[3], the Riemann Curvature Tensor can be decomposed in principally 2 different parts: a conformal part, which represents the causal structure, and a projective part, which represents the geometry of time-space. Both are usually defined on top of a suitable Differential Manifold, such that –including the latter- we have 3 basic structural ingredients.

Stachel observes that both the conformal and the projective part may be decomposed further, into a something completely combinatorial –the structures- and a something that provide measurement or metrics. In this way combining causal structure with volume or geometry with metric and applying suitable compatibly conditions, the original Riemann Curvature Tensor may be reconstructed.

A pure combinatorial, background independent approach hence would forget about inherent metrics and focus only on the combinatorial aspects of conformal and projective structure. Combinatorial models for geometry are known since the antique times, actually they were geometry until the 19th century.
Interesting enough, there is a very large class of purely combinatorial models for geometries –finite or discrete or continuous- which allows an algebraic representation by means of coordinates, something again that the ancient already knew about yet to its full extent was formalized not until the 19th and 20th century.  
For the conformal part we will present a combinatorial model for less restrictive than the normally used partial orders as it still allows time-loops.

Yet the underlying framework Differential Manifold itself comes with inherent metric properties twice. First it inherits topologic properties like regular and Haussdorff. which make the underlying topology metrizable. Second it inherits locally the Euclidean metric, which in turn is implicitly but intensively used in the Differential Calculi.  So for a pure combinatorial model we will have to drop some topological properties, while retaining what we think –from a combinatorial point of view- as essential.

Finally aiming “on long shot” on a Sum-over-history approach, we have to be aware that we will have to talk finally about not one but rather classes of causal structures and their respective probabilities.  Different causal structures may require different topologies, such that we will have to start with causal structures and introduce suitable topologies as an additional property. (In our last intent, we still started with topology and went from there to individual geometries and Q-orders).

[1]Fotini Markopoulou,Conserved quantities in background independent theories,Journal of Physics: Conference Series 67 (2007) 012019
[2] Hermann Weyl, Raum-Zeit-Materie, Julius Springer, Berlin 1919
[3] John Stachel, Projective and Conformal Structures in General Relativity, Loops ’07, Morelia June 25-30, 2007

The Axioms for Combinatorial Pre-Weyl-Spaces

The above sets the course for the following 3 sets of Axioms. The first set –Q-Spaces- defines Q-Orders as our representation for causal structures, i.e. the conformal part without metric. The second set –Topological Q-Spaces- constructs a topology based on Q-Orders and claims axiomatically two important topological properties of Topological Manifolds: being connected and being locally path connected. The third set –Geometric Q-Spaces- introduces local Incidence-Geometries as our model for the combinatorial aspects of the projective structure, sufficiently rich to be embedded point-wise into Projective Geometries.
The objects –sets and classes- defined by the 3 axiom-sets try to resemble the combinatorial qualities (or properties) of Pre-Weyl-spaces, i.e. the combination of a conformal and a projective combinatorial structure, yet without an equivalent for a connection nor any metric yet.

{Something about notation: we use pure Set-Theory, large letters denote sets, fractional large letters sets of sets and finally large Greek Letters sets of sets of sets. Bold letters are distinguished, named sets, which may be referenced later by their letter. Small letters denote elements of the base-set S. Except the usual letter (N) for natural numbers, there shouldn’t be any undefined name.
Sets are defined by denoting the element, sometimes giving the initial  originating domain (before |) and the condition a specific element satisfies (after |)
:= and :<> are used to introduce definitions: the left expression/symbol is defined by the right expression. The left side might be a predicate with arguments written as dyadic-operator (small Greek letter).
All and Existence quantifiers a represented a usual, with the element-variables before ‘:’ and the expression that combines the variables after “:”. Sometimes a restricting condition is included before “:”. Formulas with non-quantified element-variables carry an implicit All-quantifier at their beginning, ranging over the base set S.}

Axiom Set 1: Q-Spaces Eqn1(a hint: by clicking opens another window with the formula-text in large).

(1) introduces the base-set S. (2) and (3) introduce the class of objects –Q-Orders- as subsets of pairs of pairs of distinct elements. (4) to (9) specify additional properties that we are demanding. The Q-relation resembles the ordering of 4 elements “on a 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) is just a simplification of the Q-Relation: We forget the order and only remember that the 4 elements are on some path.
(6) Defines as predicate a rule of interference to combine different paths.  
(6.1) whenever all pairs of 3 points can be found on some path –regardless their configuration there-, there should be a 4th element giving some path, where all 4 may be positioned.
(6.2) 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).
(7) defines as predicate τ a rule for separation and a rule for connectivity:
(7.1) requires that different elements may be separated by the Q-Relation. Be aware that only in the 1-dimensional case this implies that between any 2 elements there is a third one.
(7.2) enables to go from a subset to its complement just by interchanging one element. It forces S to have at least 5 elements.
(8) introduces the concept of double-cover to define in (9) a predicate ο that claims the existence of an global orientation.  4 points on a path may be traversed just in one of 2 cyclic directions, clockwise or counterclockwise. (9) extends this concept to whole Q respectively S. (The entry Going around in Circles III provides some heuristics).
(10) defines the class Θ of all Q-Orders on S that is of all sets Q that satisfy the axioms defined as predicates in (4), (6), (7) and (9).
(11) finally claims that Θ shall not be empty, i.e. S shall allow at least one Q-Order.

Notes: in Q-Spaces - Examples there are examples of Q-Spaces just with one Q-Order shown/defined. The entry as such still awaits its update to our most recent version. Yet for those familiar with causal structures on Lorentz-Manifolds: think about the Q-Order as the arrangement of for 4 points on a time-like curve …. and it becomes quite plausible that time-orientable Lorentz-Manifolds are indeed Q-spaces. A formal proof is still due.

Axiom Set 2: Topological Q-Spaces Eqn2

As next step we will construct topologies from Q-Orders.
(1) defines a Q-Space as starting point. Be aware that Θ is a class of Q-Orders, hence (2) to (8) apply to members of this class (or are parameterized by its members).
(2) defines the set of Alexandrov-sets for a given Q-Order, i.e. all elements that are at one side between 2 points a and b, including the end points. These sets resemble the Alexandrov-topology for Lorentzian Manifolds.
(3) specializes the relation “be element of” into “be contained in” i.e. not only the element is element but there is a “left” and “right” neighbor.
(4) singles out Places among the elements of S: whenever a place is contained in two different A-sets, there is a third A-set containing the place and contained in the intersection of both. As heuristics, think about A-sets as neighborhoods, then the intersection of any two neighborhoods of a place contains a neighborhood for that place.
(5) With Places singled out, we define straight forward a topology O. The Open Sets for this topology are all those sets which for all their places contain a containing A-set. No proof needed to see that arbitrary joins of Open Sets and the intersection of two Open sets are open.
(6) defines a connectivity-predicate ω, true if a set can not be split into to two open, disjoint subsets i.e. this set is topologically connected.
(7) introduce the combinatorial equivalent of a Jordan-Curve, normally defined as the injective, continuous mapping of a circle, yet as injective carries the Hausdorff-property to the target-space, not applicable in our case. We define Paths as those connected subsets, which fall apart removing one element except at most 2, the possible endpoints.
(8) defines a stronger connectivity-predicate ω, true if any subset of a set and its complement can be connected by a path.
(9) selects into the class Ω all those Q-Orders from Θ that define a connected and locally path connected A-topology.
(10) finally claims that Ω shall not be empty, i.e. S shall allow at least one topological Q-Order.

Notes: All examples are topological Q-Spaces. In the discrete case, refining the A-Topology into the Path- or Hawking-topology, Q-Spaces turn out to be Petri-Nets. Finally, a topological manifold by definition is connected and locally path-connected. Yet only with additional constraints –for instance strongly casual- this topology and our A-Topology for Lorentz-Manifolds are equivalent. The exact investigation of these constraints for Q-Spaces has yet to be done.

Axiom Set 3: Geometric Q-Spaces Eqn3

(1) defines a topological Q-Space as starting point. Be aware that Ω is a class of Q-Orders, hence (2) to (8) apply to members of this class (or are parameterized by its members). Please note that G is a global relation, even if we define its additional properties locally. This ensure that a property existing in on block remains valid in other blocks, which contain the same points.
(2) introduces the well know closure of a set and the closured A-sets as building-Blocks.
(3) introduces the objects or relations, we will use: subsets of the set of sets with 3 distinct elements.
(4) defines as predicate γ three properties, we will require locally i.e. within a block B.
(4.1) may be called the line property: 2 points define a line i.e. if two different points c and d are collinear to the same points a and b, then they are on the same line.
(4.2) may be called the separation property: different points differ in at least one line.
(4.3) may be called the continuity property: if there is a collinear third point anywhere, than there is a local representative.
The combination of (4.1) and (4.2) imply that locally G satisfies the axioms for Incidence geometry: all lines have at least 2 points, two points define exactly one line and finally any 2 different points are on at least 2 different lines, in our case even stronger: 2 different lines with each at least 3 points.
(5) defines linear sets i.e. those that with any 2 points contain all locally collinear point. The <linear set generated by a set of points> is the smallest linear set that contains them all. The concept of a generated linear set will allow later to introduce combinatorial the concept of linear independence.
(6) introduces the set of 1-dimensional linear sets or lines and the subset of those lines that go through a single point, some times called the star belonging to a point.
(7) the predicate λ transcribed literally claims that the star of any point contains an at most countable subset of lines whose points generate the whole block.
As for its heuristics, assume that a block carries an Euclidian geometry with finite dimension, then at each point one may find a finite number of lines that may serve as coordinates taking that point as origin i.e. generate the whole block starting at that point. More general, assume that a block carries the geometry resulting from separable linear vector-space, then at each point we may find a countable base for that vector-space. 
Different to the above examples, we still do not specify the procedure of how to reconstruct the whole block just starting from coordinates at a specific point. We only claim that these coordinates exist, what ever the procedure might be to generate the whole block.
Reversely, as all points are in this respect equivalent, moving from one point to another means changing the set of coordinates, or more precisely changing sets of coordinates at the origin by sets of coordinates at the destiny, as there might be more than one set at each point, procedures similar to those used when working with frame-bundles, yet again we still do not specify how-to.  
(8) introduces as predicate κ a simple compatibility condition between underlying Q-Order and Geometry: if a line contains 2 Q-related elements, then the whole line is part of Q, heuristically a time-like geodesic somewhere stays time-like all the way.
(9) defines the class Γ of all G-orders, for which there exists a Q-Order such that the axioms defined as predicates in (4), (7) and (8) are satisfied.
(10) finally claims that Γ shall not be empty, i.e. S shall allow at least one topological Q-Order and one complying G-order or Geometry.

Notes: with respect to Lorentz-manifolds, the normal or geodesic coordinates  define a Geometry  in the above sense, taking as blocks the closure of a totally normal neighborhood, in which any two points are connected be a geodesic. 3 points on such a geodesic from a g-triple {a,b,c}. The formal proof is still outstanding. Yet it appears as if all Lorentz-Manifolds are Geometric Q-Spaces. For Petri-Nets I have not found a concept similar to Geometry beyond the example Petri uses –the 2 dimensional Grid as combinatorial equivalent for an 2-dimensional Minkowski space. Yet in Minkowski-Space all and every thing is linear and in 2 dimensions there isn't a true conformal structure beyond triviality either (conformally flat).

Well, there is a lot still missing namely the definition of connections -that is the actual procedure or mapping when moving from point to point-, as the proper embedding of stars into Projective Geometries alas Vector-spaces, which will be required to define metrics and connections.

Again the examples appear to be valid even though I haven't revised them already one by one. Interesting the geometry for Génesis is the smallest Geometry possible .. and it’s most logical projective embedding appears to be the Fano-Plane, the smallest possible Projective Geometry.

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.

[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.

Tuesday, December 14, 2010

Einstein Spaces

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

  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.

[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.

 

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 SpaceEqn1

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.

  1. 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.
  2. 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. 
  3. 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.
  4. 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.
  5. 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  Eqn2

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 3 P–Space     Path SpaceEqn3

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 pathEqn4

(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.

Axiom 4 T–Space Time SpaceEqn6

(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 GeometryEqn7

(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 CategoryEqn8

(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. geo-001

geo-002

{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