Name it - Tame it

After almost 3 years of silence -while I was like the mouse in the cage, wheeling around without getting anywhere, here is something new announcing that finally I found something that looks like the combinatorial equivalent of a Pseudo-Riemann Differential Manifolds.
It appears as if all boiled down to find the most appropriate universe of discourse to name the abstract objects I was struggling with. 
Here is the story:

Name it – tame it

When creating new Mathematics -new structures, properties, relations and operations- sooner or later you face a non-mathematical but non-trivial problem: how to name them.

If you use already established names, obviously you have to show that your objects belong to the same category of objects already baptized or at least to the same family that is sharing some typical characteristics while dropping or generalizing others, else an already familiarized reader would be misled intuitively into wrong assumptions and worse false conclusions. 

The most radical approach would be to invent the names themselves that is even new words as sequences of letters that never have been used before. Yet words in common language do not come isolated but as a vocabulary belonging to some common problem domain, a for that problem domain.

Much of Mathematics itself consists in taking some of these words as terms defining their initial meaning that is converting intuition by means of axioms into Mathematics, and then proceeds to derive properties, relations and operations using the tools of mathematical logic while preserving o précising the intuitive meaning of the corresponding term in the original vocabulary.

Geometry itself is a mathematical subject, which started in the Euclidian Geometry using some terms of the Anschaung as still informally defined axioms yet a rigorous argumentation-scheme to derive properties, relations and operations.

It’s formalization starting the early XIX century did not only provide us with new types of geometries, not covered by Euclid himself, but finally with an extremely powerful mathematical tool, Differential Manifolds, which is right at the heart of General Relativity as we know it now.  

However in this context Sheaf-theory shows another way Mathematics adopts common language terminology not as only isolated words but as a vocabulary of meaning-connected words.  Here the imported vocabulary itself is not the object of study but serves or at least aims to serve as an intuitive guidance to comprehend the otherwise completely abstract, purely mathematical structures. I do suspect it was even more and earlier: the imported language served as mind-guidance to its creators while creating the new theory itself.

At its beginning, the mathematical language for General Relativity that Einstein used was Tensor-Calculus, which in turn was developed during the XIX century starting with Gauss. Its real problem-domain was literally Geometry not as an abstract mathematical subject but as a technology used to create the first large-scale exact maps. As the areas were by far greater than Attica, pure Euclidian Geometry and its algebraization by Descartes were insufficient as they did not take into account the curvature of earth. Hence much of the new names for the new mathematical objects were an import from the language of the land-surveyors.

The now most frequently used mathematical language for advanced GRT is the Calculus of Differential Forms as developed by Cartan at the beginning of the XX century. While maintaining Differential Manifolds as a fundamental concept, the Cartan Formalism provides a concise yet very compact hence very elegant language to express complicated relations, yet its full theoretical background includes many concepts from many areas, from Topology to Lie-Groups. That’s sufficient stuff to fill a complete curriculum for postgraduate studies in Mathematics. Therefore most introductory courses into GRT still rely on the original Tensor-Calculus. Yet on the other hand almost all attempts aimed at Quantum-Gravitation start with the Einstein-Cartan form of GRT.

GRT as such combines two rather different concepts: Causality and its rules as defined by Special Relativity and Gravity as modification of the space-time geometry, both connected by Einstein’s field equitation, where -as John Archibald Wheeler put it- "Space-time tells matter how to move; matter tells space-time how to curve."

A closer look on Causality first by Stephen Hawking then by David B. Malament moreover revealed that it comes with its own geometry, named causal structure, whose topology is somewhat related yet not equivalent neither to the topology created by gravity nor the topology of the underlying Differential Manifold. Similar, the effects of Gravity can be completely described by the geodetics of free falling particles, which are both causal paths w.r.t. Causality and extremal paths w.r.t. Gravity. Seen as Geometries, the first correspond to conformal Geometry, the latter to projective Geometry, which together completely define a Pseudo-Riemann Differential Manifold.

The concepts and tools provided by the above 2 structures are not sufficient to express neither of Einstein’s famous equations i.e. neither E = M c² of SRT nor the Field equation of GRT. Worse yet they do not allow to express the special equivalence principle i.e. inertial mass = gravitating mass, leaving it buried inside the formalism of Tensor-Calculus; the more general idea that Physics is the same – anywhere, anytime degenerates into covariance, which was meant by Einstein as an expression for the former not it’s founding.

While it’s rather straightforward to find combinatorial structures to model Causality and the effects of Gravity, actually the causal-structural itself is a combinatorial structure, it turned out to be quite difficult to find an combinatorial substitute for the Differential Manifold itself and its construction and almost impossible in the vocabulary of surveyor’s geometry where the constituents are just maps, alas descriptions. It’s already almost impossible to grasp using this vocabulary the conceptual difference between active and passive Diffeomorphisms as introduced by Carlos Rovelli. Yet without that substitute there is no way neither to find a combinatorial substitute for the Cartan Formalism, even if the latter already eliminated algebraic invariance using geometrical symmetry instead. And without that substitute there is, in my humble understanding, no Quantum-Gravity either.

May be due to my laziness –I did not invest the equivalent of a postgrad-curriculum- may be due to my other limitations, I made countable many attempts to find this substitute, always staying within the traditional vocabulary for Differential Manifolds. Finally I decided to drop this vocabulary all together and to switch to another taken from textiles. Now everything fits neatly, at least up to the combinatorial equivalent of a Pseudo-Riemann Differential Manifold, yet still –and fortunately- without metric.

As mathematical structure beneath we use connected locally path-connected Topologies, yet weaker than the usual Haussdorff-Space of Differential Manifolds. Our category of topologies includes discrete models, alas Petri-Nets. What in the Haussdorff case are the paths of Jordan-Curves is generalized into ropes.

In my framework for new terminology, you may think about Einstein’s Universe as a patchwork of many fabric pieces, tied together by compatible seams. The seaming process and its compatibility condition resemble the classical construction of a Differential Manifold. Even more: using proper strands that is subsets of ropes -the paths of continuously differentiable Jordan-Curves- and specific seams -patch-wise isomorphic- the resulting quilt is the combinatorial equivalent of a Differential Manifold.

Classical fabric is made out of 2 quite different components, the warp and the weft, where the warp defines a fixed, pre-tensed setting that the weft may transverse connecting; or catching up Wheeler, warps will resemble effects of Gravity, wefts Causality. A weaving process puts both together, where shedding is formally expressed by strands that belong to both structures, as were the geodetics of free falling particles.  

As a result, using the above mentioned proper strands and specific seams, the woven fabric is the combinatorial equivalent of a Pseudo-Riemann Differential Manifold yet without metric.

Up to here the ideas are completed (to follow soon).