**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

*Anschaung*as still informally defined axioms yet a rigorous argumentation-scheme to derive properties, relations and operations.

*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.

*Causality*and its rules as defined by S

*pecial 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."

*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.

^{2}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.

*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.

*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.*

*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*.

*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.

*woven fabric*

**is**the combinatorial equivalent of a Pseudo-Riemann Differential Manifold yet without metric.