A Spacetime Model for the Matter Common-State Conjecture under Causal Ordering
Year: 2026
Type: preprint
Abstract: Additional Abstract This manuscript is an unfinished draft within a broader research program on the matter common-state conjecture. We propose a series of methods for investigating the common-spacetime-state problem and believe that an effective framework can be formulated. However, developing and presenting the theory takes time, and the version currently made available remains a draft. In the static model, our approach is to use light cones to determine the interfaces. After this partition is made, the prescribed spacetime ordering determines the causal order of the iterations within each spacetime block. The calculations are then carried out block by block, and the resulting pieces are assembled into a set. In the dynamic model, determining interfaces from light-cone data is considerably more complex, but is intended as a generalization of the static model. Volume I is the part completed to date; Volume II will focus primarily on the interface problem, which is substantially more intricate than initially anticipated. As a heuristic picture, consider two overlapping quadratic functions that meet at a point. If the left-hand quadratic is regarded as an interior whole, the portion entering it from the other region can be pictured as a curve. At the intersection, this incoming portion is rotated relative to the tangent direction and expressed in an orthogonal coordinate frame adapted to that tangent. The idea is that, after rotation and superposition, the combined result may be continuous on the interface and on both sides of it, although the precise continuity conditions require careful analysis. More generally, the portions entering one another are envisaged as spacetime regions governed by the causal-ordering method. The interface acts as an anchor: the incoming parts undergo a causal-order rotation, or are propagated again relative to the interface, so that their combined output may form a continuous spacetime. This is only a conceptual picture. The proposed exchange behavior and the resulting continuity still require rigorous algebraic proof. Original Abstract We introduce spacetime-order marking to address the labeling part of the matter common-state problem for many causally ordered sources. In the static model, simultaneous point emissions at fixed positions in a Euclidean reference space define fixed pairwise perpendicular-bisector interfaces and fixed order blocks. Within each block, the gravitational field is updated through the prescribed source stages without moving or recomputing those interfaces. The optical metric remains an auxiliary tool for causal estimates; it does not define the static partition. Rather than re-sort all sources independently in every region, the W-marking rule propagates a known complete block label across realized simple walls by the corresponding adjacent transposition. It identifies the neighboring block and its primary arrival signature, but does not transport field values or impose interface matching. Each regular chamber carries an ordered active-frame solution fiber, and their product with common source and seed data is a candidate static common-state space. We prove, under the axiom of choice, that this product is nonempty when every regional fiber is nonempty, and that countable inverse systems of nonempty compact Hausdorff spaces with continuous bonding maps have a nonempty compact inverse limit. These conclusions remain conditional on their stated fiber and compactness assumptions; neither proves existence or interface compatibility of physical Einstein–matter solutions. A law governing interface motion is outside the fixed-interface static model and is deferred to Volume II. Keywords Causal ordering; matter common-state conjecture; spacetime-order marking; static spacetime; order blocks; Einstein–matter systems; interface compatibility; causal arrival times
Author: Wang, Kianming
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222894664
Cites:
Cited by:
Related to: