^{1,a)}and Newshaw Bahreyni

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### Abstract

We present a novel derivation of both the Minkowski metric and Lorentz transformations from the consistent quantification of a causally ordered set of events with respect to an embedded observer. Unlike past derivations, which have relied on assumptions such as the existence of a 4-dimensional manifold, symmetries of space-time, or the constant speed of light, we demonstrate that these now familiar mathematics can be derived as the unique means to consistently quantify a network of events. This suggests that space-time need not be physical, but instead the mathematics of space and time emerges as the unique way in which an observer can consistently quantify events and their relationships to one another. The result is a potential foundation for emergent space-time.

Kevin Knuth would like to thank Keith Earle, Ariel Caticha, Seth Chaiken, Adom Giffin, Philip Goyal, Jeffrey Jewell, Carlos Rodríguez, Jeff Scargle, John Skilling, and Michael Way for many insightful discussions and comments. He would also like to thank Rockne, Ann and Emily Knuth for their faith and support, Henry Knuth for suggesting that he “try using a J,” and Lucy Knuth for decorating his notes. Newshaw Bahreyni would like to thank Shahram Pourmand for his helpful discussions and Mahshid Zahiri, Mohammad Bahreyni, and Shima Bahreyni for their continued support. The authors would also like to thank Giacomo Mauro D’Ariano, Alessandro Tosini, Joshua Choinsky, Oleg Lunin, Margaret May, Patrick O’Keefe, Matthew Sarker, Cristi Stoica, and James Lyons Walsh for valuable comments that have improved the quality of this work. This work was supported, in part, by a grant from the John Templeton Foundation supported, in part, by a grant from the John Templeton Foundation (Knuth) and a University at Albany Benevolent Association Benevolent Award (Bahrenyi).

I. INTRODUCTION II. EVENTS, CHAINS, AND OBSERVERS III. QUANTIFYING A CHAIN A. Valuations B. Closed intervals IV. QUANTIFICATION OF A POSET BY CHAIN PROJECTION A. Chain projection mapping B. Generalized intervals V. CHAIN-INDUCED STRUCTURE A. Induced subspaces 1. Collinearity 2. Proper collinearity B. Coordinated chains 1. Compatibility 2. Coordinated VI. QUANTIFICATION OF GENERALIZED INTERVALS A. The interval pair B. Quantifying coordinated chains C. The symmetric-antisymmetric decomposition D. Interval classes E. Orthogonal subspaces F. Joining generalized intervals G. Scalar quantification of intervals H. Pair transformations VII. THE SPACE-TIME PICTURE A. Space-time coordinates B. Motion C. Coordinates and the pythagorean decomposition VIII. SUBSPACE PROJECTION IX. CONCLUSION

### Key Topics

- Subspaces
- 48.0
- Lorentz group
- 14.0
- Quasicrystals
- 9.0
- Spacetime topology
- 9.0
- Philosophy of science
- 6.0

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### Abstract

We present a novel derivation of both the Minkowski metric and Lorentz transformations from the consistent quantification of a causally ordered set of events with respect to an embedded observer. Unlike past derivations, which have relied on assumptions such as the existence of a 4-dimensional manifold, symmetries of space-time, or the constant speed of light, we demonstrate that these now familiar mathematics can be derived as the unique means to consistently quantify a network of events. This suggests that space-time need not be physical, but instead the mathematics of space and time emerges as the unique way in which an observer can consistently quantify events and their relationships to one another. The result is a potential foundation for emergent space-time.

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