Black Holes and Complexity via Constructible Universe

Jerzy Król,Paweł Klimasara

doi:10.3390/universe6110198

Jerzy Król, Paweł Klimasara

Open Access

https://doi.org/10.3390/universe6110198

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Abstract

The relation of randomness and classical algorithmic computational complexity is a vast and deep subject by itself. However, already, 1-randomness sequences call for quantum mechanics in their realization. Thus, we propose to approach black hole’s quantum computational complexity by classical computational classes and randomness classes. The model of a general black hole is proposed based on formal tools from Zermelo–Fraenkel set theory like random forcing or minimal countable constructible model Lα. The Bekenstein–Hawking proportionality rule is shown to hold up to a multiplicative constant. Higher degrees of randomness and algorithmic computational complexity are derived in the model. Directions for further studies are also formulated. The model is designed for exploring deep quantum regime of spacetime.

Highlights

Introduction and MotivationsSusskind et al (e.g., [1,2,3]) in a series of papers have approached the complexity of quantum systems
Their findings are interesting in the case of black holes (BH) and the horizons which are formed in spacetime. This program can be seen as the effort toward understanding the regime where quantum mechanics (QM) and spacetime overlap each other as deeply involved physical structures
It is fair to say that there is a great variety of forcing procedures known for mathematicians but precisely one—the so-called Solovay forcing, which is responsible for randomness, is realized in QM [8,18]

Summary

Introduction and Motivations

Susskind et al (e.g., [1,2,3]) in a series of papers have approached the complexity of quantum systems. We expect that the degree of randomness it carries is high enough for giving an alternative to quantum computing point of view in the context of BHs. As we mentioned already, such expectation is supported by the result that classical 1-random behavior of the coin-tossing generating infinite binary sequence cannot be realized as a deterministic process and, it requires QM (e.g., [6]). Known from the algorithmic computability theory (see Appendix B), one finds that the relative to 0(n) 1-randomness (i.e., n-randomness) is realized by generic BHs. again, the source of the results above lies in the structure of the QM lattice L that governs the change of the ZFC models by random forcing. More thorough and complete treatment of the relation of QM and randomness will be the topic of a separate publication

Mathematical Formalism of QM in the Minimal Constructive Model of ZFC

Models of ZFC and Computational Complexity of BH’s State Space

BH Horizon

Computational Complexity and BHs

Discussion