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Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity

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The pentalogy (Mallios, A. and Raptis, I. (2001). International Journal of Theoretical Physics 40, 1885; Mallios, A. and Raptis, I. (2002). International Journal of Theoretical Physics 41, 1857; Mallios, A. and Raptis, I. (2003).International Journal of Theoretical Physics 42, 1479; Mallios, A. and Raptis, I. (2004). ‘paper-book’/research monograph); I. Raptis (2005). International Journal of Theoretical Physics (to appear)is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantumcausal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure . We show that the category of finitary differential triads is a finitary instance of an elementary topos proper in the original sense dueto Lawvere and Tierney. We present in the light of Abstract Differential Geometry (ADG) a Grothendieck-type of generalization of Sorkin’s finitary substitutes of continuous spacetime manifoldtopologies, the latter’s topological refinement inverse systems of locally finite coverings and their associated coarse graining sieves, the upshot being that is also a finitary example of a Grothendieck topos. In the process, we discover that the subobject classifier Ω fcq of is a Heyting algebra type of object, thus we infer that the internal logic of our finitary topos is intuitionistic, as expected. We also introduce the new notion of ‘finitary differential geometric morphism’ which, as befits ADG, gives a differential geometric slant to Sorkin’s purely topological acts of refinement (:coarse graining). Based on finitary differential geometric morphisms regarded as natural transformations of the relevant sheaf categories, we observe that the functorial ADG-theoretic version of the principle of general covariance of GeneralRelativity is preserved under topological refinement. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research.

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Authors and Affiliations

  1. Algebra and Geometry Section, Department of Mathematics, University of Athens, Panepistimioupolis, Athens, 157 84, Greece

    Ioannis Raptis (European Commission Marie Curie Reintegration Research Fellow, Visiting Researcher)

  2. Theoretical Physics Group, Blackett Laboratory, Imperial College of Science, Technology and Medicine, Prince Consort Road, South Kensington, London, SW7 2BZ, UK

    Ioannis Raptis (European Commission Marie Curie Reintegration Research Fellow, Visiting Researcher)

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Correspondence to Ioannis Raptis.

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PACS numbers: 04.60.-m, 04.20.Gz, 04.20.-q

Posted at the General Relativity and Quantum Cosmology (gr-qc) electronic archive (www.arXiv.org), as: gr-qc/0507100.

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Raptis, I. Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity. Int J Theor Phys 46, 688–739 (2007). https://doi.org/10.1007/s10773-006-9240-y

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