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Universe 2017, 3(1), 8; doi:10.3390/universe3010008

Convexity and the Euclidean Metric of Space-Time

Carnegie Mellon University in Qatar, Education City, P.O. Box 24866 Doha, Qatar
Academic Editors: Stephon Alexander, Jean-Michel Alimi, Elias C. Vagenas and Lorenzo Iorio
Received: 1 November 2016 / Revised: 2 February 2017 / Accepted: 2 February 2017 / Published: 8 February 2017
View Full-Text   |   Download PDF [328 KB, uploaded 8 February 2017]

Abstract

We address the reasons why the “Wick-rotated”, positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces. View Full-Text
Keywords: space-time metric; convexity; smoothness; Hilbert spaces; Banach spaces; dualities space-time metric; convexity; smoothness; Hilbert spaces; Banach spaces; dualities
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Kalogeropoulos, N. Convexity and the Euclidean Metric of Space-Time. Universe 2017, 3, 8.

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