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Entropy 2018, 20(2), 83; https://doi.org/10.3390/e20020083

On the Holographic Bound in Newtonian Cosmology

Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, 46022 Valencia, Spain
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Received: 11 December 2017 / Revised: 17 January 2018 / Accepted: 23 January 2018 / Published: 25 January 2018
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Abstract

The holographic principle sets an upper bound on the total (Boltzmann) entropy content of the Universe at around 10 123 k B ( k B being Boltzmann’s constant). In this work we point out the existence of a remarkable duality between nonrelativistic quantum mechanics on the one hand, and Newtonian cosmology on the other. Specifically, nonrelativistic quantum mechanics has a quantum probability fluid that exactly mimics the behaviour of the cosmological fluid, the latter considered in the Newtonian approximation. One proves that the equations governing the cosmological fluid (the Euler equation and the continuity equation) become the very equations that govern the quantum probability fluid after applying the Madelung transformation to the Schroedinger wavefunction. Under the assumption that gravitational equipotential surfaces can be identified with isoentropic surfaces, this model allows for a simple computation of the gravitational entropy of a Newtonian Universe. In a first approximation, we model the cosmological fluid as the quantum probability fluid of free Schroedinger waves. We find that this model Universe saturates the holographic bound. As a second approximation, we include the Hubble expansion of the galaxies. The corresponding Schroedinger waves lead to a value of the entropy lying three orders of magnitude below the holographic bound. Current work on a fully relativistic extension of our present model can be expected to yield results in even better agreement with empirical estimates of the entropy of the Universe. View Full-Text
Keywords: Boltzmann entropy; holographic bound; Newtonian cosmology Boltzmann entropy; holographic bound; Newtonian cosmology
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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Isidro, J.M.; Fernández de Córdoba, P. On the Holographic Bound in Newtonian Cosmology. Entropy 2018, 20, 83.

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