{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:55:52Z","timestamp":1760241352074,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,1,25]],"date-time":"2018-01-25T00:00:00Z","timestamp":1516838400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2019s 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.<\/jats:p>","DOI":"10.3390\/e20020083","type":"journal-article","created":{"date-parts":[[2018,1,25]],"date-time":"2018-01-25T12:25:49Z","timestamp":1516883149000},"page":"83","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On the Holographic Bound in Newtonian Cosmology"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0720-9945","authenticated-orcid":false,"given":"Jos\u00e9","family":"Isidro","sequence":"first","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universidad Polit\u00e9cnica de Valencia, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0347-7280","authenticated-orcid":false,"given":"Pedro","family":"Fern\u00e1ndez de C\u00f3rdoba","sequence":"additional","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universidad Polit\u00e9cnica de Valencia, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1007\/s10714-014-1673-7","article-title":"General Relativity from a Thermodynamic Perspective","volume":"46","author":"Padmanabhan","year":"2014","journal-title":"Gen. Relativ. Grav."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1540007","DOI":"10.1142\/S0217732315400076","article-title":"Emergent Gravity Paradigm: Recent Progress","volume":"30","author":"Padmanabhan","year":"2015","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2236","DOI":"10.18520\/cs\/v109\/i12\/2236-2242","article-title":"Gravity and\/is Thermodynamics","volume":"109","author":"Padmanabhan","year":"2015","journal-title":"Curr. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1630020","DOI":"10.1142\/S0218271816300202","article-title":"The Atoms of Space, Gravity and the Cosmological Constant","volume":"D25","author":"Padmanabhan","year":"2016","journal-title":"Int. J. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Linnemann, N., and Visser, M. (arXiv, 2017). Hints towards the Emergent Nature of Gravity, arXiv.","DOI":"10.1016\/j.shpsb.2018.04.001"},{"key":"ref_6","unstructured":"Moustos, D. (arXiv, 2017). Gravity as a Thermodynamic Phenomenon, arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/JHEP04(2011)029","article-title":"On the Origin of Gravity and the Laws of Newton","volume":"1104","author":"Verlinde","year":"2011","journal-title":"J. High Energy Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1073\/pnas.15.3.168","article-title":"A Relation between Distance and Radial Velocity among Extra\u2013Galactic Nebulae","volume":"15","author":"Hubble","year":"1929","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_9","first-page":"1351","article-title":"Cosmology from Type Ia Supernovae","volume":"29","author":"Perlmutter","year":"1997","journal-title":"Bull. Am. Astron. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1086\/300499","article-title":"Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant","volume":"116","author":"Riess","year":"1998","journal-title":"Astron. J."},{"key":"ref_11","unstructured":"Bondi, H. (2010). Cosmology, Dover."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/S0034-4877(06)80006-6","article-title":"Dark Energy States from Quantization of Boson Fields in a Universe with Unstable Modes","volume":"57","author":"Broadbridge","year":"2006","journal-title":"Rep. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Weinberg, S. (2008). Cosmology, Oxford University Press.","DOI":"10.1093\/oso\/9780198526827.001.0001"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Cabrera, D., de C\u00f3rdoba, P.F., and Isidro, J.M. (2017). Boltzmann Entropy of a Newtonian Universe. Entropy, 19.","DOI":"10.3390\/e19050212"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"L71","DOI":"10.1086\/187073","article-title":"Using the Schr\u00f6dinger Equation to Simulate Collisionless Matter","volume":"416","author":"Widrow","year":"1993","journal-title":"Astrophys. J. Lett."},{"key":"ref_16","unstructured":"Planck Collaboration (2016). Planck 2015 Results. XIII. Cosmological Parameters. Astron. Astrophys., 594, A13."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"092104","DOI":"10.1063\/1.4997418","article-title":"Lp\u2013Spectrum of the Schr\u00f6dinger Operator with Inverted Harmonic Oscillator Potential","volume":"58","author":"Finster","year":"2017","journal-title":"J. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1825","DOI":"10.1088\/0004-637X\/710\/2\/1825","article-title":"A Larger Estimate of the Entropy of the Universe","volume":"710","author":"Egan","year":"2010","journal-title":"Astrophs. J."},{"key":"ref_19","unstructured":"De C\u00f3rdoba, P.F., and Isidro, J.M. (2018). Boltzmann Entropy of the FLRW Universe, in preparation."},{"key":"ref_20","unstructured":"Lebedev, N. (1972). Special Functions and Their Applications, Dover Publications."},{"key":"ref_21","unstructured":"Mart\u00ednez-Merino, A., Obreg\u00f3n, O., and Ryan, M. (arXiv, 2016). Newtonian Black Holes: Particle Production, \u201cHawking\" Temperature, Entropies and Entropy Field Equations, arXiv."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"012034","DOI":"10.1088\/1742-6596\/171\/1\/012034","article-title":"Symmetry Aspects in Emergent Quantum Mechanics","volume":"171","author":"Elze","year":"2009","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_23","unstructured":"Torrom\u00e9, R.G. (arXiv, 2014). A Theory of Emergent Quantum Mechanics and Classical Emergent Gravity, arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1023\/A:1021243926749","article-title":"Equivalence Postulate and Quantum Origin of Gravitation","volume":"15","author":"Matone","year":"2002","journal-title":"Found. Phys. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/2\/83\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:52:29Z","timestamp":1760194349000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/2\/83"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,25]]},"references-count":24,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,2]]}},"alternative-id":["e20020083"],"URL":"https:\/\/doi.org\/10.3390\/e20020083","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,1,25]]}}}