{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T00:33:09Z","timestamp":1783557189699,"version":"3.55.0"},"reference-count":10,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,12,25]],"date-time":"2017-12-25T00:00:00Z","timestamp":1514160000000},"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>Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by     S = ln \u03c7 ( x )     with     \u03c7 ( x )     being the distance between two nearby geodesics. We derive an equation for the entropy, which by transformation to a Riccati-type equation becomes similar to the Jacobi equation. We further show that the geodesic equation for a null geodesic in a double-warped spacetime leads to the same entropy equation. By applying a Robertson\u2013Walker metric for a flat three-dimensional Euclidean space expanding as a function of time, we again reach the entropy equation stressing the connection between the chosen entropy measure and time. We finally turn to the Raychaudhuri equation for expansion, which also is a Riccati equation similar to the transformed entropy equation. Those Riccati-type equations have solutions of the same form as the Jacobi equation. The Raychaudhuri equation can be transformed to a harmonic oscillator equation, and it has been shown that the geodesic deviation equation of Jacobi is essentially equivalent to that of a harmonic oscillator. The Raychaudhuri equations are strong geometrical tools in the study of general relativity and cosmology. We suggest a refined entropy measure applicable in cosmology and defined by the average deviation of the geodesics in a congruence.<\/jats:p>","DOI":"10.3390\/e20010006","type":"journal-article","created":{"date-parts":[[2017,12,26]],"date-time":"2017-12-26T03:06:38Z","timestamp":1514257598000},"page":"6","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Entropy Measures as Geometrical Tools in the Study of Cosmology"],"prefix":"10.3390","volume":"20","author":[{"given":"Gilbert","family":"Weinstein","sequence":"first","affiliation":[{"name":"Physics Department, Ariel University, Ariel 40700, Israel"},{"name":"Department of Mathematics, Ariel University, Ariel 40700, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yosef","family":"Strauss","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ben Gurion University, Be\u2019er Sheva 84105, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sergey","family":"Bondarenko","sequence":"additional","affiliation":[{"name":"Physics Department, Ariel University, Ariel 40700, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6679-3320","authenticated-orcid":false,"given":"Asher","family":"Yahalom","sequence":"additional","affiliation":[{"name":"Department of Electrical and Electronic Engineering, Ariel University, Ariel 40700, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Meir","family":"Lewkowicz","sequence":"additional","affiliation":[{"name":"Physics Department, Ariel University, Ariel 40700, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lawrence","family":"Horwitz","sequence":"additional","affiliation":[{"name":"Physics Department, Ariel University, Ariel 40700, Israel"},{"name":"School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel"},{"name":"Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jacob","family":"Levitan","sequence":"additional","affiliation":[{"name":"Physics Department, Ariel University, Ariel 40700, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,12,25]]},"reference":[{"key":"ref_1","unstructured":"Tabor, M. (1989). Chaos and Integrability in Nonlinear Dynamics: An Introduction, Wiley."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1921","DOI":"10.1103\/PhysRevA.13.1921","article-title":"Numerical computations on a stochastic parameter related to the Kolmogorov entropy","volume":"13","author":"Casartelli","year":"1976","journal-title":"Phys. Rev. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2338","DOI":"10.1103\/PhysRevA.14.2338","article-title":"Kolmogorov entropy and numerical experiments","volume":"14","author":"Benettin","year":"1976","journal-title":"Phys. Rev. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"234301","DOI":"10.1103\/PhysRevLett.98.234301","article-title":"Geometry of Hamiltonian Chaos","volume":"98","author":"Horwitz","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Gutzwiller, M. (1990). Chaos in Classical and Quantum Mechanics, Springer.","DOI":"10.1007\/978-1-4612-0983-6"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1007\/BF00661958","article-title":"On the existence of an entropy-like quantity","volume":"96","author":"Evangelidis","year":"1983","journal-title":"Astrophys. Space Sci."},{"key":"ref_7","unstructured":"Carroll, S. (2004). Spacetime and Geometry: An Introduction to General Relativity, Addison Wesley."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s12043-007-0110-9","article-title":"The Raychaudhuri equations: A brief review","volume":"69","author":"Kar","year":"2007","journal-title":"Pramana"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"072701","DOI":"10.1063\/1.4918614","article-title":"Quantum Field Theory of Classically Unstable Hamiltonian Dynamics","volume":"56","author":"Strauss","year":"2015","journal-title":"J. Math. Phys."},{"key":"ref_10","first-page":"47","article-title":"On the average size of polytopes in a convex set","volume":"13","author":"Groemer","year":"1982","journal-title":"Geometr. Dedic."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/6\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:55:26Z","timestamp":1760208926000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,25]]},"references-count":10,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010006"],"URL":"https:\/\/doi.org\/10.3390\/e20010006","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,12,25]]}}}