Geometry as Thermodynamics: Deriving Gravitational Dynamics from Entropic Principles

Geometry as Thermodynamics: Deriving Gravitational Dynamics from Entropic Principles | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 27 August 2025 V1 Latest version Share on Geometry as Thermodynamics: Deriving Gravitational Dynamics from Entropic Principles Author : Wen-Xiang Chen 0000-0002-0560-8280 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175631312.21292805/v1 464 views 95 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We present a conceptual and mathematical framework in which spacetime geometry and Einstein's gravitational dynamics emerge from thermodynamic or entropic principles. In this view, the Einstein field equations are obtained as an extremization condition of a suitable entropy functional, rather than from a traditional action principle. We introduce an entropy functional S[g, ξ µ ] = (∇ µ ξ ν)(∇ µ ξ ν) √ −g d 4 x depending on the spacetime metric g ab and an auxiliary vector field ξ µ, and show that its extremum (for all null ξ µ) reproduces the vacuum Einstein equations (with a cosmological constant) as well as the appropriate generalization in presence of matter. In particular, the variational principle δS = 0 yields the geometric field equations R µν − 1 2 Rg µν + Λg µν = 8πG T µν, highlighting gravity as an equation of state of spacetime thermodynamics [4, 5]. Furthermore, by considering a complex analytic extension of the entropy functional, we relate the Laurent series expansion of a potential function f (z) = exp(S[g, ξ µ ]) to gravitational dynamics. We demonstrate that the absence of higher-order poles in this expansion is equivalent to the satisfaction of Einstein's equations, while the coefficient a −1 of the simple pole (the entropy residue) corresponds to conserved Noether charges such as the horizon entropy. In particular, the entropy residue is shown to be proportional to the horizon area and yields the black hole entropy, consistent with the Bekenstein-Hawking area law and Wald's Noether charge formalism. These results provide a novel characterization of gravitational field equations as conditions of thermodynamic equilibrium (no entropy production), reinforcing Jacobson's, Padmanabhan's, and Verlinde's perspectives on gravity as an emergent, thermodynamic phenomenon [4, 6, 7]. Our derivations are formulated in a self-contained manner using the tools of variational calculus, Laurent series, and residue theory. We situate our approach in the context of prior work by Jacobson on the thermodynamic origin of Einstein equations [4], Padmanabhan on entropy extremization and emergent gravity [5], and Verlinde on entropic forces [7]. Supplementary Material File (sss (1).pdf) Download 378.74 KB Information & Authors Information Version history V1 Version 1 27 August 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords black hole gravity mathematics physics thermodynamic geometry Authors Affiliations Wen-Xiang Chen 0000-0002-0560-8280 [email protected] School of Physics, Materials Science, Guangzhou University View all articles by this author Metrics & Citations Metrics Article Usage 464 views 95 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Wen-Xiang Chen. Geometry as Thermodynamics: Deriving Gravitational Dynamics from Entropic Principles. Authorea . 27 August 2025. DOI: https://doi.org/10.22541/au.175631312.21292805/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. 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