{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T14:49:10Z","timestamp":1774882150205,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,29]],"date-time":"2023-01-29T00:00:00Z","timestamp":1674950400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2022R27"],"award-info":[{"award-number":["PNURSP2022R27"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The goal of the present research paper is to study how a spacetime manifold evolves when thermal flux, thermal energy density and thermal stress are involved; such spacetime is called a thermodynamical fluid spacetime (TFS). We deal with some geometrical characteristics of TFS and obtain the value of cosmological constant \u039b. The next step is to demonstrate that a relativistic TFS is a generalized Ricci recurrent TFS. Moreover, we use TFS with thermodynamic matter tensors of Codazzi type and Ricci cyclic type. In addition, we discover the solitonic significance of TFS in terms of the Ricci metric (i.e., Ricci soliton RS).<\/jats:p>","DOI":"10.3390\/axioms12020138","type":"journal-article","created":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T02:01:18Z","timestamp":1675044078000},"page":"138","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Geometrical Structure in a Relativistic Thermodynamical Fluid Spacetime"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1713-6831","authenticated-orcid":false,"given":"Mohd. Danish","family":"Siddiqi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2116-7382","authenticated-orcid":false,"given":"Fatemah","family":"Mofarreh","sequence":"additional","affiliation":[{"name":"Mathematical Science Department, Faculty of Science, Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3895-7548","authenticated-orcid":false,"given":"Aliya Naaz","family":"Siddiqui","sequence":"additional","affiliation":[{"name":"Division of Mathematics, School of Basic & Applied Sciences, Galgotias University, Greater Noida 203201, Uttar Pradesh, India"}]},{"given":"Shah Alam","family":"Siddiqui","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,29]]},"reference":[{"key":"ref_1","unstructured":"Eisenhart, L.P. (1909). A Treatise on the Differential Geometry of Curves and Surfaces, Ginn."},{"key":"ref_2","unstructured":"Hawking, S.W., and Ellis, G.F.R. (1973). The Large Scale Structure of Spacetime, Cambridge University Press. Cambridge Monographs on Mathematical Physics, 1."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1086\/156533","article-title":"The stability of a rotating universe","volume":"225","author":"Novello","year":"1978","journal-title":"Astrophys. J."},{"key":"ref_4","unstructured":"Stephani, H. (1982). General Relativity-An Introduction to the Theory of Gravitational Field, Cambridge University Press."},{"key":"ref_5","unstructured":"O\u2019Neill, B. (1983). Semi-Riemannian Geometry with Applications to Relativity, Academic Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"L263","DOI":"10.1088\/0264-9381\/6\/12\/005","article-title":"Energy-momentum tensor in the general scalar tensor theory","volume":"6","author":"Pimental","year":"1989","journal-title":"Class. Quantum Grav."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3292","DOI":"10.1103\/PhysRevD.9.3292","article-title":"Generalized second law of thermodynamics in blackhole physics","volume":"9","author":"Bekestein","year":"1974","journal-title":"Phy. Rev. D"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1260","DOI":"10.1103\/PhysRevLett.75.1260","article-title":"Thermodynamic of Spacetime: The Einstein equation of state","volume":"75","author":"Jacobson","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s004070000025","article-title":"Lambda: The constant that refuses to die","volume":"55","author":"Earman","year":"2001","journal-title":"Arch. Hist. Exact Sci."},{"key":"ref_10","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-Galactic Nebulae","volume":"15","author":"Hubble","year":"1929","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1007\/s10714-007-0557-5","article-title":"The cosmological constant","volume":"40","author":"Bousso","year":"2008","journal-title":"Gen. Rel. Grav."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1140\/epjc\/s10052-020-08588-2","article-title":"Structure scalars and their evolution for massive objects in f(R) gravity","volume":"81","author":"Bhatti","year":"2021","journal-title":"Eur. Phys. J. C"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1140\/epjp\/s13360-020-00408-6","article-title":"Complexity for self-gravitating fluid distributions in f(G,T) gravity","volume":"135","author":"Yousaf","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1007\/BF00755985","article-title":"A Critical density cosmological model with varying gravitational and cosmological constants","volume":"22","year":"1990","journal-title":"Gen. Rel. Grav."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1103\/RevModPhys.75.559","article-title":"The cosmological constant and dark energy","volume":"75","author":"Peebles","year":"2003","journal-title":"Rev. Modern Phys."},{"key":"ref_16","first-page":"063507","article-title":"Viscous dark fluid cosmology","volume":"82","author":"Velten","year":"2010","journal-title":"Phys. Rev."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/0003-4916(79)90128-3","article-title":"Relativistic thermodynamics of fluids. I","volume":"118","author":"Havas","year":"1979","journal-title":"Ann. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1103\/PhysRev.58.919","article-title":"The Thermodynamics of Irreversible Processes III. Relativistic Theory of the simple fluid","volume":"58","author":"Eckart","year":"1940","journal-title":"Phys. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1027","DOI":"10.1007\/BF02302387","article-title":"Spacetimes with covariant constant energy momentum tensor","volume":"35","author":"Chaki","year":"1996","journal-title":"Int. J. Theoret. Phys."},{"key":"ref_20","first-page":"915","article-title":"Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetime","volume":"30","author":"Alias","year":"1995","journal-title":"Gen Rel. Grav."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"049901","DOI":"10.1063\/1.4945591","article-title":"A condition for a perfect fluid spacetime to be a generalized Robertson-Walker spacetimes, a survey","volume":"57","author":"Mantica","year":"2016","journal-title":"J. Math. Phys."},{"key":"ref_22","first-page":"526","article-title":"On pseudo Ricci-symmetric manifolds","volume":"6","author":"Arslan","year":"2001","journal-title":"Balkan J. Geom. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.na.2004.09.034","article-title":"Conformal Killing vector field on spacetime solutions of Einstein\u2019s equation and initial data","volume":"63","author":"Duggal","year":"2005","journal-title":"Nonlinear Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1007\/s10773-015-2692-1","article-title":"A study of generalized quasi-Einstein spacetimes with applications in general relativity","volume":"55","author":"Guler","year":"2016","journal-title":"Int. J. Theor. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1090\/conm\/071\/954419","article-title":"The Ricci flow on surfaces","volume":"17","author":"Hamilton","year":"1988","journal-title":"Contemp. Math."},{"key":"ref_26","first-page":"75","article-title":"Ricci Solitons and Symmetries of Space time manifold of general relativity","volume":"1","author":"Ali","year":"2014","journal-title":"J. Adv. Res. Class. Mod. Geom."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1216\/rmj.2020.50.41","article-title":"Solitons and geometrical structures in a perfect fluid spacetime","volume":"50","author":"Blaga","year":"2020","journal-title":"Rocky Mountain J. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1007\/s13370-019-00679-y","article-title":"Ricci solitons and Geometrical structure in a Perfect fluid spacetime with Torse-forming vector filed","volume":"30","author":"Venkatesha","year":"2019","journal-title":"Afr. Math."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Chow, B., Chu, S.C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., and Ni, L. (2007). The Ricci Flow: Techniques and Applications, AMS. Part I: Geometric Aspects.","DOI":"10.1090\/surv\/144"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Deshmukh, S., and Alsodais, H. (2020). A Note on Ricci Solitons. Symmetry, 12.","DOI":"10.3390\/sym12020289"},{"key":"ref_31","first-page":"163","article-title":"Ricci \u03c1-soliton and geometrical structure in a dust fluid and viscous fluid sapcetime","volume":"46","author":"Siddiqi","year":"2019","journal-title":"Bulg. J. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2050083","DOI":"10.1142\/S0219887820500838","article-title":"Conformal Ricci soliton and Geometrical structure in a perfect fluid spacetime","volume":"17","author":"Siddiqi","year":"2020","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_33","first-page":"126","article-title":"Almost Ricci-Bourguignon solitons and geometrical structure in a relativistic perfect fluid spacetime","volume":"26","author":"Siddiqui","year":"2021","journal-title":"Balkan J. Geom. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"104370","DOI":"10.1016\/j.geomphys.2021.104370","article-title":"Relativistic magneto-fluid spacetimes","volume":"170","author":"Siddiqi","year":"2021","journal-title":"J. Geom. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Siddiqi, M.D., Chaubey, S.K., and Khan, M.N.I. (2021). f(R,T)-Gravity Model with Perfect Fluid Admitting Einstein Solitons. Mathematics, 10.","DOI":"10.3390\/math10010082"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11005-021-01493-z","article-title":"Relativistic perfect fluid spacetimes and Ricci\u2013Yamabe solitons","volume":"112","author":"Siddiqi","year":"2022","journal-title":"Lett. Math. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3265","DOI":"10.1090\/S0002-9939-2013-11616-3","article-title":"Codazzi tensor with two eigenvalues functions","volume":"141","author":"Merton","year":"2013","journal-title":"Proc. Amer. Math. Soc."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1112\/plms\/s3-47.1.15","article-title":"Codazzi tensor fileds, curvature and Pontryagir forms","volume":"47","author":"Derdzinski","year":"1983","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_39","first-page":"15","article-title":"A pseudo-quasi-conformal curvature tensor on a Riemannian manifold","volume":"4","author":"Shaikh","year":"2005","journal-title":"South East Asian J. Math. Math. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10474-008-8049-y","article-title":"On gebralized recurrent Riemannian manifolds","volume":"123","author":"Arslan","year":"2009","journal-title":"Acta Math. Hungar."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1112\/jlms\/s1-27.3.287","article-title":"Some theorems on Ricci-recurrent spaces","volume":"27","author":"Patterson","year":"1952","journal-title":"J. Lond. Math. Soc."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Schoutten, J.A. (1954). Ricci Calculus, Springer.","DOI":"10.1007\/978-3-662-12927-2"},{"key":"ref_43","first-page":"385","article-title":"\u03c6(Ric)-vector fields in Riemannian spaces","volume":"44","author":"Hinterleitner","year":"2008","journal-title":"Arch. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/138\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:19:17Z","timestamp":1760120357000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/138"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,29]]},"references-count":43,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["axioms12020138"],"URL":"https:\/\/doi.org\/10.3390\/axioms12020138","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,29]]}}}