{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,27]],"date-time":"2025-11-27T10:49:49Z","timestamp":1764240589286,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T00:00:00Z","timestamp":1665964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Prince Sultan University, Riyadh, Saudi Arabia"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we find all nonstatic plane symmetric spacetime metrics whose corresponding Lagrangians possess Noether symmetries. The set of determining equations is analyzed through a Maple algorithm that restricts the metric coefficients to satisfy certain conditions. These restrictions on metric coefficients, while using them to solve the determining equations, give rise to a number of plane symmetric metrics admitting 4-, 5-, 6-, 7-, 8-, 9-, 11-, and 17-dimensional Noether algebras. The Noether theorem is used to find a conserved quantity corresponding to each Noether symmetry. Some physical implications are discussed by finding bounds for different energy conditions for the obtained metrics.<\/jats:p>","DOI":"10.3390\/sym14102174","type":"journal-article","created":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T05:08:02Z","timestamp":1665983282000},"page":"2174","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Noether Symmetries and Conservation Laws in Non-Static Plane Symmetric Spacetime"],"prefix":"10.3390","volume":"14","author":[{"given":"Muhammad","family":"Farhan","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Peshawar, Peshawar 25000, Khyber Pakhtunkhwa, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2303-760X","authenticated-orcid":false,"given":"Tahir","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Peshawar, Peshawar 25000, Khyber Pakhtunkhwa, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9275-0965","authenticated-orcid":false,"given":"Fatima","family":"Azmi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 12435, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 12435, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Stephani, H., Kramer, D., Maccallum, M., Hoenselaers, C., and Herlt, E. (2003). Exact Solutions of Einstein\u2019s Field Equations, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511535185"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hall, G.S. (2004). Symmetries and Curvature Structure in General Relativity, World Scientific.","DOI":"10.1142\/1729"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Feroze, T., Qadir, A., and Ziad, M. (2001). The classification of plane symmetric spacetimes by isometries. J. Math. Phys., 42.","DOI":"10.1063\/1.1385175"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Deshmukh, S., and Belova, O. (2021). On killing vector fields on Riemannian manifolds. Sigma Math., 9.","DOI":"10.3390\/math9030259"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Nikonorov, Y.G. (2019). Spectral properties of Killing vector fields of constant length. J. Geom. Phys., 145.","DOI":"10.1016\/j.geomphys.2019.103485"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bokhari, A.H., and Qadir, A. (1990). Killing vectors of static spherically symmetric metrics. J. Math. Phys., 31.","DOI":"10.1063\/1.528737"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bokhari, A.H., and Qadir, A. (1987). Symmetries of static, spherically symmetric spacetimes. J. Math. Phys., 28.","DOI":"10.1063\/1.527594"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ahmad, D., and Ziad, M. (1997). Homothetic motions of spherically symmetric spacetimes. J. Math. Phys., 38.","DOI":"10.1063\/1.531994"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hall, G.S., and Steele, J.D. (1990). Homothety groups in space-time. Gen. Relativ. Gravit., 22.","DOI":"10.1007\/BF00756152"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bokhari, A.H., Hussain, T., Khan, J., and Nasib, U. (2021). Proper homothetic vector fields of Bianchi type I spacetimes via Rif tree approach. Results Phys., 25.","DOI":"10.1016\/j.rinp.2021.104299"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Usmani, A.A., Rahaman, F., Ray, S., Nandi, K.K., Kuhfittig, P.K.F., Rakib, S.A., and Hasan, Z. (2011). Charged gravastars admitting conformal motion. Phys. Lett. B, 701.","DOI":"10.1016\/j.physletb.2011.06.001"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Moopanar, S., and Maharaj, S.D. (2010). Conformal symmetries of spherical spacetimes. Int. J. Theor. Phys., 49.","DOI":"10.1007\/s10773-010-0366-6"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Maartens, R., and Maharaj, S.D. (1986). Conformal killing vectors in Robertson-Walker spacetimes. Class. Quantum Gravit., 3.","DOI":"10.1088\/0264-9381\/3\/5\/027"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Saifullah, K., and Azdan, S.Y. (2009). Conformal motions in plane symmetric static space\u2013times. Int. J. Mod. Phys. D, 18.","DOI":"10.1142\/S0218271809014340"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Coley, A.A., and Tupper, B.O.J. (1990). Spherically symmetric spacetimes admitting inheriting conformal Killing vector fields. Class. Quantum Gravit., 7.","DOI":"10.1088\/0264-9381\/7\/12\/005"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Maartens, R., Mason, D.P., and Tsamparlis, M. (1986). Kinematic and dynamic properties of conformal Killing vectors in anisotropic fluids. J. Math. Phys., 27.","DOI":"10.1063\/1.527225"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Noether, E. (1971). Invariant variation problems. Transp. Theory Stat. Phys., 1.","DOI":"10.1080\/00411457108231446"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bluman, G., and Kumei, S. (1989). Symmetries and Differential Equations, Springer.","DOI":"10.1007\/978-1-4757-4307-4"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Wafo, S.C., and Mahomed, F.M. (2001). Linearization criteria for a system of second-order ordinary differential equations. Int. J. Non-Linear Mech., 36.","DOI":"10.1016\/S0020-7462(00)00032-9"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ibragimov, N.H., and Maleshko, S.V. (2005). Linearization of third-order ordinary differential equations by point and contact transformations. J. Math. Anal. Appl., 308.","DOI":"10.1016\/j.jmaa.2005.01.025"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Hickman, M., and Yazdan, S. (2017). Noether symmetries of Bianchi type II spacetimes. Gen. Relativ. Gravit., 49.","DOI":"10.1007\/s10714-017-2228-5"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bokhari, A.H., and Kara, A.H. (2007). Noether versus Killing symmetry of conformally flat Friedmann metric. Gen. Relativ. Gravit., 39.","DOI":"10.1007\/s10714-007-0501-8"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Bokhari, A.H., Kara, A.H., Kashif, A.R., and Zaman, F.D. (2006). Noether symmetries versus Killing vectors and isometries of spacetimes. Int. J. Theor. Phys., 45.","DOI":"10.1007\/s10773-006-9096-1"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Camci, U., Jamal, S., and Kara, A.H. (2014). Invariances and conservation laws based on some FRWuniverses. Int. J. Theor. Phys., 53.","DOI":"10.1007\/s10773-013-1948-x"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ali, F., Feroze, T., and Ali, S. (2015). Complete classification of spherically symmetric static space-times via Noether symmetries. Theor. Math. Phys., 184.","DOI":"10.1007\/s11232-015-0310-2"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ali, S., and Hussain, I. (2016). A study of positive energy condition in Bianchi V spacetimes via Noether symmetries. Eur. Phys. J. C, 76.","DOI":"10.1140\/epjc\/s10052-016-3903-5"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Capozziello, S., Marmo, G., Rubano, C., and Scudellaro, P. (1997). Noether symmetries in Bianchi universes. Int. J. Mod. Phys. D, 6.","DOI":"10.1142\/S0218271897000297"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Capozziello, S., Ritis, R.D., Rubano, C., and Scudellaro, P. (1996). Noether symmetries in cosmology. Riv. Nuovo Cim., 19.","DOI":"10.1007\/BF02742992"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Capozziello, S., Piedipalumbo, E., Rubano, C., and Scudellaro, P. (2009). Noether symmetry approach in phantom quintessence cosmology. Phys. Rev. D, 80.","DOI":"10.1103\/PhysRevD.80.104030"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Hansraj, S., Govender, M., and Mewalal, N. (2018). Expanding, shearing and accelerating isotropic plane symmetric universe with conformal Kasner geometry. Mod. Phys. Lett. A, 33.","DOI":"10.1142\/S0217732318501432"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bedran, M.L., Calvao, M.O., Paiva, F.M., and Soares, I.D. (1997). Taub\u2019s plane-symmetric vacuum spacetime reexamined. Phys. Rev. D, 55.","DOI":"10.1103\/PhysRevD.55.3431"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Coley, A.A., and Tupper, B.O.J. (1994). Spherically symmetric anisotropic fluid ICKV spacetimes. Class. Quantum Gravit., 11.","DOI":"10.1088\/0264-9381\/11\/10\/015"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2174\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:55:31Z","timestamp":1760144131000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2174"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,17]]},"references-count":32,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102174"],"URL":"https:\/\/doi.org\/10.3390\/sym14102174","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,10,17]]}}}