{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T15:00:56Z","timestamp":1768402856540,"version":"3.49.0"},"reference-count":77,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,11,28]],"date-time":"2019-11-28T00:00:00Z","timestamp":1574899200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Finding spherically symmetric exact solutions in modified gravity is usually a difficult task. In this paper, we use Noether symmetry approach for a modified teleparallel theory of gravity labeled as     f ( T , B )     gravity where T is the scalar torsion and B the boundary term. Using Noether theorem, we were able to find exact spherically symmetric solutions for different forms of the function     f ( T , B )     coming from Noether symmetries.<\/jats:p>","DOI":"10.3390\/sym11121462","type":"journal-article","created":{"date-parts":[[2019,11,28]],"date-time":"2019-11-28T10:54:10Z","timestamp":1574938450000},"page":"1462","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Exact Spherically Symmetric Solutions in Modified Teleparallel Gravity"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6235-120X","authenticated-orcid":false,"given":"Sebastian","family":"Bahamonde","sequence":"first","affiliation":[{"name":"Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia"},{"name":"Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3431-2574","authenticated-orcid":false,"given":"Ugur","family":"Camci","sequence":"additional","affiliation":[{"name":"Siteler Mahallesi, 1307 Sokak, Ahmet Kartal Konutlari, A-1 Blok, No:7\/2, Konyaalti 07070, Antalya, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1753","DOI":"10.1142\/S021827180600942X","article-title":"Dynamics of dark energy","volume":"D15","author":"Copeland","year":"2006","journal-title":"Int. J. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.physrep.2004.08.031","article-title":"Particle dark matter: Evidence, candidates and constraints","volume":"405","author":"Bertone","year":"2005","journal-title":"Phys. Rept."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1016\/j.crhy.2012.04.008","article-title":"Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask)","volume":"13","author":"Martin","year":"2012","journal-title":"C. R. Phys."},{"key":"ref_4","unstructured":"Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A.J., Barreiro, R.B., Bartolo, N., and Basak, S. (2018). Planck 2018 results. VI. Cosmological parameters. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"136","DOI":"10.3847\/1538-4357\/aaadb7","article-title":"New Parallaxes of Galactic Cepheids from Spatially Scanning the Hubble Space Telescope: Implications for the Hubble Constant","volume":"855","author":"Riess","year":"2018","journal-title":"Astrophys. J."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wong, K.C., Suyu, S.H., Chen, G.C.-F., Rusu, C.E., Millon, M., Sluse, D., Bonvin, V., Fassnacht, C.D., Taubenberger, S., and Auger, M.W. (2019). H0LiCOW XIII. A 2.4% measurement of H0 from lensed quasars: 5.3\u03c3 tension between early and late-Universe probes. arXiv.","DOI":"10.1093\/mnras\/stz3094"},{"key":"ref_7","unstructured":"The LIGO Scientific Collaboration, and The Virgo Collaboration (2016). Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett., 116, 061102."},{"key":"ref_8","unstructured":"The Event Horizon Telescope Collaboration (2019). First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophys. J., 875, L1."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2012.01.001","article-title":"Modified Gravity and Cosmology","volume":"513","author":"Clifton","year":"2012","journal-title":"Phys. Rept."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2018.11.006","article-title":"A systematic approach to generalisations of General Relativity and their cosmological implications","volume":"796","author":"Heisenberg","year":"2019","journal-title":"Phys. Rept."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s41114-018-0017-4","article-title":"Testing General Relativity in Cosmology","volume":"22","author":"Ishak","year":"2019","journal-title":"Living Rev. Rel."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2017.06.001","article-title":"Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution","volume":"692","author":"Nojiri","year":"2017","journal-title":"Phys. Rept."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Aldrovandi, R., and Pereira, J.G. (2013). Teleparallel Gravity, Springer.","DOI":"10.1007\/978-94-007-5143-9"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"106901","DOI":"10.1088\/0034-4885\/79\/10\/106901","article-title":"f(T) teleparallel gravity and cosmology","volume":"79","author":"Cai","year":"2016","journal-title":"Rept. Prog. Phys."},{"key":"ref_15","first-page":"084031","article-title":"Modified teleparallel gravity: Inflation without inflaton","volume":"D75","author":"Ferraro","year":"2007","journal-title":"Phys. Rev."},{"key":"ref_16","first-page":"124019","article-title":"Dark torsion as the cosmic speed-up","volume":"D79","author":"Bengochea","year":"2009","journal-title":"Phys. Rev."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"215011","DOI":"10.1088\/0264-9381\/28\/21\/215011","article-title":"Matter Bounce Cosmology with the f(T) Gravity","volume":"28","author":"Cai","year":"2011","journal-title":"Class. Quant. Grav."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"021","DOI":"10.1088\/1475-7516\/2011\/01\/021","article-title":"Equation of state for dark energy in f(T) gravity","volume":"1101","author":"Bamba","year":"2011","journal-title":"JCAP"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"009","DOI":"10.1088\/1475-7516\/2011\/01\/009","article-title":"f(T) gravity mimicking dynamical dark energy. Background and perturbation analysis","volume":"1101","author":"Dent","year":"2011","journal-title":"JCAP"},{"key":"ref_20","first-page":"083518","article-title":"Spherically symmetric static spacetimes in vacuum f(T) gravity","volume":"D84","author":"Ferraro","year":"2011","journal-title":"Phys. Rev."},{"key":"ref_21","first-page":"064012","article-title":"Forecast constraints on f(T) gravity with gravitational waves from compact binary coalescences","volume":"D100","author":"Nunes","year":"2019","journal-title":"Phys. Rev."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.physletb.2013.10.022","article-title":"No further gravitational wave modes in F(T) gravity","volume":"B727","author":"Bamba","year":"2013","journal-title":"Phys. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Cai, Y.-F., Li, C., Saridakis, E.N., and Xue, L. (2018). f(T) gravity after GW170817 and GRB170817A. arXiv.","DOI":"10.1103\/PhysRevD.97.103513"},{"key":"ref_24","first-page":"104055","article-title":"New observational constraints on f(T) gravity through gravitational-wave astronomy","volume":"D98","author":"Nunes","year":"2018","journal-title":"Phys. Rev."},{"key":"ref_25","first-page":"124064","article-title":"Gravitational Waves in Modified Teleparallel Theories","volume":"D97","author":"Farrugia","year":"2018","journal-title":"Phys. Rev."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1140\/epjc\/s10052-016-4118-5","article-title":"Astrophysical flows near f(T) gravity black holes","volume":"C76","author":"Ahmed","year":"2016","journal-title":"Eur. Phys. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"245020","DOI":"10.1088\/0264-9381\/28\/24\/245020","article-title":"Existence of relativistic stars in f(T) gravity","volume":"28","author":"Boehmer","year":"2011","journal-title":"Class. Quant. Grav."},{"key":"ref_28","first-page":"104014","article-title":"Weak-Field Spherically Symmetric Solutions in f(T) gravity","volume":"D91","author":"Ruggiero","year":"2015","journal-title":"Phys. Rev."},{"key":"ref_29","first-page":"124025","article-title":"Spherically symmetric vacuum in covariant F(T) = T + \u03b12T2 + O(T\u03b3) gravity theory","volume":"D94","author":"DeBenedictis","year":"2016","journal-title":"Phys. Rev."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Bahamonde, S., Flathmann, K., and Pfeifer, C. (2019). Photon sphere and perihelion shift in weak f(T) gravity. arXiv.","DOI":"10.1103\/PhysRevD.100.084064"},{"key":"ref_31","first-page":"044033","article-title":"Wormhole geometries in modified teleparralel gravity and the energy conditions","volume":"D85","author":"Boehmer","year":"2012","journal-title":"Phys. Rev."},{"key":"ref_32","first-page":"084042","article-title":"Scalar-Tensor Teleparallel Wormholes by Noether Symmetries","volume":"D94","author":"Bahamonde","year":"2016","journal-title":"Phys. Rev."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"035023","DOI":"10.1088\/0264-9381\/32\/3\/035023","article-title":"Nonminimally coupled scalar field in teleparallel gravity: Boson stars","volume":"32","author":"Horvat","year":"2015","journal-title":"Class. Quant. Grav."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1140\/epjc\/s10052-017-4637-8","article-title":"Quark stars in f(T, T) gravity","volume":"C77","author":"Pace","year":"2017","journal-title":"Eur. Phys. J."},{"key":"ref_35","first-page":"104042","article-title":"New Schwarzschild-like solutions in f(T) gravity through Noether symmetries","volume":"D89","author":"Paliathanasis","year":"2014","journal-title":"Phys. Rev."},{"key":"ref_36","first-page":"104042","article-title":"Modified teleparallel theories of gravity","volume":"D92","author":"Bahamonde","year":"2015","journal-title":"Phys. Rev."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Abedi, H., and Capozziello, S. (2018). Gravitational waves in modified teleparallel theories of gravity. arXiv.","DOI":"10.1140\/epjc\/s10052-018-5967-x"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.dark.2017.12.005","article-title":"Thermodynamics and cosmological reconstruction in f(T, B) gravity","volume":"19","author":"Bahamonde","year":"2018","journal-title":"Phys. Dark Univ."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s10714-018-2400-6","article-title":"Cartan symmetries and global dynamical systems analysis in a higher-order modified teleparallel theory","volume":"50","author":"Karpathopoulos","year":"2018","journal-title":"Gen. Rel. Grav."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"027","DOI":"10.1088\/1475-7516\/2017\/08\/027","article-title":"de Sitter and Scaling solutions in a higher-order modified teleparallel theory","volume":"1708","author":"Paliathanasis","year":"2017","journal-title":"JCAP"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02742992","article-title":"Noether symmetries in cosmology","volume":"19N4","author":"Capozziello","year":"1996","journal-title":"Riv. Nuovo Cim."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1840007","DOI":"10.1142\/S0219887818400078","article-title":"Noether Symmetries as a geometric criterion to select theories of gravity","volume":"15","author":"Dialektopoulos","year":"2018","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"016","DOI":"10.1088\/1475-7516\/2008\/08\/016","article-title":"f(R) cosmology by Noether\u2019s symmetry","volume":"0808","author":"Capozziello","year":"2008","journal-title":"JCAP"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2016","DOI":"10.1140\/epjc\/s10052-012-2016-z","article-title":"f(T) cosmology via Noether symmetry","volume":"C72","author":"Atazadeh","year":"2012","journal-title":"Eur. Phys. J."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1881","DOI":"10.1007\/s10714-012-1367-y","article-title":"New Spherically Symmetric Solutions in f(R)-gravity by Noether Symmetries","volume":"44","author":"Capozziello","year":"2012","journal-title":"Gen. Rel. Grav."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1140\/epjc\/s10052-017-5283-x","article-title":"Constraining Generalized Non-local Cosmology from Noether Symmetries","volume":"C77","author":"Bahamonde","year":"2017","journal-title":"Eur. Phys. J."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"065013","DOI":"10.1088\/1361-6382\/ab0510","article-title":"Noether symmetries and boundary terms in extended Teleparallel gravity cosmology","volume":"36","author":"Bahamonde","year":"2019","journal-title":"Class. Quant. Grav."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1140\/epjc\/s10052-017-4677-0","article-title":"Noether Symmetry Approach in f(T, B) teleparallel cosmology","volume":"C77","author":"Bahamonde","year":"2017","journal-title":"Eur. Phys. J."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"016","DOI":"10.1088\/1475-7516\/2019\/02\/016","article-title":"New Exact Spherically Symmetric Solutions in f(R, \u03d5, X) gravity by Noether\u2019s symmetry approach","volume":"1902","author":"Bahamonde","year":"2019","journal-title":"JCAP"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1140\/epjc\/s10052-019-7106-8","article-title":"Noether symmetries in Symmetric Teleparallel Cosmology","volume":"C79","author":"Dialektopoulos","year":"2019","journal-title":"Eur. Phys. J."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1140\/epjc\/s10052-018-5939-1","article-title":"Classification of the Horndeski cosmologies via Noether Symmetries","volume":"C78","author":"Capozziello","year":"2018","journal-title":"Eur. Phys. J."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1140\/epjc\/s10052-016-4491-0","article-title":"Noether symmetries in Gauss\u2013Bonnet-teleparallel cosmology","volume":"C76","author":"Capozziello","year":"2016","journal-title":"Eur. Phys. J."},{"key":"ref_53","first-page":"123514","article-title":"Constraints and analytical solutions of f(R) theories of gravity using Noether symmetries","volume":"D84","author":"Paliathanasis","year":"2011","journal-title":"Phys. Rev."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Aldrovandi, R., and Pereira, J. (1995). An Introduction to Geometrical Physics, World Scientific. Available online: https:\/\/books.google.com.mt\/books?id=w8hBT4DV1vkC.","DOI":"10.1142\/9789812831026"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1023\/B:GERG.0000006696.98824.4d","article-title":"Gravitation without the equivalence principle","volume":"36","author":"Aldrovandi","year":"2004","journal-title":"Gen. Rel. Grav."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.physletb.2017.10.026","article-title":"New classes of modified teleparallel gravity models","volume":"B775","author":"Bahamonde","year":"2017","journal-title":"Phys. Lett."},{"key":"ref_57","first-page":"044009","article-title":"Good and bad tetrads in f(T) gravity","volume":"D86","author":"Tamanini","year":"2012","journal-title":"Phys. Rev."},{"key":"ref_58","first-page":"084002","article-title":"Modified teleparallel theories of gravity in symmetric spacetimes","volume":"D100","author":"Hohmann","year":"2019","journal-title":"Phys. Rev."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"115009","DOI":"10.1088\/0264-9381\/33\/11\/115009","article-title":"The covariant formulation of f(T) gravity","volume":"33","author":"Saridakis","year":"2016","journal-title":"Class. Quant. Grav."},{"key":"ref_60","unstructured":"Weinberg, S. (2019, November 27). Gravitation and Cosmology, Available online: http:\/\/www-spires.fnal.gov\/spires\/find\/books\/www?cl=QC6.W431."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Nunes, R.C. (2018). Structure formation in f(T) gravity and a solution for H0 tension. arXiv.","DOI":"10.1088\/1475-7516\/2018\/05\/052"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1140\/epjc\/s10052-018-6028-1","article-title":"Galactic Rotation Dynamics in f(T) gravity","volume":"C78","author":"Finch","year":"2018","journal-title":"Eur. Phys. J."},{"key":"ref_63","unstructured":"Escamilla-Rivera, C., and Levi Said, J. (2019). Cosmological viable models in f(T,B) gravity as solutions to the H0 tension. arXiv."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1140\/epjp\/i2018-12252-2","article-title":"Energy constraints and the phenomenon of cosmic evolution in the f(T, B) framework","volume":"133","author":"Zubair","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1140\/epjc\/s10052-015-3734-9","article-title":"Accretion Processes for General Spherically Symmetric Compact Objects","volume":"C75","author":"Bahamonde","year":"2015","journal-title":"Eur. Phys. J."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"056","DOI":"10.1088\/1475-7516\/2018\/04\/056","article-title":"Generalised teleparallel quintom dark energy non-minimally coupled with the scalar torsion and a boundary term","volume":"1804","author":"Bahamonde","year":"2018","journal-title":"JCAP"},{"key":"ref_67","first-page":"064018","article-title":"Can Horndeski Theory be recast using Teleparallel Gravity?","volume":"D100","author":"Bahamonde","year":"2019","journal-title":"Phys. Rev."},{"key":"ref_68","first-page":"084034","article-title":"Teleparallel quintessence with a nonminimal coupling to a boundary term","volume":"D92","author":"Bahamonde","year":"2015","journal-title":"Phys. Rev."},{"key":"ref_69","first-page":"064004","article-title":"Scalar-torsion theories of gravity III: Analogue of scalar-tensor gravity and conformal invariants","volume":"D98","author":"Hohmann","year":"2018","journal-title":"Phys. Rev."},{"key":"ref_70","first-page":"064003","article-title":"Scalar-torsion theories of gravity II: L(T, X, Y, \u03d5) theory","volume":"D98","author":"Hohmann","year":"2018","journal-title":"Phys. Rev."},{"key":"ref_71","first-page":"064002","article-title":"Scalar-torsion theories of gravity I: General formalism and conformal transformations","volume":"D98","author":"Hohmann","year":"2018","journal-title":"Phys. Rev."},{"key":"ref_72","first-page":"104011","article-title":"Covariant formulation of scalar-torsion gravity","volume":"D97","author":"Hohmann","year":"2018","journal-title":"Phys. Rev."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1140\/epjc\/s10052-017-5043-y","article-title":"Generalized Second Law of Thermodynamic in Modified Teleparallel Theory","volume":"C77","author":"Zubair","year":"2017","journal-title":"Eur. Phys. J."},{"key":"ref_74","first-page":"124013","article-title":"Growth of matter perturbations in nonminimal teleparallel dark energy","volume":"D98","author":"Luongo","year":"2018","journal-title":"Phys. Rev."},{"key":"ref_75","first-page":"084008","article-title":"Effective gravitational coupling in modified teleparallel theories","volume":"D97","author":"Abedi","year":"2018","journal-title":"Phys. Rev."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1140\/epjc\/s10052-017-5210-1","article-title":"Nonlocal Teleparallel Cosmology","volume":"C77","author":"Bahamonde","year":"2017","journal-title":"Eur. Phys. J."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1140\/epjc\/s10052-018-5793-1","article-title":"Generalised nonminimally gravity-matter coupled theory","volume":"C78","author":"Bahamonde","year":"2018","journal-title":"Eur. Phys. J."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/12\/1462\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:38:27Z","timestamp":1760189907000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/12\/1462"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,28]]},"references-count":77,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["sym11121462"],"URL":"https:\/\/doi.org\/10.3390\/sym11121462","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,11,28]]}}}