{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T15:38:56Z","timestamp":1768405136386,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T00:00:00Z","timestamp":1698624000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)","award":["UIDB\/04434\/2020"],"award-info":[{"award-number":["UIDB\/04434\/2020"]}]},{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)","award":["UIDP\/04434\/2020"],"award-info":[{"award-number":["UIDP\/04434\/2020"]}]},{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)","award":["CERN\/FIS-PAR\/0037\/2019"],"award-info":[{"award-number":["CERN\/FIS-PAR\/0037\/2019"]}]},{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)","award":["PTDC\/FIS-AST\/0054\/2021"],"award-info":[{"award-number":["PTDC\/FIS-AST\/0054\/2021"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Universe"],"abstract":"<jats:p>The recently obtained special Buchdahl-inspired metric Phys. Rev. D 107, 104008 (2023) describes asymptotically flat spacetimes in pure Ricci-squared gravity. The metric depends on a new (Buchdahl) parameter k\u02dc of higher-derivative characteristic, and reduces to the Schwarzschild metric, for k\u02dc=0. For the case k\u02dc\u2208(\u22121,0), it was shown that it describes a traversable Morris\u2013Thorne\u2013Buchdahl (MTB) wormhole Eur. Phys. J. C 83, 626 (2023), where the weak energy condition is formally violated. In this paper, we briefly review the special Buchdahl-inspired metric, with focuses on the construction of the Kruskal\u2013Szekeres (KS) diagram and the situation for a wormhole to emerge. Interestingly, the MTB wormhole structure appears to permit the formation of closed timelike curves (CTCs). More specifically, a CTC straddles the throat, comprising of two segments positioned in opposite quadrants of the KS diagram. The closed timelike loop thus passes through the wormhole throat twice, causing two reversals in the time direction experienced by the (timelike) traveller on the CTC. The key to constructing a CTC lies in identifying any given pair of antipodal points (T,X) and (\u2212T,\u2212X)\u00a0on the wormhole throat in the KS diagram as corresponding to the same spacetime event. It is interesting to note that the Campanelli\u2013Lousto metric in Brans\u2013Dicke gravity is known to support two-way traversable wormholes, and the formation of the CTCs presented herein is equally applicable to the Campanelli\u2013Lousto solution.<\/jats:p>","DOI":"10.3390\/universe9110467","type":"journal-article","created":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T13:20:07Z","timestamp":1698672007000},"page":"467","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Closed Timelike Curves Induced by a Buchdahl-Inspired Vacuum Spacetime in R2 Gravity"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2343-0508","authenticated-orcid":false,"given":"Hoang Ky","family":"Nguyen","sequence":"first","affiliation":[{"name":"Department of Physics, Babe\u015f\u2013Bolyai University, 400084 Cluj-Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9388-8373","authenticated-orcid":false,"given":"Francisco S. N.","family":"Lobo","sequence":"additional","affiliation":[{"name":"Instituto de Astrof\u00edsica e Ci\u00eancias do Espa\u00e7o, Faculdade de Ci\u00eancias da Universidade de Lisboa, Campo Grande, Edif\u00edcio C8, 1749-016 Lisbon, Portugal"},{"name":"Departamento de F\u00edsica, Faculdade de Ci\u00eancias da Universidade de Lisboa, Campo Grande, Edif\u00edcio C8, 1749-016 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1119\/1.15620","article-title":"Wormholes in spacetime and their for interstellar travel: A tool for teaching general relativity","volume":"56","author":"Morris","year":"1988","journal-title":"Am. J. Phys."},{"key":"ref_2","unstructured":"Visser, M. (1996). Lorentzian Wormholes: From Einstein to Hawking, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/978-3-319-55182-1_11","article-title":"Wormholes, Warp Drives and Energy Conditions","volume":"189","author":"Alcubierre","year":"2017","journal-title":"Fundam. Theor. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"104012","DOI":"10.1103\/PhysRevD.80.104012","article-title":"Wormhole geometries in f(R) modified theories of gravity","volume":"80","author":"Lobo","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"067504","DOI":"10.1103\/PhysRevD.87.067504","article-title":"Modified-gravity wormholes without exotic matter","volume":"87","author":"Harko","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1446","DOI":"10.1103\/PhysRevLett.61.1446","article-title":"Wormholes, Time Machines, and the Weak Energy Condition","volume":"61","author":"Morris","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","first-page":"3","article-title":"An analysis of the operation of a time machine","volume":"68","author":"Novikov","year":"1989","journal-title":"Sov. Phys. JETP"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1103\/PhysRevD.42.1057","article-title":"Physical effects in wormholes and time machines","volume":"42","author":"Frolov","year":"1990","journal-title":"Phys. Rev. D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1915","DOI":"10.1103\/PhysRevD.42.1915","article-title":"Cauchy problem in spacetimes with closed timelike curves","volume":"42","author":"Friedman","year":"1990","journal-title":"Phys. Rev. D"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1103\/RevModPhys.21.447","article-title":"An Example of a New Type of Cosmological Solution of Einstein\u2019s Field Equations of Gravitation","volume":"21","year":"1949","journal-title":"Rev. Mod. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1007\/BF00756366","article-title":"Time travel in G\u00f6del\u2019s space","volume":"13","author":"Pfarr","year":"1981","journal-title":"Gen. Rel. Grav."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1103\/PhysRevLett.37.879","article-title":"Causality violation in asymptotically flat space-times","volume":"37","author":"Tipler","year":"1976","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/BF00760150","article-title":"Time machine and geodesic motion in Kerr metric","volume":"9","author":"Calvani","year":"1978","journal-title":"Gen. Rel. Grav."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1126","DOI":"10.1103\/PhysRevLett.66.1126","article-title":"Closed Timelike Curves Produced by Pairs of Moving Cosmic Strings: Exact Solutions","volume":"66","author":"Gott","year":"1991","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2517","DOI":"10.1103\/PhysRevLett.71.2517","article-title":"Must time machine construction violate the weak energy condition?","volume":"71","author":"Ori","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"021101","DOI":"10.1103\/PhysRevLett.95.021101","article-title":"A class of time-machine solutions with a compact vacuum core","volume":"95","author":"Ori","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"L73","DOI":"10.1088\/0264-9381\/11\/5\/001","article-title":"The warp drive: Hyperfast travel within general relativity","volume":"11","author":"Alcubierre","year":"1994","journal-title":"Class. Quant. Grav."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"7365","DOI":"10.1103\/PhysRevD.53.7365","article-title":"Warp drive and causality","volume":"53","author":"Everett","year":"1996","journal-title":"Phys. Rev. D"},{"key":"ref_19","unstructured":"Lobo, F.S.N. (2010). Closed timelike curves and causality violation. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"104008","DOI":"10.1103\/PhysRevD.107.104008","article-title":"Beyond Schwarzschild-de Sitter spacetimes: II. An exact non-Schwarzschild metric in pure R2 gravity and new anomalous properties of R2 spacetimes","volume":"107","author":"Nguyen","year":"2023","journal-title":"Phys. Rev. D"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/BF02733549","article-title":"On the Gravitational Field Equations Arising from the Square of the Gaussian Curvature","volume":"23","author":"Buchdahl","year":"1962","journal-title":"Nuovo Cimento"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1140\/epjc\/s10052-023-11805-3","article-title":"Traversable Morris-Thorne-Buchdahl wormholes in quadratic gravity","volume":"83","author":"Nguyen","year":"2023","journal-title":"Eur. Phys. J. C"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.physletb.2010.03.029","article-title":"Radial motion motion into an Einstein\u2013Rosen bridge","volume":"687","year":"2010","journal-title":"Phys. Lett. B"},{"key":"ref_24","unstructured":"Nguyen, H.K., and Azreg-A\u00efnou, M. (2023). Revisiting Weak Energy Condition and wormholes in Brans-Dicke gravity. arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"104004","DOI":"10.1103\/PhysRevD.106.104004","article-title":"Beyond Schwarzschild-de Sitter spacetimes: A new exhaustive class of metrics inspired by Buchdahl for pure R2 gravity in a compact form","volume":"106","author":"Nguyen","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Nguyen, H.K. (2023). Buchdahl-inspired spacetimes and wormholes: Unearthing Hans Buchdahl\u2019s other \u2018hidden\u2019 treasure trove. Int. J. Mod. Phys. D, 2342007.","DOI":"10.1142\/S0218271823420075"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3197","DOI":"10.1103\/PhysRevD.44.3197","article-title":"Quantum mechanics near closed timelike lines","volume":"44","author":"Deutsch","year":"1991","journal-title":"Phys. Rev. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4470","DOI":"10.1103\/PhysRevD.46.4470","article-title":"Simple quantum systems in spacetimes with closed timelike curves","volume":"46","author":"Politzer","year":"1992","journal-title":"Phys. Rev. D"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1088\/0264-9381\/15\/4\/020","article-title":"A singularity-free WEC-respecting time machine","volume":"15","author":"Krasnikov","year":"1998","journal-title":"Class. Quant. Grav."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1007\/BF02105077","article-title":"Topology Change without any Pathology","volume":"27","author":"Krasnikov","year":"1995","journal-title":"Gen. Rel. Gravit."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2011","DOI":"10.1103\/PhysRevD.51.2011","article-title":"Wormholes in the Brans-Dicke theory of gravitation","volume":"51","author":"Agnese","year":"1995","journal-title":"Phys. Rev. D"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Sidharth, B.G., and Altaisky, M.V. (2001). Frontiers of Fundamental Physics 4, Kluwer Academic\/Plenum Publishers.","DOI":"10.1007\/978-1-4615-1339-1"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1134\/S0202289310030047","article-title":"Notes on wormhole existence in scalar-tensor and F(R) gravity","volume":"16","author":"Bronnikov","year":"2010","journal-title":"Grav. Cosmol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2497","DOI":"10.1103\/PhysRevD.55.2497","article-title":"Brans wormhole","volume":"55","author":"Nandi","year":"1997","journal-title":"Phys. Rev. D"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"084031","DOI":"10.1103\/PhysRevD.86.084031","article-title":"Campanelli-Lousto and veiled spacetimes","volume":"86","author":"Vanzo","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Earman, J. (1995). Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes, Oxford University Press.","DOI":"10.1093\/oso\/9780195095913.001.0001"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1103\/PhysRevD.44.1077","article-title":"Billiard Balls in Wormhole Spacetimes with Closed Timelike Curves: Classical Theory","volume":"44","author":"Echeverria","year":"1991","journal-title":"Phys. Rev. D"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1989","DOI":"10.1103\/PhysRevD.45.1989","article-title":"Time machine and self-consistent evolution in problems with self-interaction","volume":"45","author":"Novikov","year":"1992","journal-title":"Phys. Rev. D"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1103\/PhysRevD.46.603","article-title":"Chronology protection conjecture","volume":"46","author":"Hawking","year":"1992","journal-title":"Phys. Rev. D"},{"key":"ref_40","unstructured":"Bronnikov, K.A., Constantinidis, C.P., Evangelista, R.L., and Fabris, J.C. (1997). Cold black holes in scalar-tensor theories. arXiv."},{"key":"ref_41","first-page":"249","article-title":"Generic wormhole throats","volume":"13","author":"Visser","year":"1997","journal-title":"Ann. Isr. Phys."}],"container-title":["Universe"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-1997\/9\/11\/467\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:14:11Z","timestamp":1760130851000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-1997\/9\/11\/467"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,30]]},"references-count":41,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["universe9110467"],"URL":"https:\/\/doi.org\/10.3390\/universe9110467","relation":{},"ISSN":["2218-1997"],"issn-type":[{"value":"2218-1997","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,30]]}}}