{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T14:25:40Z","timestamp":1761402340547,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,6]],"date-time":"2024-08-06T00:00:00Z","timestamp":1722902400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11973025","NCET-10-0702"],"award-info":[{"award-number":["11973025","NCET-10-0702"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004602","name":"Program for New Century Excellent Talents in University","doi-asserted-by":"publisher","award":["11973025","NCET-10-0702"],"award-info":[{"award-number":["11973025","NCET-10-0702"]}],"id":[{"id":"10.13039\/501100004602","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We calculate the metric of a self-gravitating and collapsing infinitely thin spherical shell in the weak-field and slow-motion limits, and we demonstrate that Birkhoff\u2019s theorem is not consistent with the theory of the post-Newtonian approximation. More importantly, it is illustrated that performing a coordinate transformation in solving Einstein field equations may change the matter energy-momentum tensor, making the resultant solution not correspond to the original problem.<\/jats:p>","DOI":"10.3390\/sym16080998","type":"journal-article","created":{"date-parts":[[2024,8,6]],"date-time":"2024-08-06T11:54:19Z","timestamp":1722945259000},"page":"998","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Revisit Birkhoff\u2019s Theorem: The Post-Newtonian Metric of a Self-Gravitating and Collapsing Thin Spherical Shell"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4282-066X","authenticated-orcid":false,"given":"Wenbin","family":"Lin","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, University of South China, Hengyang 421001, China"},{"name":"School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,6]]},"reference":[{"key":"ref_1","unstructured":"Birkhoff, G. (1923). Relativity and Modern Physics, Harvard University Press."},{"key":"ref_2","unstructured":"Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1357325","DOI":"10.1080\/23311940.2017.1357325","article-title":"Does Birkhoff\u2019s theorem really hold?","volume":"4","author":"Lin","year":"2017","journal-title":"Cogent. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02710419","article-title":"Singular hypersurfaces and thin shells in general relativity","volume":"44","author":"Israel","year":"1966","journal-title":"Nuovo C. B"},{"key":"ref_5","unstructured":"Lightman, A., Press, W., Price, R., and Teukolsky, S. (1975). Problem Book in Relativity and Gravitation, Princeton Univiversity Press."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hawking, S., and Ellis, G. (1973). The Large Scale Structure of Space-Time, Cambridge University Press.","DOI":"10.1017\/CBO9780511524646"},{"key":"ref_7","unstructured":"Poisson, E. (2004). A Relativist\u2019s Toolkit: The Mathematics of Black-Hole Mechanics, Cambridge University Press."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5938","DOI":"10.1073\/pnas.1103127108","article-title":"On the unreasonable effectiveness of the post-Newtonian approximation in gravitational physics","volume":"108","author":"Will","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Poisson, E., and Will, C. (2014). Gravity\u2013Newtonian, Post-Newtonian, Relativistic, Cambridge University Press.","DOI":"10.1017\/CBO9781139507486"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"064025","DOI":"10.1103\/PhysRevD.65.064025","article-title":"Gravitomagnetic effects in the propagation of electromagnetic waves in variable gravitational fields of arbitrary-moving and spinning bodies","volume":"65","author":"Kopeikin","year":"2002","journal-title":"Phys. Rev. D"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/S0375-9601(02)01361-0","article-title":"Gravitational lensing by spinning and escaping lenses","volume":"305","author":"Sereno","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"063001","DOI":"10.1103\/PhysRevD.69.063001","article-title":"Deflection of light and particles by moving gravitational lenses","volume":"69","author":"Wucknitz","year":"2004","journal-title":"Phys. Rev. D"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"L19","DOI":"10.1111\/j.1745-3933.2005.00026.x","article-title":"On gravitational lensing by deflectors in motion","volume":"359","author":"Sereno","year":"2005","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1007\/s10714-007-0483-6","article-title":"Gravimagnetism, causality, and aberration of gravity in the gravitational light-ray deflection experiments","volume":"39","author":"Kopeikin","year":"2007","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1088\/0253-6102\/61\/2\/21","article-title":"Exact Harmonic Metric for a Uniformly Moving Schwarzschild Black Hole","volume":"61","author":"He","year":"2014","journal-title":"Commun. Theor. Phys."},{"key":"ref_16","first-page":"1015","article-title":"Violation of Birkhoff\u2019s theorem for pure quadratic gravity action","volume":"338","author":"Kehm","year":"2017","journal-title":"Astron. Notes"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"138544","DOI":"10.1016\/j.physletb.2024.138544","article-title":"Is Birkhoff\u2019s theorem valid in Einstein-Aether theory?","volume":"850","author":"Chan","year":"2024","journal-title":"Phys. Lett. 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