Time dilation (English Wikipedia)

See equations 3, 4, 6 and 9 of Iorio, Lorenzo (2005). "An analytical treatment of the Clock Paradox in the framework of the Special and General Theories of Relativity". Foundations of Physics Letters. 18 (1): 1–19. arXiv:physics/0405038. Bibcode:2005FoPhL..18....1I. doi:10.1007/s10702-005-2466-8. S2CID 15081211.
Burns, M. Shane; Leveille, Michael D.; Dominguez, Armand R.; Gebhard, Brian B.; Huestis, Samuel E.; Steele, Jeffrey; Patterson, Brian; Sell, Jerry F.; Serna, Mario; Gearba, M. Alina; Olesen, Robert; O'Shea, Patrick; Schiller, Jonathan (18 September 2017). "Measurement of gravitational time dilation: An undergraduate research project". American Journal of Physics. 85 (10): 757–762. arXiv:1707.00171. Bibcode:2017AmJPh..85..757B. doi:10.1119/1.5000802. S2CID 119503665.

books.google.com (Global: 3^rd place; English: 3^rd place)

Hraskó, Péter (2011). Basic Relativity: An Introductory Essay (illustrated ed.). Springer Science & Business Media. p. 60. ISBN 978-3-642-17810-8. Extract of page 60
Cutner, Mark Leslie (2003). Astronomy, A Physical Perspective. Cambridge University Press. p. 128. ISBN 978-0-521-82196-4.
Lerner, Lawrence S. (1996). Physics for Scientists and Engineers, Volume 2. Jones and Bartlett. pp. 1051–1052. ISBN 978-0-7637-0460-5.
Ellis, George F. R.; Williams, Ruth M. (2000). Flat and Curved Space-times (2nd ed.). Oxford University Press. pp. 28–29. ISBN 978-0-19-850657-7.
Cassidy, David C.; Holton, Gerald James; Rutherford, Floyd James (2002). Understanding Physics. Springer-Verlag. p. 422. ISBN 978-0-387-98756-9.
Forshaw, Jeffrey; Smith, Gavin (2014). Dynamics and Relativity. John Wiley & Sons. ISBN 978-1-118-93329-9.
Adams, Steve (1997). Relativity: An introduction to space-time physics. CRC Press. p. 54. ISBN 978-0-7484-0621-0.
Stewart, J. V. (2001). Intermediate electromagnetic theory. World Scientific. p. 705. ISBN 978-981-02-4470-5.
Blaszczak, Z. (2007). Laser 2006. Springer. p. 59. ISBN 978-3540711131.
Petkov, Vesselin (2009). Relativity and the Nature of Spacetime (2nd, illustrated ed.). Springer Science & Business Media. p. 87. ISBN 978-3-642-01962-3. Extract of page 87
Kaplan, Elliott; Hegarty, Christopher (2005). Understanding GPS: Principles and Applications. Artech House. p. 306. ISBN 978-1-58053-895-4. Extract of page 306

bourbaphy.fr (Global: low place; English: low place)

Darrigol, Olivier (2005). "The Genesis of the Theory of Relativity". Einstein, 1905–2005 (PDF). Vol. 1. pp. 1–22. doi:10.1007/3-7643-7436-5_1. ISBN 978-3-7643-7435-8. {{cite book}}: |work= ignored (help)

doi.org (Global: 2^nd place; English: 2^nd place)

Ashby, Neil (2003). "Relativity in the Global Positioning System". Living Reviews in Relativity. 6 (1): 16. Bibcode:2003LRR.....6....1A. doi:10.12942/lrr-2003-1. PMC 5253894. PMID 28163638.
Hughes, Theo; Kersting, Magdalena (5 January 2021). "The invisibility of time dilation". Physics Education. 56 (2): 025011. Bibcode:2021PhyEd..56b5011H. doi:10.1088/1361-6552/abce02.
Darrigol, Olivier (2005). "The Genesis of the Theory of Relativity". Einstein, 1905–2005 (PDF). Vol. 1. pp. 1–22. doi:10.1007/3-7643-7436-5_1. ISBN 978-3-7643-7435-8. {{cite book}}: |work= ignored (help)
Larmor, Joseph (1897). "On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with Material Media" . Philosophical Transactions of the Royal Society. 190: 205–300. Bibcode:1897RSPTA.190..205L. doi:10.1098/rsta.1897.0020.
Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik (in German). 322 (10): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004.. See also: English translation.
Galli, J. Ronald; Amiri, Farhang (Apr 2012). "The Square Light Clock and Special Relativity". The Physics Teacher. 50 (4). American Association of Physics Teachers: 212. Bibcode:2012PhTea..50..212G. doi:10.1119/1.3694069. S2CID 120089462.
Bailey, J.; et al. (1977). "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit". Nature. 268 (5618): 301. Bibcode:1977Natur.268..301B. doi:10.1038/268301a0. S2CID 4173884.
Hasselkamp, D.; Mondry, E.; Scharmann, A. (1979). "Direct observation of the transversal Doppler-shift". Zeitschrift für Physik A. 289 (2): 151–155. Bibcode:1979ZPhyA.289..151H. doi:10.1007/BF01435932. S2CID 120963034.
Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. (2010). "Optical Clocks and Relativity". Science. 329 (5999): 1630–1633. Bibcode:2010Sci...329.1630C. doi:10.1126/science.1192720. PMID 20929843. S2CID 206527813.
See equations 3, 4, 6 and 9 of Iorio, Lorenzo (2005). "An analytical treatment of the Clock Paradox in the framework of the Special and General Theories of Relativity". Foundations of Physics Letters. 18 (1): 1–19. arXiv:physics/0405038. Bibcode:2005FoPhL..18....1I. doi:10.1007/s10702-005-2466-8. S2CID 15081211.
Bailey, H.; Borer, K.; Combley, F.; Drumm, H.; Krienen, F.; Lange, F.; Picasso, E.; von Ruden, W.; Farley F. J. M.; Field J. H.; Flegel W. & Hattersley P. M. (1977). "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit". Nature. 268 (5618): 301–305. Bibcode:1977Natur.268..301B. doi:10.1038/268301a0. S2CID 4173884.
Roos, C. E.; Marraffino, J.; Reucroft, S.; Waters, J.; Webster, M. S.; Williams, E. G. H. (1980). "σ+/- lifetimes and longitudinal acceleration". Nature. 286 (5770): 244–245. Bibcode:1980Natur.286..244R. doi:10.1038/286244a0. S2CID 4280317.
Burns, M. Shane; Leveille, Michael D.; Dominguez, Armand R.; Gebhard, Brian B.; Huestis, Samuel E.; Steele, Jeffrey; Patterson, Brian; Sell, Jerry F.; Serna, Mario; Gearba, M. Alina; Olesen, Robert; O'Shea, Patrick; Schiller, Jonathan (18 September 2017). "Measurement of gravitational time dilation: An undergraduate research project". American Journal of Physics. 85 (10): 757–762. arXiv:1707.00171. Bibcode:2017AmJPh..85..757B. doi:10.1119/1.5000802. S2CID 119503665.
Pound, R. V.; Snider J. L. (November 2, 1964). "Effect of Gravity on Nuclear Resonance". Physical Review Letters. 13 (18): 539–540. Bibcode:1964PhRvL..13..539P. doi:10.1103/PhysRevLett.13.539.
Ashby, Neil (2002). "Relativity in the Global Positioning System". Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583. PMC 5253894. PMID 28163638.
See equations 2 & 3 (combined here and divided throughout by c²) at pp. 35–36 in Moyer, T. D. (1981). "Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference". Celestial Mechanics. 23 (1): 33–56. Bibcode:1981CeMec..23...33M. doi:10.1007/BF01228543. hdl:2060/19770007221. S2CID 118077433.
A version of the same relationship can also be seen at equation 2 inAshbey, Neil (2002). "Relativity and the Global Positioning System" (PDF). Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583.
Rousseau, V.G. (2025). "Einstein’s Cat: A Thought Experiment on the Universality of Time Dilation". Physics Education. 60 (4). IOP Publishing. doi:10.1088/1361-6552/adedf1.

fourmilab.ch (Global: 8,613^th place; English: 6,778^th place)

Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik (in German). 322 (10): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004.. See also: English translation.
Einstein, A. (1905). "On the electrodynamics of moving bodies". Fourmilab.

framerated.co.uk (Global: low place; English: low place)

Collins, Frank (26 June 2017). "DOCTOR WHO, 10.11 – 'World Enough and Time'". Frame Rated.

futura-sciences.com (Global: 3,507^th place; English: low place)

blogs.futura-sciences.com

Luminet, Jean-Pierre (16 January 2016). "The Warped Science of Interstellar (4/6) : Time dilation and Penrose process". e-LUMINESCIENCES.

gsu.edu (Global: 1,911^th place; English: 1,817^th place)

hyperphysics.phy-astr.gsu.edu

Nave, C. R. (22 August 2005). "Hafele and Keating Experiment". HyperPhysics. Retrieved 2013-08-05.

handle.net (Global: 102^nd place; English: 76^th place)

hdl.handle.net

See equations 2 & 3 (combined here and divided throughout by c²) at pp. 35–36 in Moyer, T. D. (1981). "Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference". Celestial Mechanics. 23 (1): 33–56. Bibcode:1981CeMec..23...33M. doi:10.1007/BF01228543. hdl:2060/19770007221. S2CID 118077433.

harvard.edu (Global: 18^th place; English: 17^th place)

ui.adsabs.harvard.edu

Ashby, Neil (2003). "Relativity in the Global Positioning System". Living Reviews in Relativity. 6 (1): 16. Bibcode:2003LRR.....6....1A. doi:10.12942/lrr-2003-1. PMC 5253894. PMID 28163638.
Hughes, Theo; Kersting, Magdalena (5 January 2021). "The invisibility of time dilation". Physics Education. 56 (2): 025011. Bibcode:2021PhyEd..56b5011H. doi:10.1088/1361-6552/abce02.
Larmor, Joseph (1897). "On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with Material Media" . Philosophical Transactions of the Royal Society. 190: 205–300. Bibcode:1897RSPTA.190..205L. doi:10.1098/rsta.1897.0020.
Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik (in German). 322 (10): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004.. See also: English translation.
Galli, J. Ronald; Amiri, Farhang (Apr 2012). "The Square Light Clock and Special Relativity". The Physics Teacher. 50 (4). American Association of Physics Teachers: 212. Bibcode:2012PhTea..50..212G. doi:10.1119/1.3694069. S2CID 120089462.
Bailey, J.; et al. (1977). "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit". Nature. 268 (5618): 301. Bibcode:1977Natur.268..301B. doi:10.1038/268301a0. S2CID 4173884.
Hasselkamp, D.; Mondry, E.; Scharmann, A. (1979). "Direct observation of the transversal Doppler-shift". Zeitschrift für Physik A. 289 (2): 151–155. Bibcode:1979ZPhyA.289..151H. doi:10.1007/BF01435932. S2CID 120963034.
Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. (2010). "Optical Clocks and Relativity". Science. 329 (5999): 1630–1633. Bibcode:2010Sci...329.1630C. doi:10.1126/science.1192720. PMID 20929843. S2CID 206527813.
See equations 3, 4, 6 and 9 of Iorio, Lorenzo (2005). "An analytical treatment of the Clock Paradox in the framework of the Special and General Theories of Relativity". Foundations of Physics Letters. 18 (1): 1–19. arXiv:physics/0405038. Bibcode:2005FoPhL..18....1I. doi:10.1007/s10702-005-2466-8. S2CID 15081211.
Bailey, H.; Borer, K.; Combley, F.; Drumm, H.; Krienen, F.; Lange, F.; Picasso, E.; von Ruden, W.; Farley F. J. M.; Field J. H.; Flegel W. & Hattersley P. M. (1977). "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit". Nature. 268 (5618): 301–305. Bibcode:1977Natur.268..301B. doi:10.1038/268301a0. S2CID 4173884.
Roos, C. E.; Marraffino, J.; Reucroft, S.; Waters, J.; Webster, M. S.; Williams, E. G. H. (1980). "σ+/- lifetimes and longitudinal acceleration". Nature. 286 (5770): 244–245. Bibcode:1980Natur.286..244R. doi:10.1038/286244a0. S2CID 4280317.
Burns, M. Shane; Leveille, Michael D.; Dominguez, Armand R.; Gebhard, Brian B.; Huestis, Samuel E.; Steele, Jeffrey; Patterson, Brian; Sell, Jerry F.; Serna, Mario; Gearba, M. Alina; Olesen, Robert; O'Shea, Patrick; Schiller, Jonathan (18 September 2017). "Measurement of gravitational time dilation: An undergraduate research project". American Journal of Physics. 85 (10): 757–762. arXiv:1707.00171. Bibcode:2017AmJPh..85..757B. doi:10.1119/1.5000802. S2CID 119503665.
Pound, R. V.; Snider J. L. (November 2, 1964). "Effect of Gravity on Nuclear Resonance". Physical Review Letters. 13 (18): 539–540. Bibcode:1964PhRvL..13..539P. doi:10.1103/PhysRevLett.13.539.
Ashby, Neil (2002). "Relativity in the Global Positioning System". Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583. PMC 5253894. PMID 28163638.
See equations 2 & 3 (combined here and divided throughout by c²) at pp. 35–36 in Moyer, T. D. (1981). "Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference". Celestial Mechanics. 23 (1): 33–56. Bibcode:1981CeMec..23...33M. doi:10.1007/BF01228543. hdl:2060/19770007221. S2CID 118077433.
A version of the same relationship can also be seen at equation 2 inAshbey, Neil (2002). "Relativity and the Global Positioning System" (PDF). Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583.

iop.org (Global: 1,725^th place; English: 1,828^th place)

iopscience.iop.org

Rousseau, V.G. (2025). "Einstein’s Cat: A Thought Experiment on the Universality of Time Dilation". Physics Education. 60 (4). IOP Publishing. doi:10.1088/1361-6552/adedf1.

ipgp.fr (Global: low place; English: low place)

A version of the same relationship can also be seen at equation 2 inAshbey, Neil (2002). "Relativity and the Global Positioning System" (PDF). Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583.

johnmeaney.com (Global: low place; English: low place)

Meaney, John (17 December 2003). "Time passages (2)". John Meaney's WebLog.

nih.gov (Global: 4^th place; English: 4^th place)

pubmed.ncbi.nlm.nih.gov

Ashby, Neil (2003). "Relativity in the Global Positioning System". Living Reviews in Relativity. 6 (1): 16. Bibcode:2003LRR.....6....1A. doi:10.12942/lrr-2003-1. PMC 5253894. PMID 28163638.
Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. (2010). "Optical Clocks and Relativity". Science. 329 (5999): 1630–1633. Bibcode:2010Sci...329.1630C. doi:10.1126/science.1192720. PMID 20929843. S2CID 206527813.
Ashby, Neil (2002). "Relativity in the Global Positioning System". Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583. PMC 5253894. PMID 28163638.

ncbi.nlm.nih.gov

Ashby, Neil (2003). "Relativity in the Global Positioning System". Living Reviews in Relativity. 6 (1): 16. Bibcode:2003LRR.....6....1A. doi:10.12942/lrr-2003-1. PMC 5253894. PMID 28163638.
Ashby, Neil (2002). "Relativity in the Global Positioning System". Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583. PMC 5253894. PMID 28163638.

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Kranking, Carlyn (31 May 2019). Wagner, Ryan (ed.). "Time travel in movies, explained". North by Northwestern.

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"Einstein" (PDF). Metromnia. National Physical Laboratory. 2005. pp. 1–4.

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Ashby, Neil (2002). "Relativity in the Global Positioning System". Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583. PMC 5253894. PMID 28163638.

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Weiner, Adam (30 April 2008). "The Science of Sci-Fi". Popular Science.

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Galli, J. Ronald; Amiri, Farhang (Apr 2012). "The Square Light Clock and Special Relativity". The Physics Teacher. 50 (4). American Association of Physics Teachers: 212. Bibcode:2012PhTea..50..212G. doi:10.1119/1.3694069. S2CID 120089462.
Bailey, J.; et al. (1977). "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit". Nature. 268 (5618): 301. Bibcode:1977Natur.268..301B. doi:10.1038/268301a0. S2CID 4173884.
Hasselkamp, D.; Mondry, E.; Scharmann, A. (1979). "Direct observation of the transversal Doppler-shift". Zeitschrift für Physik A. 289 (2): 151–155. Bibcode:1979ZPhyA.289..151H. doi:10.1007/BF01435932. S2CID 120963034.
Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. (2010). "Optical Clocks and Relativity". Science. 329 (5999): 1630–1633. Bibcode:2010Sci...329.1630C. doi:10.1126/science.1192720. PMID 20929843. S2CID 206527813.
See equations 3, 4, 6 and 9 of Iorio, Lorenzo (2005). "An analytical treatment of the Clock Paradox in the framework of the Special and General Theories of Relativity". Foundations of Physics Letters. 18 (1): 1–19. arXiv:physics/0405038. Bibcode:2005FoPhL..18....1I. doi:10.1007/s10702-005-2466-8. S2CID 15081211.
Bailey, H.; Borer, K.; Combley, F.; Drumm, H.; Krienen, F.; Lange, F.; Picasso, E.; von Ruden, W.; Farley F. J. M.; Field J. H.; Flegel W. & Hattersley P. M. (1977). "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit". Nature. 268 (5618): 301–305. Bibcode:1977Natur.268..301B. doi:10.1038/268301a0. S2CID 4173884.
Roos, C. E.; Marraffino, J.; Reucroft, S.; Waters, J.; Webster, M. S.; Williams, E. G. H. (1980). "σ+/- lifetimes and longitudinal acceleration". Nature. 286 (5770): 244–245. Bibcode:1980Natur.286..244R. doi:10.1038/286244a0. S2CID 4280317.
Burns, M. Shane; Leveille, Michael D.; Dominguez, Armand R.; Gebhard, Brian B.; Huestis, Samuel E.; Steele, Jeffrey; Patterson, Brian; Sell, Jerry F.; Serna, Mario; Gearba, M. Alina; Olesen, Robert; O'Shea, Patrick; Schiller, Jonathan (18 September 2017). "Measurement of gravitational time dilation: An undergraduate research project". American Journal of Physics. 85 (10): 757–762. arXiv:1707.00171. Bibcode:2017AmJPh..85..757B. doi:10.1119/1.5000802. S2CID 119503665.
See equations 2 & 3 (combined here and divided throughout by c²) at pp. 35–36 in Moyer, T. D. (1981). "Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference". Celestial Mechanics. 23 (1): 33–56. Bibcode:1981CeMec..23...33M. doi:10.1007/BF01228543. hdl:2060/19770007221. S2CID 118077433.

sf-encyclopedia.com (Global: 3,802^nd place; English: 2,631^st place)

Langford, David; Stableford, Brian M (20 August 2018). Clute, John; Langford, David; Nicholls, Peter; Sleight, Graham (eds.). "Relativity". The Encyclopedia of Science Fiction.

thedailybeast.com (Global: 330^th place; English: 222^nd place)

Tyson, Neil deGrasse (12 July 2017). "Neil deGrasse Tyson Breaks Down 'Interstellar': Black Holes, Time Dilations, and Massive Waves". The Daily Beast (Interview). Interviewed by Marlow Stern.

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Walter, Damien (22 February 2018). "It's about time: how sci-fi has described Einstein's universe". The Guardian.

unlp.edu.ar (Global: 4,596^th place; English: 9,174^th place)

sedici.unlp.edu.ar

Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik (in German). 322 (10): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004.. See also: English translation.

washington.edu (Global: 1,067^th place; English: 749^th place)

npl.washington.edu

Cramer, John G. (20 August 1989). "The Twin Paradox Revisited". Analog Science Fiction and Fact. No. March-1990 – via University of Washington.

wikisource.org (Global: 27^th place; English: 51^st place)

en.wikisource.org

Larmor, Joseph (1897). "On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with Material Media" . Philosophical Transactions of the Royal Society. 190: 205–300. Bibcode:1897RSPTA.190..205L. doi:10.1098/rsta.1897.0020.
Cohn, Emil (1904), "Zur Elektrodynamik bewegter Systeme II" [On the Electrodynamics of Moving Systems II], Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (in German and English), vol. 1904/2, no. 43, pp. 1404–1416
Minkowski, Hermann (1908) [1907], "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern" [The Fundamental Equations for Electromagnetic Processes in Moving Bodies], Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German and English), pp. 53–111

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Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. (2010). "Optical Clocks and Relativity". Science. 329 (5999): 1630–1633. Bibcode:2010Sci...329.1630C. doi:10.1126/science.1192720. PMID 20929843. S2CID 206527813.

Time dilation (English Wikipedia)

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nytimes.com (Global: 7^th place; English: 7^th place)

phys.org (Global: 1,283^rd place; English: 1,130^th place)

popsci.com (Global: 2,941^st place; English: 2,021^st place)

semanticscholar.org (Global: 11^th place; English: 8^th place)

sf-encyclopedia.com (Global: 3,802^nd place; English: 2,631^st place)

thedailybeast.com (Global: 330^th place; English: 222^nd place)

theguardian.com (Global: 12^th place; English: 11^th place)

unlp.edu.ar (Global: 4,596^th place; English: 9,174^th place)

washington.edu (Global: 1,067^th place; English: 749^th place)

wikisource.org (Global: 27^th place; English: 51^st place)

zenodo.org (Global: 621^st place; English: 380^th place)