愛因斯坦的廣義相對論預言了處於轉動狀態的質量會對其周圍的時空產生拖拽的現象，這種現象被稱作參考系拖拽（英文：）或慣性系拖拽（英文：）。因轉動而產生的參考系拖拽的相應理論最早由奧地利物理學家與於 1918 年透過廣義相對論推匯出，因此參考系拖拽也常常被叫做冷澤 提爾苓效應Thirring H Über die Wirkung rotierender ferner Massen in der Einsteinschen Gravitationstheorie Physikalische Zeitschrift 19 33 1918 On the Effect of Rotating Distant Masses in Einsteins Theory of Gravitation Thirring H Berichtigung zu meiner Arbeit Über die Wirkung rotierender Massen in der Einsteinschen Gravitationstheorie Physikalische Zeitschrift 22 29 1921 Correction to my paper On the Effect of Rotating Distant Masses in Einsteins Theory of Gravitation Lense J and Thirring H Über den Einfluss der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie Physikalische Zeitschrift 19 15663 1918 On the Influence of the Proper Rotation of Central Bodies on the Motions of Planets and Moons According to Einsteins Theory of Gravitation。冷澤與提爾苓預言物體的轉動會導致其周圍時空參考系的改變，從而使周圍物體的位置和牛頓力學下的經典結果產生偏差。不過理論預言這種偏差將非常之小，大約只有幾萬億分之一。想要在實驗中觀測到這種現象，需要對大質量的旋轉物體（例如行星或恆星）使用高靈敏度的儀器進行探測，例如圍繞地球公轉的人造衛星軌道會因地球自轉而產生細微的變化。更一般的，這種由加速質量而產生的引力場變化可以歸結到引力磁學的研究領域。 參考系拖拽的分類 轉動的參考系拖拽（冷澤 提爾苓效應）：廣義相對論和引力磁學等有關替代理論都預言處於轉動的大質量物體周圍會產生參考系的拖拽。根據冷澤 提爾苓效應，一個處於遠處的觀察者將看到圍繞中心物體轉動的參考系內的時鐘走得最快；這也說明在觀察者看來，與物體轉動速度方向相同的光將比與轉動速度方向相反的光走得更快。冷澤 提爾苓效應是最著名的廣義相對論效應之一，這要部分歸功於引力探測器 B 的實驗觀測。 直線的參考系拖拽：類似於轉動產生的參考系拖拽，移動也會產生類似的效應。儘管這種效應是否和轉動的效應一樣在理論上成立有著爭議，但實驗驗證的高度困難導致其相關討論比轉動效應少得多，也常常在討論參考系拖拽的文章中被忽略，不過愛因斯坦在其 1921 年的 著 作《相對論的意義》中敘述了這種效應 Einstein A The Meaning of Relativity contains transcripts of his 1921 Princeton lectures。 靜止質量的增加：這是愛因斯坦在同一 著 作中提到的第三種效應 Einstein A The Meaning of Relativity pp9596 Chapman and Hall 1987：當週圍有質量存在時，一個物體的慣性會增加。儘管嚴格說這不是一種參考系的拖拽（而愛因斯坦在論文中並未使用參考系拖拽一詞來概括這幾種效應），愛因斯坦在廣義相對論下使用了與以上兩種參考系拖拽效應相同的推導過程得到了這種效應。同樣，這也是一種非常微小的效應，在實驗中難以觀測。 參考系拖拽的實驗驗證 1976 年範帕騰和伊夫利特在他們的論文中提出了一個專門用於觀測冷澤 提爾苓效應的實驗設想 Van Patten RA Everitt CWF Possible Experiment with Two CounterOrbiting DragFree Satellites to Obtain a New Test of Einsteinss General Theory of Relativity and Improved Measurements in Geodesy Phys Rev Lett 36 629632 1976Van Patten RA Everitt CWF A possible experiment with two counterrotating dragfree satellites to obtain a new test of Einsteins general theory of relativity and improved measurements in geodesy Celest Mech Dyn Astron 13 429447 1976：測量一對置於地球極軌道上的相向繞轉的航天器的節點的冷澤 提爾苓進動，而航天器上的測量儀器需要滿足無拖拽條件。1986 年，丘弗里尼提出了一個基本等效但更廉價的版本 Ciufolini I Measurement of LenseThirring Drag on HighAltitude LaserRanged Artificial Satellites Phys Rev Lett 56 278281 1986，他提議發射一枚被動的沿測地軌道執行的衛星，其軌道和 1976 年發射的衛星的軌道相比除了軌道平面相差 180 度外完全相同，這枚衛星和的共同構造被稱作蝴蝶式配置。在這種情況下，系統的可觀測量為衛星和這枚新衛星（其後在不同場合下被稱作為、或）的節點總數。儘管這種設想被很多研究小組進行過詳細的研究 Ries JC Eanes RJ Watkins MM Tapley B Joint NASAASI Study on Measuring the LenseThirring Precession Using a Second LAGEOS Satellite CSR893 Center for Space Research The University of Texas at Austin 1989Iorio L Lucchesi DM and Ciufolini I The LARES Mission Revisited An Alternative Scenario Class Quantum Grav 19 43114325 2002，至今還沒有將這個設想付諸實踐。原則上，在蝴蝶式配置下人們不僅可以測量節點的總數，還能夠觀測到近地點之間相差的距離 Iorio L A new proposal for measuring the LenseThirring effect with a pair of supplementary satellites in the gravitational field of the Earth Phys Lett A 308 8184 2003Iorio L On a new observable for measuring the LenseThirring effect with Satellite Laser Ranging Gen Relativ Gravit 35 15831595 2003Iorio L Lucchesi DM LAGEOStype Satellites in Critical Supplementary Orbital Configuration and the LenseThirring Effect Detection Class Quantum Grav 20 24772490 2003，雖說在實際執行中系統的開普勒軌道引數受到的來自非引力的微擾影響會更大，例如來自太陽的輻射壓。這也是在實際執行中需要實現動態的無拖拽技術的原因。其他被提出的設想包括髮射一顆處於近極軌道且較低高度的衛星 Lucchesi DM Paolozzi A A cost effective approach for LARES satellite paper presented at XVI Congresso Nazionale AIDAA 2428 Sept 2001 Palermo 2001Ciufolini I On the orbit of the LARES satellite Preprint httpwwwarxivorgabsgrqc0609081 2006，但這種設想已經被證明不可行 Peterson GE Estimation of the LenseThirring precession using laserranged satellites CSR971 Center for Space Research The University of Texas at Austin 1997Iorio L A critical approach to the concept of a polar lowaltitude LARES satellite Class Quantum Grav 19 L175L183 2002Iorio L A comment on the paper On the orbit of the LARES satellite by I Ciufolini Planet Space Sci 55 11981200 2007。近年來為了提高實際架設蝴蝶式配置的可能性，有人提出衛星將有能力驗證 Ciufolini I LARESWEBERSAT framedragging and fundamental physics Preprint httparxivorgabsgrqc0412001 2004 由、和建立的多維膜世界模型 Dvali G Gabadadze G Porrati M 4D Gravity on a Brane in 5D Minkowski Space Phys lett B 485 208214 2000，並能夠將現今對等效原理驗證的精確度提高兩個數量級 Ciufolini I Frame Dragging and LenseThirring Effect Gen Relativ Gravit 36 22572270 2004，不過這種說法已經被證明非常不現實 Iorio L On the possibility of testing the BraneWorld scenario with orbital motions in the Solar System J Cosmol Astrpart Phys 7 8 2005Iorio L LARESWEBERSAT and the equivalence principle Europhys Lett 80 40007 2007 See also this preprint。 現在我們只侷限於討論有關已發射衛星的研究的情形，最早提出利用衛星以及衛星雷射測距技術來測量冷澤 提爾苓效應的設想可追溯至 19771978 年 Cugusi L Proverbio E Relativistic effects on the Motion of the Earths Satellites paper presented at the International Symposium on Satellite Geodesy in Budapest from June 28 to July 1 1977 J of Geodesy 51 249252 1977Cugusi L Proverbio E Relativistic Effects on the Motion of Earths Artificial Satellites Astron Astrophys 69 321325 1978。而真正有效的實驗測試則是在 1996 年透過和完成的 Ciufolini I Lucchesi DM Vespe F Mandiello A Measurement of Dragging of Inertial Frames and Gravitomagnetic Field Using LaserRanged Satellites Il Nuovo Cimento A 109 575590 1996，實驗中採用了對兩枚衛星的節點和的近地點位置的恰當組合 Ciufolini I On a new method to measure the gravitomagnetic field using two orbiting satellites Il Nuovo Cimento A 109 17091720 1996。最近的一次來自衛星的探測是在 20042006 年 Ciufolini I and Pavlis EC A confirmation of the general relativistic prediction of the LenseThirring effect Nature 431 958960 2004Ciufolini I Pavlis EC and Peron R Determination of framedragging using Earth gravity models from CHAMP and GRACE New Astron 11 527550 2006，觀測中沒有使用的近地點，而只採用了對兩架航天器節點觀測結果的線性組合 Pavlis EC Geodetic contributions to gravitational experiments in space In Cianci R Collina R Francaviglia M Fré P Eds Recent Developments in General Relativity 14th SIGRAV Conference on General Relativity and Gravitational Physics Genova Italy September 1822 2000 Springer Milano pp 217233 2002Iorio L Morea A The impact of the new Earth gravity models on the measurement of the LenseThirring effect Gen Relativ Gravit 36 13211333 2004 Preprint httpwwwarxivorgabsgrqc0304011Iorio L The new Earth gravity models and the measurement of the LenseThirring effect In Novello M Bergliaffa SP Ruffini R Eds On Recent Developments in Theoretical and Experimental General Relativity Gravitation and Relativistic Field Theories World Scientific Singapore pp 10111020 2003 Preprint httpwwwarxivorgabsgrqc0308022Iorio L The impact of the new CHAMP and GRACE Earth gravity models on the measurement of the general relativistic LenseThirring effect with the LAGEOS and LAGEOS II satellites In Reigber Ch Luehr H Schwintzer P Wickert J Eds Earth Observation with CHAMP Results from Three Years in Orbit SpringerVerlag Berlin pp 187192 2003 Preprint httparxivorgabsgrqc0309092 Ries JC Eanes RJ Tapley BD LenseThirring Precession Determination from Laser Ranging to Artificial Satellites In Ruffini R Sigismondi C Eds Nonlinear Gravitodynamics The LenseThirring Effect World Scientific Singapore pp 201211 2003aRies JC Eanes RJ Tapley BD Peterson GE Prospects for an Improved LenseThirring Test with SLR and the GRACE Gravity Mission In Noomen R Klosko S Noll C Pearlman M Eds Proceedings of the 13th International Laser Ranging Workshop NASA CP 2003212248 NASA Goddard Greenbelt 2003b Preprint httpcddisagsfcnasagovlw13lw_proceedingshtmlscience。實驗結果和廣義相對論的預言並不矛盾，然而計算得到的觀測總誤差卻引發了一些爭論 Iorio L On the reliability of the so far performed tests for measuring the LenseThirring effect with the LAGEOS satellites New Astron 10 603615 2005Ciufolini I and Pavlis EC On the Measurement of the LenseThirring effect Using the Nodes of the LAGEOS Satellites in reply to On the reliability of the sofar performed tests for measuring the LenseThirring effect with the LAGEOS satellites by L Iorio New Astron 10 636651 2005Lucchesi DM The impact of the even zonal harmonics secular variations on the LenseThirring effect measurement with the two Lageos satellites Int J of Mod Phys D 14 19892023 2005Iorio L A critical analysis of a recent test of the LenseThirring effect with the LAGEOS satellites J of Geodesy 80 128136 2006Iorio L An assessment of the measurement of the LenseThirring effect in the Earth gravity field in reply to On the measurement of the LenseThirring effect using the nodes of the LAGEOS satellites in reply toOn the reliability of the so far performed tests for measuring the LenseThirring effect with the LAGEOS satellitesby L Iorio by I Ciufolini and E Pavlis Planet Space Sci 55 503511 2007Iorio L On some critical issues of the LAGEOSLAGEOS II LenseThirring experiment 2007 Preprint httparxivorgabs07101022。另一個在火星引力場中觀測冷澤 提爾苓效應的測試結果是根據對火星全球探勘者號的位置資料進行恰當分析後得到的 Iorio L A note on the evidence of the gravitomagnetic field of Mars Class Quantum Grav 23 54515454 2006，不過這個結果也引發了爭議 Iorio L Testing framedragging with the Mars Global Surveyor spacecraft in the gravitational field of Mars In Iorio L Ed The Measurement of Gravitomagnetism A Challenging Enterprise Nova publishers Hauppauge NY pp 177187 2007 Preprint httpwwwarxivorgabsgrqc0701042 2007 Krogh K Comment on Evidence of the gravitomagnetic field of Mars Class Quantum Grav 24 57095715 2007Iorio L Reply to Iorioshighprecision measurementof frame dragging with the Mars Global Surveyor by Kris Krogh J Graviit Phys at press 2008 Preprint httpwwwarxivorgabsgrqc0701146。此外還有觀測太陽自轉對太陽系內行星軌道產生的冷澤 提爾苓效應的嘗試 Iorio L First preliminary tests of the general relativistic gravitomagnetic field of the Sun and new constraints on a Yukawalike fifth force from planetary data Planet Space Sci 55 12901298 2007，廣義相對論對冷澤 提爾苓進動的預言和觀測到的行星近日點進動的修正相符合 Pitjeva EV Relativistic Effects and Solar Oblateness from Radar Observations of Planets and Spacecraft Astron Lett 31 340349 2005，但誤差依然比較大；以及對環繞木星的伽利略衛星的觀測 Iorio L and Lainey V The LenseThirring effect in the Jovian system of the Galilean satellites and its measurability Int J Mod Phys D 14 20392050 2005，這同時也是最初由冷澤和提爾苓提出的實驗建議。 引力探測器 BEveritt CWF The Gyroscope Experiment I General Description and Analysis of Gyroscope Performance In Bertotti B Ed Proc Int School Phys Enrico Fermi Course LVI New Academic Press New York pp 331360 1974 Reprinted in Ruffini RJ Sigismondi C Eds Nonlinear Gravitodynamics The LenseThirring Effect World Scientific Singapore pp 439468 2003Everitt CWF et al Gravity Probe B Countdown to Launch In Laemmerzahl C Everitt CWF Hehl FW Eds Gyros Clocks Interferometers Testing Relativistic Gravity in Space Springer Berlin pp 5282 2001 正在對另一種引力磁學的效應即希夫進動進行實驗觀測 Pugh GE Proposal for a Satellite Test of the Coriolis Prediction of General Relativity WSEG Research Memorandum No 11 1959 Reprinted in Ruffini RJ Sigismondi C Eds Nonlinear Gravitodynamics The LenseThirring Effect World Scientific Singapore pp 414426 2003Schiff L On Experimental Tests of the General Theory of Relativity Am J of Phys 28 340343 1960，並期望達到 1 或更好的誤差。然而目前來看這個目標還是個奢望，2007 年 4 月的最初結果表明現在能達到的誤差範圍在 256128 之間 Muhlfelder B Mac Keiser G and Turneaure J Gravity Probe B Experiment Error poster L100027 presented at the American Physical Society APS meeting in Jacksonville Florida on 1417 April 2007 2007，並期望在 2007 年 12 月時能夠縮小至 13 StanfordNews 41407 downloadable at httpeinsteinstanfordedu 。2005 年物理學家艾奧里奧指出 Iorio L The impact of the new Earth gravity models on the measurement of the LenseThirring effect with a new satellite New Astron 10 616635 2005，如要在地球引力場中觀測冷澤 提爾苓效應達到 1 的誤差水平，還需要發射至少兩個新的衛星，並且最好能夠採用偏心率很高的軌道以及一些能夠抵消非引力微擾的動態機制。最近，義大利航天局宣佈將於 2008 年底使用運載火箭發射衛星，其目標是將地球引力場中冷澤 提爾苓效應的測量達到 1 的誤差精度，但有人對這一計劃能否成功表示懷疑 Iorio L LARES approved towards a 1 measurement of frame dragging Preprint httparxivorgabs08022031 Iorio L Perspectives in measuring framedragging after the approval of the LARES mission Preprint httparxivorgabs08033278。近來還有墨菲等人報告了一個透過月球雷射測距對引力磁性的間接觀測實驗，並聲稱其誤差小於 01 Murphy TW Nordtvedt K and Turyshev SG The Gravitomagnetic Influence on Gyroscopes and on the Lunar Orbit Phys Rev Lett 98 071102 2007；但物理學家科佩金對月球雷射測距技術能否感應到引力磁性的存在表示疑問 Kopeikin Comment on The gravitomagnetic influence on gyroscopes and on the lunar orbitPhys Rev Lett 98 229001 2007 。 天體物理學的證據 相對論性噴流或許可以為參考系拖拽的存在提供直接證據：在一個旋轉黑洞的能層內由冷澤 提爾苓效應產生的引力磁力 Williams R K 1995 May 15 Extracting X rays Ύ rays and relativistic ee pairs from supermassive Kerr black holes using the Penrose mechanism Physical Review 51 10 53875427Williams R K 2004 August 20 Collimated escaping vortical polar ee jets intrinsically produced by rotating black holes and Penrose processes The Astrophysical Journal 611 952963 和由羅傑彭羅斯爵士提出的能量抽取機制 Penrose R 1969 Gravitational collapse The role of general relativity Nuovo Cimento Rivista Numero Speciale 1 252276 一起可以被用來解釋對相對論性噴流觀測到的性質。由雷瓦凱威廉姆斯發展的引力磁學模型預言了已被觀測到的由類星體和活動星系核發射的高能粒子（～吉電子伏特）、X 射線和伽瑪射線的光子抽取、平行於極軸的相對論性噴流以及噴流相對於軌道平面的非對稱結構等現象的存在。 參考系拖拽的數學推導 描述參考系拖拽的最直接方法是使用克爾度規，克爾度規描述了一個質量為 M，具有角動量 J 的旋轉黑洞周圍的時空幾何。 c2 dtau2 left 1 frac r_s rrho2 right c2 dt2 frac rho2Lambda2 dr2 rho2 dtheta2 left r2 a2 frac r_s r alpha2rho2 sin2 theta right sin2 theta dphi2 frac 2r_s ralpharho2 dphi dt 這裡 rs 是史瓦西半徑。 r_s frac 2GMc2 度規中其他變數的意義為 alpha frac JMc rho2 r2 alpha2 cos2 theta Lambda2 r2 r_s r alpha2 在非相對論極限下，當質量 M 或等效地，rs 趨於零時，克爾度規成為扁球面座標系下的正交度規形式： c2 dtau2 c2 dt2 frac rho2r2 alpha2 dr2 rho2 dtheta2 left r2 alpha2 right sin2theta dphi2 我們可以將它重新寫作如下形式 c2 dtau2 left g_tt frac g_tphi2g_phiphi right dt2 g_rr dr2 g_thetatheta dtheta2 g_phiphi left dphi frac g_tphig_phiphi dt right2 這個度規和一個以角速度 Ω 旋轉的座標系等價，其中角速度 Ω 和半徑 r 及餘緯 θ 有關： Omega frac g_tphig_phiphi frac r_s alpha rrho2 left r2 alpha2 right r_s alpha2 r sin2theta 在赤道面上可以簡化為： Omega frac r_s alpha cr3 alpha2 r r_s alpha2 從而我們看到，一個處於旋轉狀態的中心質量能夠導致周圍慣性參考系的轉動，這就是參考系拖拽。 一個旋轉黑洞的能層內的參考系拖拽效應是一個極端的版本。從數學形式上看，克爾度規存在有兩個奇性的面，內表面對應著一個和史瓦西度規類似的球面視界： r_inner frac r_s sqrt r_s2 4alpha22 在這個半徑上度規的純徑向分量 grr 趨於無窮大。 奇性的外表面不是一個球面，而是一個扁球面。這個扁球面在其旋轉軸的極點處和內表面相重合，這時的餘緯度 θ 等於 0 或 π；而其半徑由下式給出： r_outer frac r_s sqrt r_s2 4alpha2 cos2theta2 在這個半徑上度規的純時間分量 gtt 由正變為負，內外奇性表面所包圍的空間被稱作能層。一般來說，沿世界線運動的運動粒子所經歷的固有時總是正的，但在時間分量 gtt 為負的能層中這無法做到，除非這個粒子同時也以不低於 Ω 的角速度圍繞中心質量 M 旋轉。當然，參考系拖拽效應發生在任何旋轉質量周圍的任何位置，而不僅僅在能層中。 參考系拖曳力的公式 根據 Lev Laudau The classic theory of fields F frac SJjr4 J 為中心質量自旋角動量 j 為周圍質量公轉角動量 S 為史氏常數 S frac 2Gc2 而旋轉參考系拖曳能量公式為 E frac SJjr3 參見 廣義相對論的實驗驗證 測地線效應 引力磁學 克爾度規 相對論性噴流 冷澤 提爾苓進動 重力磁性 德西特進動 參考文獻 外部連結 NASA RELEASE 04351 As The World Turns It Drags Space And Time New Scientist press release of the MGS test by Iorio in the gravitational field of Mars Paper by Giampiero Sindoni Claudio Paris and Paolo Ialongo about the MarsMGS test Paper by G Felici about the MarsMGS test Paper by Kris Krogh about the MarsMGS test Reply by Ignazio Ciufolini and Erricos Pavlis about some criticisms by Iorio Frame dragging applied to relativistic jets Frame Dragging Duke University press release General Relativistic Frame Dragging MSNBC report on Xray observations Ciufolini et al LAGEOS paper 1997 25 error Ciufolini update Sep 2002 20 error Press release regarding LAGEOS study Preprint by Ries et al Ciufolini and Pavlis Nature new article on 2004 reanalysis of the LAGEOS data Iorio New Astronomy general paper with full references Iorio J of Geodesy paper on the impact of the secular variations of the even zonal harmonics of the geopotential Iorio Planetary Space Science paper The Naked Singularity Category 廣義相對論 Category 參考系 Category 物理現象