参考系拖拽

爱因斯坦的广义相对论预言了处于转动状态的质量会对其周围的时空产生拖拽的现象，这种现象被称作参考系拖拽（英文：）或惯性系拖拽（英文：）。因转动而产生的参考系拖拽的相应理论最早由奧地利物理学家與於1918年通过广义相对论推导出，因此参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 Einstein's 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 Einstein's 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 19 156-63 (1918) [On the Influence of the Proper Rotation of Central Bodies on the Motions of Planets and Moons According to Einstein's Theory of Gravitation]。冷澤與提爾苓预言物体的转动会导致其周围时空参考系的改变，从而使周围物体的位置和牛顿力学下的经典结果产生偏差。不过理论预言这种偏差将非常之小，大约只有几万亿分之一。想要在实验中观测到这种现象，需要对大质量的旋转物体（例如行星或恒星）使用高灵敏度的仪器进行探测，例如围绕地球公转的人造卫星轨道会因地球自转而产生细微的变化。更一般的，这种由加速质量而产生的引力场变化可以归结到引力磁学的研究领域。

 参考系拖拽的分类 

转动的参考系拖拽（冷澤-提爾苓效应）：广义相对论和引力磁学等有关替代理论都预言处于转动的大质量物体周围会产生参考系的拖拽。根据冷澤-提爾苓效应，一个处于远处的观察者将看到围绕中心物体转动的参考系内的时钟走得最快；这也说明在观察者看来，与物体转动速度方向相同的光将比与转动速度方向相反的光走得更快。冷澤-提爾苓效应是最著名的广义相对论效应之一，这要部分归功于引力探测器B的实验观测。

直线的参考系拖拽：类似于转动产生的参考系拖拽，移動也会产生类似的效应。尽管这种效应是否和转动的效应一樣在理論上成立有著争议，但实验验证的高度困难導致其相關討論比轉動效應少得多，也常常在討論参考系拖拽的文章中被忽略，不过爱因斯坦在其1921年的-{著}-作《相对论的意义》中叙述了这种效应Einstein, A The Meaning of Relativity (contains transcripts of his 1921 Princeton lectures).。

静止质量的增加：这是爱因斯坦在同一-{著}-作中提到的第三种效应Einstein, A The Meaning of Relativity, pp95-96, Chapman and Hall, 1987：当周围有质量存在时，一个物体的惯性会增加。尽管严格说这不是一种参考系的拖拽（而爱因斯坦在论文中并未使用参考系拖拽一词来概括这几种效应），爱因斯坦在广义相对论下使用了与以上两种参考系拖拽效应相同的推导过程得到了这种效应。同样，这也是一种非常微小的效应，在实验中难以观测。

 参考系拖拽的实验验证 
1976年范·帕腾（）和伊夫利特（）在他们的论文中提出了一个专门用于观测冷澤-提爾苓效应的实验设想Van Patten, R.A., Everitt, C.W.F., Possible Experiment with Two Counter-Orbiting Drag-Free Satellites to Obtain a New Test of Einsteins's General Theory of Relativity and Improved Measurements in Geodesy, Phys. Rev. Lett., 36, 629-632, 1976.Van Patten, R.A., Everitt, C.W.F., A possible experiment with two counter-rotating drag-free satellites to obtain a new test of Einstein’s general theory of relativity and improved measurements in geodesy, Celest. Mech. Dyn. Astron., 13, 429-447, 1976.：测量一对置于地球极轨道上的相向绕转的航天器的节点的冷澤-提爾苓進動，而航天器上的测量仪器需要满足无拖拽条件。1986年，丘弗里尼（）提出了一个基本等效但更廉价的版本Ciufolini I., Measurement of Lense-Thirring Drag on High-Altitude Laser-Ranged Artificial Satellites, Phys. Rev. Lett., 56, 278-281, 1986.，他提议发射一枚被动的沿测地轨道运行的卫星，其轨道和1976年发射的卫星的轨道相比除了轨道平面相差180度外完全相同，这枚卫星和的共同构造被称作蝴蝶式配置。在这种情况下，系统的可观测量为卫星和这枚新卫星（其后在不同场合下被称作为、或）的节点总数。尽管这种设想被很多研究小组进行过详细的研究Ries, J.C., Eanes, R.J., Watkins, M.M., Tapley, B., Joint NASA/ASI Study on Measuring the Lense-Thirring Precession Using a Second LAGEOS Satellite, CSR-89-3, Center for Space Research, The University of Texas at Austin, 1989.Iorio, L., Lucchesi, D.M., and Ciufolini, I., The LARES Mission Revisited: An Alternative Scenario, Class. Quantum Grav., 19, 4311-4325, 2002.，至今还没有将这个设想付诸实践。原则上，在蝴蝶式配置下人们不仅可以测量节点的总数，还能够观测到近地点之间相差的距离Iorio, L., A new proposal for measuring the Lense-Thirring effect with a pair of supplementary satellites in the gravitational field of the Earth, Phys. Lett. A, 308, 81-84, 2003.Iorio, L., On a new observable for measuring the Lense-Thirring effect with Satellite Laser Ranging, Gen. Relativ. Gravit., 35, 1583-1595, 2003.Iorio, L.,  Lucchesi, D.M., LAGEOS-type Satellites in Critical Supplementary Orbital Configuration and the Lense--Thirring Effect Detection, Class. Quantum Grav., 20, 2477-2490, 2003.，虽说在实际运行中系统的开普勒轨道参数受到的来自非引力的微扰影响会更大，例如来自太阳的辐射压。这也是在实际运行中需要实现动态的无拖拽技术的原因。其他被提出的设想包括发射一颗处于近极轨道且较低高度的卫星Lucchesi, D.M., Paolozzi, A., A cost effective approach for LARES satellite, paper presented at XVI Congresso Nazionale AIDAA (24-28 Sept. 2001, Palermo), 2001.Ciufolini, I., On the orbit of the LARES satellite, (Preprint http://www.arxiv.org/abs/gr-qc/0609081), 2006.，但这种设想已经被证明不可行Peterson, G.E., Estimation of the Lense-Thirring precession using laser-ranged satellites, CSR-97-1, Center for Space Research, The University of Texas at Austin, 1997.Iorio, L., A critical approach to the concept of a polar, low-altitude LARES satellite, Class. Quantum Grav., 19, L175-L183, 2002.Iorio, L., A comment on the paper "On the orbit of the LARES satellite", by I. Ciufolini, Planet. Space Sci., 55, 1198-1200, 2007.。近年来为了提高实际架设蝴蝶式配置的可能性，有人提出卫星将有能力验证Ciufolini, I., LARES/WEBER-SAT, frame-dragging and fundamental physics, (Preprint http://arxiv.org/abs/gr-qc/0412001 ), 2004.由、和建立的多维膜世界模型Dvali, G., Gabadadze, G., Porrati, M., 4D Gravity on a Brane in 5D Minkowski Space, Phys. lett. B, 485, 208-214, 2000.，并能够将现今对等效原理验证的精确度提高两个数量级Ciufolini, I., Frame Dragging and Lense-Thirring Effect, Gen. Relativ. Gravit., 36, 2257-2270, 2004.，不过这种说法已经被证明非常不现实Iorio, L., On the possibility of testing the Brane-World scenario with orbital motions in the Solar System, J. Cosmol. Astrpart. Phys., 7, 8, 2005.Iorio, L., LARES/WEBER-SAT and the equivalence principle, Europhys. Lett., 80, 40007, 2007.  See also this  preprint。

现在我们只局限于讨论有关已发射卫星的研究的情形，最早提出利用卫星以及卫星激光测距（）技术来测量冷澤-提爾苓效应的设想可追溯至1977-1978年Cugusi, L., Proverbio E. Relativistic effects on the Motion of the Earth's. Satellites, paper presented at the International Symposium on Satellite Geodesy in Budapest from June 28 to July 1, 1977,  J. of Geodesy, 51, 249-252, 1977.Cugusi, L., Proverbio, E., Relativistic Effects on the Motion of Earth's Artificial Satellites, Astron. Astrophys, 69, 321-325, 1978.。而真正有效的实验测试则是在1996年通过和完成的Ciufolini, I., Lucchesi, D.M., Vespe, F., Mandiello, A., Measurement of Dragging of Inertial Frames and Gravitomagnetic Field Using Laser-Ranged Satellites, Il Nuovo Cimento A, 109, 575-590, 1996.，实验中采用了对两枚卫星的节点和的近地点位置的恰当组合Ciufolini, I.,  On a new method to measure the gravitomagnetic field using two orbiting satellites., Il Nuovo Cimento A, 109, 1709-1720, 1996.。最近的一次来自卫星的探测是在2004-2006年Ciufolini, I., and Pavlis, E.C., A confirmation of the general relativistic prediction of the Lense-Thirring effect, Nature, 431, 958-960, 2004Ciufolini, I., Pavlis, E.C., and Peron, R., Determination of frame-dragging using Earth gravity models from CHAMP and GRACE, New Astron., 11, 527-550, 2006.，观测中没有使用的近地点，而只采用了对两架航天器节点观测结果的线性组合Pavlis, E.C., 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 18-22, 2000. Springer, Milano, pp. 217-233, 2002.Iorio, L.,  Morea, A., The impact of the new Earth gravity models on the measurement of the Lense-Thirring effect, Gen. Relativ. Gravit., 36, 1321-1333, 2004. (Preprint http://www.arxiv.org/abs/gr-qc/0304011 ).Iorio, L., The new Earth gravity models and the measurement of the Lense-Thirring effect. In: Novello, M.,  Bergliaffa, S.P., Ruffini, R. (Eds.), On Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories, World Scientific, Singapore, pp. 1011-1020, 2003. (Preprint http://www.arxiv.org/abs/gr-qc/0308022).Iorio, L., The impact of the new CHAMP and GRACE Earth gravity models on the measurement of the general relativistic Lense--Thirring 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, Springer-Verlag, Berlin, pp. 187-192, 2003. (Preprint http://arxiv.org/abs/gr-qc/0309092 )Ries, J.C., Eanes, R.J., Tapley, B.D.,  Lense-Thirring Precession Determination from Laser Ranging to Artificial Satellites. In: Ruffini, R., Sigismondi, C. (Eds.), Nonlinear Gravitodynamics. The Lense-Thirring Effect, World Scientific, Singapore, pp. 201-211, 2003a.Ries, J.C., Eanes, R.J., Tapley, B.D., Peterson, G.E., Prospects for an Improved Lense-Thirring 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 2003-212248,  NASA Goddard, Greenbelt, 2003b. (Preprint http://cddisa.gsfc.nasa.gov/lw13/lw$\_${proceedings}.html$\#$science).。实验结果和广义相对论的预言并不矛盾，然而计算得到的观测总误差却引发了一些争论Iorio, L., On the reliability of the so far performed tests for measuring the Lense-Thirring effect with the LAGEOS satellites, New Astron., 10, 603-615, 2005.Ciufolini, I., and Pavlis, E.C., On the Measurement of the Lense-Thirring effect Using the Nodes of the LAGEOS Satellites in reply to "On the reliability of the so-far performed tests for measuring the Lense-Thirring effect with the LAGEOS satellites" by L. Iorio, New Astron., 10, 636-651, 2005.Lucchesi, D.M., The impact of the even zonal harmonics secular variations on the Lense-Thirring effect measurement with the two Lageos satellites, Int. J. of Mod. Phys. D, 14, 1989-2023, 2005.Iorio, L., A critical analysis of a recent test of the Lense-Thirring effect with the LAGEOS satellites, J. of Geodesy, 80, 128-136, 2006.Iorio, L., An assessment of the measurement of the Lense-Thirring effect in the Earth gravity field, in reply to: ``On the measurement of the Lense-Thirring effect using the nodes of the LAGEOS satellites, in reply to ``On the reliability of the so far performed tests for measuring the Lense-Thirring effect with the LAGEOS satellites" by L. Iorio," by I. Ciufolini and E. Pavlis, Planet. Space Sci., 55, 503-511, 2007.Iorio, L., On some critical issues of the LAGEOS/LAGEOS II Lense-Thirring experiment,  2007. (Preprint http://arxiv.org/abs/0710.1022 ).。另一个在火星引力场中观测冷澤-提爾苓效应的测试结果是根据对火星全球探勘者号（）的位置数据进行恰当分析后得到的Iorio, L., A note on the evidence of the gravitomagnetic field of Mars, Class. Quantum Grav., 23, 5451-5454, 2006.，不过这个结果也引发了争议Iorio, L., Testing frame-dragging 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. 177-187, 2007. (Preprint http://www.arxiv.org/abs/gr-qc/0701042 ), 2007.Krogh, K., Comment on 'Evidence of the gravitomagnetic field of Mars',  Class. Quantum Grav.,  24 , 5709-5715, 2007.Iorio, L., Reply to "Iorio's "high-precision measurement" of frame dragging with the Mars Global Surveyor", by Kris Krogh, J. Graviit. Phys., at press, 2008 (Preprint http://www.arxiv.org/abs/gr-qc/0701146 ).。此外还有观测太阳自转对太阳系内行星轨道产生的冷澤-提爾苓效应的尝试Iorio, L., First preliminary tests of the general relativistic gravitomagnetic field of the Sun and new constraints on a Yukawa-like fifth force from planetary data, Planet. Space Sci., 55, 1290-1298, 2007.，广义相对论对冷澤-提爾苓进动的预言和观测到的行星近日点进动的修正相符合Pitjeva, E.V., Relativistic Effects and Solar Oblateness from Radar Observations of Planets and Spacecraft. Astron. Lett., 31, 340-349, 2005.，但误差依然比较大；以及对环绕木星的伽利略卫星的观测Iorio, L., and Lainey, V., The Lense-Thirring effect in the Jovian system of the Galilean satellites and its measurability, Int. J. Mod. Phys. D, 14, 2039-2050,  2005.，这同时也是最初由冷澤和提爾苓提出的实验建议。

引力探测器BEveritt, C.W.F, 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. 331-360, 1974. Reprinted in: Ruffini, R.J., Sigismondi, C. (Eds.), Nonlinear Gravitodynamics. The Lense-Thirring Effect. World Scientific, Singapore, pp. 439-468, 2003.Everitt, C.W.F.,  et al., Gravity Probe B: Countdown to Launch. In: Laemmerzahl, C., Everitt, C.W.F., Hehl, F.W. (Eds.), Gyros, Clocks, Interferometers...: Testing Relativistic Gravity in Space. Springer, Berlin,  pp. 52-82, 2001.正在对另一种引力磁学的效应即希夫进动进行实验观测Pugh, G.E., Proposal for a Satellite Test of the Coriolis Prediction of General Relativity, WSEG, Research Memorandum No. 11, 1959. Reprinted in:   Ruffini, R.J., Sigismondi, C. (Eds.), Nonlinear Gravitodynamics. The Lense-Thirring Effect. World Scientific, Singapore, pp. 414-426, 2003.Schiff, L., On Experimental Tests of the General Theory of Relativity, Am. J. of Phys., 28, 340-343, 1960.，并期望达到1%或更好的误差。然而目前来看这个目标还是个奢望，2007年4月的最初结果表明现在能达到的误差范围在256-128%之间Muhlfelder, B., Mac Keiser, G.,  and Turneaure, J., Gravity Probe B Experiment Error, poster L1.00027 presented at the American Physical Society (APS) meeting in Jacksonville, Florida, on 14-17 April 2007, 2007.，并期望在2007年12月时能够缩小至13%StanfordNews 4/14/07, downloadable at http://einstein.stanford.edu/ 。2005年物理学家艾奥里奥（）指出Iorio, L., The impact of the new Earth gravity models on the measurement of the Lense–Thirring effect with a new satellite, New Astron., 10 , 616-635, 2005.，如要在地球引力场中观测冷澤-提爾苓效应达到1%的误差水平，还需要发射至少两个新的卫星，并且最好能够采用偏心率很高的轨道以及一些能够抵消非引力微扰的动态机制。最近，意大利航天局（）宣布将于2008年底使用运载火箭发射卫星，其目标是将地球引力场中冷澤-提爾苓效应的测量达到1%的误差精度，但有人对这一计划能否成功表示怀疑Iorio, L., LARES approved: towards a 1% measurement of frame dragging?, (Preprint http://arxiv.org/abs/0802.2031 )Iorio, L., Perspectives in measuring frame-dragging after the approval of the LARES mission, (Preprint http://arxiv.org/abs/0803.3278 )。近来还有墨菲等人报告了一个通过月球激光测距对引力磁性的间接观测实验，并声称其误差小于0.1%Murphy, T.W., Nordtvedt, K., and Turyshev, S.G., The Gravitomagnetic Influence on Gyroscopes and on the Lunar Orbit, Phys. Rev. Lett., 98, 071102, 2007.；但物理学家科佩金（）对月球激光测距技术能否感应到引力磁性的存在表示疑问Kopeikin, Comment on "The gravitomagnetic influence on gyroscopes and on the lunar orbit",Phys. Rev. Lett.,  98, 229001, 2007.。

 天体物理学的证据 

相对论性喷流或许可以为参考系拖拽的存在提供直接证据：在一个旋转黑洞的能层内由冷澤-提爾苓效应产生的引力磁力Williams, R. K. (1995, May 15).  Extracting X rays, Ύ rays, and relativistic e-e+ pairs from supermassive Kerr black holes using the Penrose mechanism.  Physical Review, 51(10), 5387-5427.Williams, R. K. (2004, August 20).  Collimated escaping vortical polar e-e+ jets intrinsically produced by rotating black holes and Penrose processes.  The Astrophysical Journal, 611, 952-963.和由罗杰·彭罗斯爵士提出的能量抽取机制Penrose, R. (1969).  Gravitational collapse: The role of general relativity.  Nuovo Cimento Rivista, Numero Speciale 1, 252-276.一起可以被用来解释对相对论性喷流观测到的性质。由雷瓦·凯·威廉姆斯发展的引力磁学模型预言了已被观测到的由类星体和活动星系核发射的高能粒子（～吉电子伏特）、X射线和伽玛射线的光子抽取、平行于极轴的相对论性喷流以及喷流相对于轨道平面的非对称结构等现象的存在。

 参考系拖拽的数学推导 

描述参考系拖拽的最直接方法是使用克尔度规，克尔度规描述了一个质量为M，具有角动量J的旋转黑洞周围的时空几何。

这里rs是史瓦西半径。

度规中其他变量的意义为

在非相对论极限下，当质量M (或等效地，rs)趋于零时，克尔度规成为扁球面坐标系下的正交度规形式：

我们可以将它重新写作如下形式

这个度规和一个以角速度Ω旋转的坐标系等价，其中角速度Ω和半径r及余纬θ有关：

在赤道面上可以简化为：

从而我们看到，一个处于旋转状态的中心质量能够导致周围惯性参考系的转动，这就是参考系拖拽。


一个旋转黑洞的能层内的参考系拖拽效应是一个极端的版本。从数学形式上看，克尔度规存在有两个奇性的面，内表面对应着一个和史瓦西度规类似的球面视界：

在这个半径上度规的纯径向分量grr趋于无穷大。

奇性的外表面不是一个球面，而是一个扁球面。这个扁球面在其旋转轴的极点处和内表面相重合，这时的余纬度θ等于0或π；而其半径由下式给出：

在这个半径上度规的纯时间分量gtt由正变为负，内外奇性表面所包围的空间被称作能层。一般来说，沿世界线运动的运动粒子所经历的固有时总是正的，但在时间分量gtt为负的能层中这无法做到，除非这个粒子同时也以不低于Ω的角速度围绕中心质量M旋转。当然，参考系拖拽效应发生在任何旋转质量周围的任何位置，而不仅仅在能层中。

參考系拖曳力的公式
根據Lev Laudau The classic theory of fields 

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

而旋轉參考系拖曳能量公式為

 参见 

广义相对论的实验验证
测地线效应
引力磁学
克尔度规
相对论性喷流
冷澤-提爾苓進動
 重力磁性
 德西特進動

 参考文献 

 外部链接 

NASA RELEASE: 04-351 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 Mars-MGS test
Paper by G. Felici about the Mars-MGS test 
Paper by Kris Krogh about the Mars-MGS 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 X-ray 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 re-analysis 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:广义相对论
Category:参考系
Category:物理现象