# Frame-dragging

Albert Einstein 's theory ofgeneral relativity predicts that rotating bodies dragspacetime around themselves in a phenomenon referred to as**frame-dragging**. The rotational frame-dragging effect was first derived from the theory of general relativity in 1918 by the Austrian physicistsJosef Lense andHans Thirring , and is also known as the**Lense-Thirring effect**. [*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"*] Lense and Thirring predicted that the rotation of an object would alter space and time, dragging a nearby object out of position compared to the predictions of Newtonian physics. The predicted effect is incredibly small — about one part in a few trillion. In order to detect it, it is necessary to look at a very massive object, or build an instrument that is incredibly sensitive. More generally, the subject of field effects caused by moving matter is known as**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]gravitomagnetism .**Frame dragging effects****Rotational frame-dragging**(the Lense-Thirring effect) appears in thegeneral principle of relativity and similar theories in the vicinity of rotating massive objects. Under the Lense-Thirring effect, the frame of reference in which a clock ticks the fastest is one which is rotating around the object as viewed by a distant observer. This also means that light traveling in the direction of rotation of the object will move around the object faster than light moving against the rotation as seen by a distant observer. It is now the best-known effect, partly thanks to theGravity Probe B experiment.**Linear frame dragging**is the similarly inevitable result of the general principle of relativity, applied to linear momentum. Although it arguably has equal theoretical legitimacy to the "rotational" effect, the difficulty of obtaining an experimental verification of the effect means that it receives much less discussion and is often omitted from articles on frame-dragging (but see Einstein, 1921). [*Einstein, A "The Meaning of Relativity" (contains transcripts of his 1921 Princeton lectures).*]**Static mass increase**is a third effect noted by Einstein in the same paper. [*Einstein, A "The Meaning of Relativity", pp95-96, Chapman and Hall, 1987*] The effect is an increase in inertia of a body when other masses are placed nearby. While not strictly a frame dragging effect (the term frame dragging is not used by Einstein), it is demonstrated by Einstein to derive from the same equation of general relativity. It is also a tiny effect that is difficult to confirm experimentally.**Experimental tests of frame-dragging**In 1976 Van Patten and Everitt [

*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.",*] proposed to implement a dedicated mission aimed to measure the Lense-Thirring node precession of a pair of counter-orbiting spacecraft to be placed in terrestrial polar orbits and endowed with drag-free apparatus. A somewhat equivalent, cheaper version of such an idea was put forth in 1986 by Ciufolini [**13**, 429-447, 1976.*Ciufolini I., Measurement of Lense-Thirring Drag on High-Altitude Laser-Ranged Artificial Satellites, "Phys. Rev. Lett.",*] who proposed to launch a passive, geodetic satellite in an orbit identical to that of the LAGEOS satellite, launched in 1976, apart from the orbital planes which should have been displaced by 180 deg apart: the so-called butterfly configuration. The measurable quantity was, in this case, the sum of the nodes of LAGEOS and of the new spacecraft, later named LAGEOS III, LARES, WEBER-SAT. Although extensively studied by various groups, [**56**, 278-281, 1986.*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.",*] such an idea has not yet been implemented. The butterfly configuration would allow, in principle, to measure not only the sum of the nodes but also the difference of the perigees, [**19**, 4311-4325, 2002.*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.",*] although such Keplerian orbital elements are more affected by the non-gravitational perturbations like the direct solar radiation pressure: the use of the active, drag-free technology would be required. Other proposed approaches involved the use of a single satellite to be placed in near polar orbit of low altitude, [**20**, 2477-2490, 2003.*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.*] but such a strategy has been shown to be unfeasible. [*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.",*] In order to enhance the possibilities of being implemented, it has been recently claimed that LARES/WEBER-SAT would be able to measure the effects [**55**, 1198-1200, 2007.*Ciufolini, I., LARES/WEBER-SAT, frame-dragging and fundamental physics, (Preprint http://arxiv.org/abs/gr-qc/0412001), 2004.*] induced by the multidimensional braneworld model by Dvali, Gabadaze and Porrati [*Dvali, G., Gabadadze, G., Porrati, M., 4D Gravity on a Brane in 5D Minkowski Space, "Phys. lett. B",*] and to improve by two orders of magnitude the present-day level of accuracy of the equivalence principle. [**485**, 208-214, 2000.*Ciufolini, I., Frame Dragging and Lense-Thirring Effect, "Gen. Relativ. Gravit.",*] Such claims have been shown to be highly unrealistic. [**36**, 2257-2270, 2004.*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 [*http://arxiv.org/abs/0706.1930 this*] preprintLimiting ourselves to the scenarios involving existing orbiting bodies, the first proposal to use the LAGEOS satellite and the Satellite Laser Ranging (SLR) technique to measure the Lense-Thirring effect dates back to 1977-1978. [

*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",*] Tests have started to be effectively performed by using the**69**, 321-325, 1978.LAGEOS andLAGEOS II satellites in 1996, [*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",*] according to a strategy [**109**, 575-590, 1996.*Ciufolini, I., On a new method to measure the gravitomagnetic field using two orbiting satellites., "Il Nuovo Cimento A",*] involving the use of a suitable combination of the nodes of both satellites and the perigee of**109**, 1709-1720, 1996.LAGEOS II . The latest tests with theLAGEOS satellites have been performed in 2004-2006 [*Ciufolini, I., and Pavlis, E.C., A confirmation of the general relativistic prediction of the Lense-Thirring effect, "Nature",*] [**431**, 958-960, 2004*Ciufolini, I., Pavlis, E.C., and Peron, R., Determination of frame-dragging using Earth gravity models from CHAMP and GRACE, "New Astron.",*] by discarding the perigee of**11**, 527-550, 2006.LAGEOS II and using a linear combination [*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).*] involving only the nodes of both the spacecraft.Although the predictions of general relativity are compatible with the experimental results, the realistic evaluation of the total error raised a debate. [*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).*] Another test of the Lense-Thirring effect in the gravitational field of Mars, performed by suitably interpreting the data of the Mars Global Surveyor (MGS) spacecraft, has been recently reported. [*Iorio, L., A note on the evidence of the gravitomagnetic field of Mars, "Class. Quantum Grav.",*] Also such a test raised a debate. [**23**, 5451-5454, 2006.*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).*] Attempts to detect the Lense-Thirring effect induced by the Sun's rotation on the orbits of the inner planets of the Solar System have been reported as well: [*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.",*] the predictions of general relativity are compatible with the estimated corrections to the perihelia precessions, [**55**, 1290-1298, 2007.*Pitjeva, E.V., Relativistic Effects and Solar Oblateness from Radar Observations of Planets and Spacecraft. "Astron. Lett.",*] although the errors are still large. However, the inclusion of the radiometric data from the Magellan orbiter recently allowed Pitjeva to greatly improve the determination of the unmodelled precession of the perihelion of Venus. It amounts to -0.0004 +/- 0.0001 arcseconds/century, while the Lense-Thirring effect for the Venus' periehlion is just -0.0003 arcseconds/century [**31**, 340-349, 2005.*Iorio, L., Advances in the measurement of the Lense-Thirring effect with planetary motions in the field of the Sun (Preprint http://arxiv.org/abs/0807.0435).*] . The system of the Galilean satellites of Jupiter was investigated as well, [*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",*] following the original suggestion by Lense and Thirring.**14**, 2039-2050, 2005.The

Gravity Probe B experiment [*Everitt, 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.*] is currently under way to experimentally measure another gravitomagnetic effect, i.e. theSchiff precession of a gyroscope, [*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.",*] to an expected 1% accuracy or better. Unfortunately, it seems that such an ambitious goal will not be achieved: indeed, first preliminary results released in April 2007 point toward a so far obtained accuracy of [**28**, 340-343, 1960.*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.*] 256-128%, with the hope of reaching about 13% in December 2007. [*StanfordNews 4/14/07, downloadable at http://einstein.stanford.edu/*] A 1% measurement of the Lense-Thirring effect in the gravitational field of the Earthcould be obtained by launching at least two entirely new satellites, preferably endowed with active mechanisms of compensation of the non-gravitational forces, in rather eccentric orbits, as stated in 2005 by Iorio. [*Iorio, L., The impact of the new Earth gravity models on the measurement of the Lense–Thirring effect with a new satellite, "New Astron.",*] Recently, the Italian Space Agency (ASI) has announced that the LARES satellite will be launched with a VEGA rocket at the end of 2008 [**10**, 616-635, 2005.*http://www.asi.it/SiteEN/MotorSearchFullText.aspx?keyw=LARES*] . The goal of LARES is to measure the Lense-Thirring effect to 1%, but there are doubts that this can be achieved [*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)*] , mainly due to the relatively low-orbit which LARES should be inserted into bringing into play more "mismodelled even zonal harmonics"Clarifyme. That is,spherical harmonics of the Earth's gravitational field caused by mass concentrations (like mountains) can drag a satellite in a way which may be difficult to distinguish from frame-dragging. Recently, an indirect test of the gravitomagnetic interaction accurate to 0.1% has been reported by Murphy et al [*Murphy, T.W., Nordtvedt, K., and Turyshev, S.G., The Gravitomagnetic Influence on Gyroscopes and on the Lunar Orbit, "Phys. Rev. Lett.",*] with the Lunar Laser Ranging (**98**, 071102, 2007.LLR ) technique, but Kopeikin [*Kopeikin, Comment on "The gravitomagnetic influence on gyroscopes and on the lunar orbit","Phys. Rev. Lett.",*] questioned the ability of**98**, 229001, 2007.LLR to be sensible to gravitomagnetism.**Astronomical evidence**Relativistic jet s may provide evidence for the reality of frame-dragging.Gravitomagnetic forces produced by the Lense-Thirring effect (frame dragging) within theergosphere ofrotating black hole s [*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*] combined with the energy extraction mechanism by Sir^{-}e^{+}jets intrinsically produced by rotating black holes and Penrose processes. "The Astrophysical Journal",**611**, 952-963.Roger Penrose [*Penrose, R. (1969). Gravitational collapse: The role of general relativity. "Nuovo Cimento Rivista", Numero Speciale***1**, 252-276. have been used to explain the observed properties ofrelativistic jet s. The gravitomagnetic model developed by Reva Kay Williams predicts the observed high energy particles (~GeV) emitted byquasars andactive galactic nuclei ; the extraction of X-ray and γ-ray photons; the collimated jets about the polar axis; and the asymmetrical formation of jets (relative to the orbital plane).**Mathematical derivation of frame-dragging***Frame-dragging may be illustrated most readily using the*] [Kerr metric ,cite journal | last = Kerr | first = RP | authorlink = Roy Kerr | year = 1963 | title = [*http://prola.aps.org/abstract/PRL/v11/i5/p237_1 Gravitational field of a spinning mass as an example of algebraically special metrics*] | journal = Physical Review Letters | volume = 11 | pages = 237–238 | doi = 10.1103/PhysRevLett.11.237*cite book | last = Landau | first = LD | authorlink = Lev Landau | coauthors = Lifshitz, EM | year = 1975 | title = The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2) | edition = revised 4th English ed. | publisher = Pergamon Press | location = New York | isbn = 978-0-08-018176-9 |pages = pp. 321–330*] which describes the geometry ofspacetime in the vicinity of a mass "M" rotating withangular momentum "J":$c^\{2\}\; d\; au^\{2\}\; =\; left(\; 1\; -\; frac\{r\_\{s\}\; r\}\{\; ho^\{2\; ight)\; c^\{2\}\; dt^\{2\}\; -\; frac\{\; ho^\{2\{Lambda^\{2\; dr^\{2\}\; -\; ho^\{2\}\; d\; heta^\{2\}$::::$-\; left(\; r^\{2\}\; +\; alpha^\{2\}\; +\; frac\{r\_\{s\}\; r\; alpha^\{2\{\; ho^\{2\; sin^\{2\}\; heta\; ight)\; sin^\{2\}\; heta\; dphi^\{2\}\; +\; frac\{2r\_\{s\}\; ralpha\; c\; sin^\{2\}\; heta\; \}\{\; ho^\{2\; dphi\; dt$

where "r"

_{"s"}is the Schwarzschild radius:$r\_\{s\}\; =\; frac\{2GM\}\{c^\{2$

and where the following shorthand variables have been introduced for brevity

:$alpha\; =\; frac\{J\}\{Mc\}$

:$ho^\{2\}\; =\; r^\{2\}\; +\; alpha^\{2\}\; cos^\{2\}\; heta,!$

:$Lambda^\{2\}\; =\; r^\{2\}\; -\; r\_\{s\}\; r\; +\; alpha^\{2\},!$

In the non-relativistic limit where "M" (or, equivalently, "r"

_{"s"}) goes to zero, the Kerr metric becomes the orthogonal metric for theoblate spheroidal coordinates :$c^\{2\}\; d\; au^\{2\}\; =\; c^\{2\}\; dt^\{2\}\; -\; frac\{\; ho^\{2\{r^\{2\}\; +\; alpha^\{2\; dr^\{2\}\; -\; ho^\{2\}\; d\; heta^\{2\}-\; left(\; r^\{2\}\; +\; alpha^\{2\}\; ight)\; sin^\{2\}\; heta\; dphi^\{2\}$

We may re-write the Kerr metric in the following form

:$c^\{2\}\; d\; au^\{2\}\; =\; left(\; g\_\{tt\}\; -\; frac\{g\_\{tphi\}^\{2\{g\_\{phiphi\; ight)\; dt^\{2\}+\; g\_\{rr\}\; dr^\{2\}\; +\; g\_\{\; heta\; heta\}\; d\; heta^\{2\}\; +\; g\_\{phiphi\}\; left(\; dphi\; +\; frac\{g\_\{tphi\{g\_\{phiphi\; dt\; ight)^\{2\}$

This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius "r" and the

colatitude θ:$Omega\; =\; -frac\{g\_\{tphi\{g\_\{phiphi\; =\; frac\{r\_\{s\}\; alpha\; r\; c\}\{\; ho^\{2\}\; left(\; r^\{2\}\; +\; alpha^\{2\}\; ight)\; +\; r\_\{s\}\; alpha^\{2\}\; r\; sin^\{2\}\; heta\}$

In the plane of the equator this simplifies to: [

*Tartaglia, A, "Detection of the gravitometric clock effect", eq.17 p11, [*]*http://arxiv.org/abs/gr-qc/9909006v2 Arxiv preprint*] , 7 Feb 2008, retrieved 4 July 2008.:$Omega\; =\; frac\{r\_\{s\}\; alpha\; c\}\{r^\{3\}\; +\; alpha^\{2\}\; r\; +\; r\_\{s\}\; alpha^\{2$

Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging.

An extreme version of frame dragging occurs within the

ergosphere of a rotatingblack hole . The Kerr metric has two surfaces on which it appears to be singular. The inner surface corresponds to a sphericalevent horizon similar to that observed in theSchwarzschild metric ; this occurs at:$r\_\{inner\}\; =\; frac\{r\_\{s\}\; +\; sqrt\{r\_\{s\}^\{2\}\; -\; 4alpha^\{2\}\{2\}$

where the purely radial component "g

_{rr}" of the metric goes to infinity. The outer surface is not a sphere, but anoblate spheroid that touches the inner surface at the poles of the rotation axis, where the colatitude θ equals 0 or π; its radius is defined by the formula:$r\_\{outer\}\; =\; frac\{r\_\{s\}\; +\; sqrt\{r\_\{s\}^\{2\}\; -\; 4alpha^\{2\}\; cos^\{2\}\; heta\{2\}$

where the purely temporal component "g

_{tt}" of the metric changes sign from positive to negative. The space between these two surfaces is called theergosphere . A moving particle experiences a positiveproper time along itsworldline , its path throughspacetime . However, this is impossible within the ergosphere, where "g_{tt}" is negative, unless the particle is co-rotating with the interior mass "M" with an angular speed at least of Ω. However, as seen above, frame-dragging occurs about every rotating mass and at every radius "r" and colatitude θ, not only within the ergosphere.**Lense-Thirring effect inside a rotating shell**Inside a rotating spherical shell the acceleration due the Lense-Thirring effect would be http://philsci-archive.pitt.edu/archive/00002681/ Herbert Pfister. On the history of the so-called Lense-Thirring effect (2005)]

$ar\{a\}\; =\; -2d\_1\; left(\; ar\{\; omega\}\; imes\; ar\; v\; ight)\; -\; d\_2\; left\; [\; ar\{\; omega\}\; imes\; left(\; ar\{\; omega\}\; imes\; ar\{r\}\; ight)\; +\; 2left(\; ar\{\; omega\}ar\{r\}\; ight)\; ar\{\; omega\}\; ight]$

where the coefficients are

$d\_1\; =\; frac\{4MG\}\{3Rc^2\}$

$d\_2\; =\; frac\{4MG\}\{15Rc^2\}$

for MG<

$d\_1=\; frac\{4\; alpha(2-\; alpha)\}\{(1+\; alpha)(3-\; alpha)\},\; alpha=frac\{MG\}\{2Rc^2\}$ The space-time inside the rotating spherical shell will not be flat. To have flat space-time inside, the rotating sphere should have non-spherical shape [

*http://www.iop.org/EJ/abstract/0264-9381/2/6/015 H. Pfister et al 1985 Class. Quantum Grav. 2 909-918 doi: 10.1088/0264-9381/2/6/015*]**See also***

Kerr metric

*Geodetic effect

*Gravitomagnetism

*Mach's principle

*Broad Iron K line

*Relativistic jet

*Lense-Thirring precession **References****External links*** [

*http://www.nasa.gov/home/hqnews/2004/oct/HQ_04351_time_drags.html NASA RELEASE: 04-351 As The World Turns, It Drags Space And Time*]

* [*http://space.newscientist.com/article/mg19325874.800-loner-stakes-claim-to-gravity-prize.html "New Scientist" press release of the MGS test by Iorio in the gravitational field of Mars*]

* [*http://arxiv.org/abs/gr-qc/0701141 Paper by Giampiero Sindoni, Claudio Paris and Paolo Ialongo about the Mars-MGS test*]

* [*http://arxiv.org/abs/gr-qc/0703020 Paper by G. Felici about the Mars-MGS test*]

* [*http://arxiv.org/abs/astro-ph/0701653 Paper by Kris Krogh about the Mars-MGS test*]

* [*http://arxiv.org/abs/gr-qc/0601015 Reply by Ignazio Ciufolini and Erricos Pavlis about some criticisms by Iorio*]

* [*http://arxiv.org/abs/astro-ph/0210139v4 Frame dragging applied to relativistic jets*]

* [*http://www.rdrop.com/users/green/school/framdrag.htm Frame Dragging*]

* [*http://www.phy.duke.edu/~kolena/framedrag.html Duke University press release: General Relativistic Frame Dragging*]

* [*http://msnbc.msn.com/id/3077887/ MSNBC report on X-ray observations*]

* [*http://xxx.lanl.gov/abs/gr-qc/9704065 Ciufolini et al. LAGEOS paper 1997 - 25% error*]

* [*http://arxiv.org/abs/gr-qc/0209109 Ciufolini update Sep 2002 - 20% error*]

* [*http://www.phy.duke.edu/~kolena/framedrag.html Press release regarding LAGEOS study*]

* [*http://cddisa.gsfc.nasa.gov/lw13/lw_proceedings.html#science Preprint by Ries et al.*]

* [*http://www.nature.com/news/2004/041018/full/041018-11.html Ciufolini and Pavlis "Nature" new article on 2004 re-analysis of the LAGEOS data*]

* [*http://www.arxiv.org/abs/gr-qc/0411024 Iorio " New Astronomy" general paper with full references*]

* [*http://www.arxiv.org/abs/gr-qc/0412057 Iorio "J. of Geodesy" paper on the impact of the secular variations of the even zonal harmonics of the geopotential*]

* [*http://www.arxiv.org/abs/gr-qc/0608119 Iorio "Planetary Space Science" paper*]

* [*http://www.rdrop.com/users/green/school/index.htm "The Naked Singularity"*] "An early version of this article was adapted from public domain material from http://science.msfc.nasa.gov/newhome/headlines/ast06nov97_1.htm "

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