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According to Albert Einstein's theory of general relativity, spacetime (space and time) get pulled out of shape near a forcibly-accelerated or rotating body 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 physicists Joseph Lense and Hans Thirring, and is also known as the Lense-Thirring effect. More generally, the subject of field effects caused by moving matter is known as gravitomagnetism.

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. This is the frame-dragging effect. The predicted effect is incredibly small — about one part in a few trillion — which means that you have to look at something very massive, or build an instrument that is incredibly sensitive.

More familiar and already-proven effects of special relativity include the equivalence of mass and energy (as seen in matter-antimatter reactions), and the Lorentz transformations which make objects near lightspeed seem to grow shorter and heavier from the point of view of an outside observer. Recent measurements of satellites in Earth orbit have shown frame dragging and represented another successful prediction of General Relativity.

Frame dragging effects

Rotational frame-dragging (Lense-Thirring effect) is the inevitable result of the general principle of relativity, applied to rotation. The relativisation of rotational effects means that a rotating body ought to pull light around with it, in a manner somewhat reminiscent of old "aether-dragging" models. It is now the best-known effect, partly thanks to the Gravity Probe B experiment.
Accelerational frame dragging is the similarly inevitable result of the general principle of relativity, applied to acceleration. 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).
Velocity frame dragging is even less well known and much more controversial. Its effects (or similar effects) do seem to appear in general relativity, but it is not usually listed as a "valid" example of frame-dragging. The reasons for this are complicated.
More complex effects can be constructed from these basic building blocks, to produce a variety of effects involving combinations of rotations and accelerations, time-variant acceleration ("jerk") and more complex rotations.

Using recent observations by X-ray astronomy satellites, including NASA's Rossi X-ray Timing Explorer, a team of astronomers announced in 1997 that they had seen evidence of frame-dragging in disks of gas swirling around a black hole. The team included Dr. Wei Cui of the Massachusetts Institute of Technology, and his colleagues, Dr. Nan Zhang, working at NASA's Marshall Space Flight Center, and Dr. Wan Chen of the University of Maryland in College Park.

Experimental test of the gravitomagnetic Schiff effect

The gyroscope-based Gravity Probe B experiment aims to detect any frame-dragging effects on the direction of spin of its gyroscopes as it orbits around the Earth. It was successfully launched on April 20, 2004 for an 18-month experiment. If this experiment is successful, it is expected to yield the most accurate measurements yet performed in this field. Indeed, an accuracy of better than 1% is expected.

Experimental tests of the Lense-Thirring effect in the gravitational fields of the Earth, Mars and the Sun

Another consequence of the gravitomagnetic field of a central rotating body is the so-called Lense-Thirring effect (Lense and Thirring 1918). It consists of small secular precessions of the longitude of the ascending node $\Omega$ and the argument of pericenter $\omega$ of the path of a test mass freely orbiting the spinning main body. de Sitter (1916) worked out the gravitomagnetic pericentre precession in the particular case of equatorial orbits. Lense and Thirring (1918) originally proposed to use the natural satellites of the gaseous giant planets of the Solar System, especially Jupiter, to detect their effect, but such a possibility is not yet viable today (Iorio and Lainey 2005). In regard to the Earth's gravitational field, Cugusi and Proverbio (1978) proposed for the first time to use the LAGEOS satellite, just launched at that time, along with the other existing terrestrial artifical satellites to measure the Lense-Thirring effect with the Satellite Laser Ranging (SLR) technique. For the nodes of the LAGEOS and LAGEOS II satellites the Lense-Thirring node rates amount to ~30 milliarcseconds per year (ms/ ${yr}$ or ms ${yr}^{-1}$ ). Such tiny precessions would totally be swamped by the much larger classical precessions induced by the even zonal harmonic coefficients $J_{\ell },\ \ell =2,4,6...$ of the multipolar expansion of the Newtonian part of the terrestrial gravitational potential. Even the most recent Earth gravity models from the dedicated CHAMP and GRACE missions would not allow to know the even zonal harmonics to a sufficiently high degree of accuracy in order to extract the Lense-Thirring effect from the analysis of the node of only one satellite.

Ciufolini (1996) proposed to overcome this problem by suitably combining the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II in order to cancel out all the static and time-dependent perturbations due to the first two even zonal harmonics $J_{2},\ J_{4}$ (Ciufolini 1996). Various analyses with the pre-CHAMP/GRACE JGM-3 and EGM96 Earth gravity models were performed by Ciufolini et al. over observational time spans of some years (Ciufolini et al. 1996; 1997; 1998). The claimed total accuracies were in the range of 20-25% (Ciufolini 2004). However, subsequent analyses by Ries et al. (2003a; 2003b) and Iorio (2003) showed that such estimates are largely optimistic. Indeed, a more conservative and realistic evaluation of the impact of the uncancelled even zonal harmonics $J_{6},\ J_{8},\ J_{10},...$ , according to the adopted EGM96 model, yield a systematic error of about 80% at 1-sigma level. Moreover, also the systematic error due to the non-gravitational perturbations mainly affecting the perigee of LAGEOS II was underestimated.

The opportunities offered by the new generation of Earth gravity models from CHAMP and, especially, GRACE allowed to discard the perigee of LAGEOS II, as pointed out by Ries et al. (2003a; 2003b). In 2003 Iorio, following the strategy put forth by Ciufolini (1996), put explicitly forth a suitable linear combination of the nodes of LAGEOS and LAGEOS II which cancels out the first even zonal harmonic $J_{2}$ (Iorio and Morea 2004). Such an observable was used by Ciufolini and Pavlis in a test performed with the 2nd generation GRACE-only EIGEN-GRACE02S Earth gravity model over a time span of 11 years (Ciufolini and Pavlis 2004). The claimed total error budget is 5% at 1-sigma level and 10% at 3-sigma level. However, Iorio (2005a; 2005b; 2006a; 2006b) criticized such results because of the neglected impact of the secular variations of the uncancelled even zonal harmonics ${\dot {J}}_{4},\ {\dot {J}}_{6}$ which would amount to about 13%. This would yield a total error of $\sim$ 20% at 1-sigma level. Moreover, the latest CHAMP/GRACE-based Earth gravity models do not yet allow for a model-independent measurement. Indeed, the systematic error due to the static part of the even zonal harmonics amounts to 4% for EIGEN-GRACE02S, 6% for EIGEN-CG01C and 9% for GGM02S at 1-sigma level. Another potential source of additional systematic bias may be represented by the cross-coupling among $J_{2}$ and the residuals of the inclination $\delta i$ , as pointed out by Iorio (2006b). Other papers on such a long-lasting, sometimes harsh, controversy are (Ciufolini and Pavlis 2005; Lucchesi 2005).

By the way, such a controversial test has recently been superseded by an unexpected result in the gravitational field of Mars. Indeed, by suitably analyzing the residuals of the Mars Global Surveyor (MGS) spacecraft currently orbiting the red planet along a polar orbit, Iorio (2006c) reported a 6% measurement, on average, of the Lense-Thirring effect. A further analysis (Iorio 2007), based on a more detailed calculation, extended time span and error analysis, has pushed the precision level of such a test below the 1% level.

Other preliminary tests of the Lense-Thirring effect induced by the Sun's gravitomagnetic field on the orbital motions of the inner planets of the Solar System can be found in (Iorio 2005c). The predictions of general relativity for the Lense-Thirring perihelion precessions are, in fact, in agreement with the latest determinations of the extra-perihelion advances of the inner planets (Pitjeva 2005a) obtained with the EPM2004 ephemerides (Pitjeva 2005b), but the errors are still large.

A final remark to the interested readers: rather surprisingly (at least, for the non-experts in the field...), in previous versions of such an article all the references of a researcher in particular have systematically and completely been censored. They refer to the criticisms to the recent test performed with the LAGEOS satellites, and to the Mars and Sun tests. It is likely that the experts of this field are sonorously laughing at this ridicolous, childish and really paranoid actions, but the non-expert readers may suffer a damage from such an unbelievable attitude which has nothing to do with the serious and genuine scientific practice. Thus, we strongly advise to carefully inspect all the relevant literature so to form an unbiased and autonomous judgement. In the meantime, we gladly note that the paranoid and ridicolous censor showed some encouraging signs of repentance; is it merit of some successful psychiatric therapy? We all hope he will continue along the route of the serious and fair scientific practice.

References

I. Ciufolini. On a new method to measure the gravitomagnetic field using two orbiting satellites. Il Nuovo Cimento A, 109, 1709-1720, (1996).
I. Ciufolini, D. Lucchesi, F. Vespe, and A. Mandiello, Measurement of Dragging of Inertial Frames and Gravitomagnetic Field Using Laser-Ranged Satellites, Il Nuovo Cimento A 109 575-590, (1996).
I. Ciufolini, F. Chieppa, D. Lucchesi, and F. Vespe. Test of Lense-Thirring orbital shift due to spin. Classical and Quantum Gravity 14, 2701-2726, (1997).
I. Ciufolini, E.C. Pavlis, F. Chieppa, E. Fernandes-Vieira, and J. Perez-Mercader, J. Test of general relativity and measurement of the Lense-Thirring effect with two Earth satellites. Science 279, 2100-2103, (1998).
I. Ciufolini. Frame Dragging and Lense-Thirring Effect, General Relativity and Gravitation 36, 2257-2270, (2004).
I. Ciufolini, E. C. Pavlis. A confirmation of the general relativistic prediction of the Lense – Thirring effect. Nature 431, 958 - 960 (21 October 2004)
I. Ciufolini, E. C. Pavlis. 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 Astronomy 10 636-651, (2005).
L. Cugusi, and E. Proverbio Relativistic Effects on the Motion of Earth's Artificial Satellites. Astronomy and Astrophysics 69 321-325 (1978).
W. de Sitter, Einstein's theory of gravitation and its astronomical consequences, Monthly Notices of the Royal Astronomical Society 76 699-728, (1916).
Einstein, A The Meaning of Relativity (contains transcripts of his 1921 Princeton lectures).
L. Iorio. The impact of the static part of the Earth's gravity field on some tests of General Relativity with Satellite Laser Ranging. Celestial Mechanics and Dynamical Astronomy 86 277-294, (2003).
L. Iorio and A. Morea. The impact of the new Earth gravity models on the measurement of the Lense-Thirring effect. General Relativity and Gravitation 36, 1321-1333, (2004). Preprint [1].
L. Iorio. On the reliability of the so-far performed tests for measuring the Lense – Thirring effect with the LAGEOS satellites. New Astronomy 10 603-615, (2005a).[2]
L. Iorio. The impact of the new Earth gravity models on the measurement of the Lense-Thirring effect with a new satellite. New Astronomy 10 616-635, (2005b).[3]
L. Iorio. First preliminary evidence of the general relativistic gravitomagnetic field of the Sun and new constraints on a Yukawa-like fifth force. (2005c). [4]
L. Iorio. A critical analysis of a recent test of the Lense-Thirring effect with the LAGEOS satellites. Journal of Geodesy 80 128-136, (2006a). [5]
L. Iorio. 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. Planetary Space Science doi: 10.1016/j.pss.2006.08.001 (2006b).[6]
L. Iorio. A note on the evidence of the gravitomagnetic field of Mars. Classical Quantum Gravity 23, 5451-5454 (2006c). [7]
L. Iorio. High-precision measurement of frame-dragging with the Mars Global Surveyor spacecraft in the gravitational field of Mars. (2007). [8]
L. Iorio, and V. Lainey, The Lense-Thirring Effect in the Jovian System of the Galilean Satellites and its Measurability, International Journal of Modern Physics D 14 2039-2049, (2005).[9]
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]
D. Lucchesi. The impact of the even zonal harmonics secular variations on the Lense-Thirring effect measurement with the two Lageos satellites. International Journal of Modern Physics D 14 1989-2023, (2005).
E.V. Pitjeva, Relativistic Effects and Solar Oblateness from Radar Observations of Planets and Spacecraft, Astronomy Letters 31, 340-349 (2005a).
E.V. Pitjeva, High-Precision Ephemerides of Planets—EPM and Determination of Some Astronomical Constants, Solar System Research 39, 176-186 (2005b).
J. C. Ries, R. J. Eanes and B. D. Tapley. Lense-Thirring Precession Determination from Laser Ranging to Artificial Satellites. Nonlinear Gravitodynamics ed. R. Ruffini and C. Sigismondi (World Scientific, Singapore, 2003a) pp. 201-211.
J. C. Ries, R. J. Eanes, B. D. Tapley and G. E. Peterson. Prospects for an Improved Lense-Thirring Test with SLR and the GRACE Gravity Mission. Proc. 13th Int. Laser Ranging Workshop NASA CP 2003-212248 ed. R. Noomen, S. Klosko, C. Noll and M. Pearlman. (NASA Goddard 2003b). Preprint [10]
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"]

External links

An early version of this article was adapted from public domain material from http://science.msfc.nasa.gov/newhome/headlines/ast06nov97_1.htm

@@ Line 30: / Line 30: @@
 Ciufolini (1996) proposed to overcome this problem by suitably combining the nodes of [[LAGEOS]] and LAGEOS II and the perigee of LAGEOS II in order to cancel out all the static and time-dependent perturbations due to the first two even zonal harmonics <math>J_2,\ J_4</math> (Ciufolini 1996). Various analyses with the pre-CHAMP/GRACE JGM-3 and EGM96 Earth gravity models were performed by Ciufolini et al. over observational time spans of some years (Ciufolini et al. 1996; 1997; 1998). The claimed total accuracies were in the range of 20-25% (Ciufolini 2004). However, subsequent analyses by Ries et al. (2003a; 2003b) and Iorio (2003) showed that such estimates are largely optimistic. Indeed, a more conservative and realistic evaluation of the impact of the uncancelled even zonal harmonics <math>J_6,\ J_8,\ J_{10},...</math>, according to the adopted EGM96 model, yield a systematic error of about 80% at 1-sigma level. Moreover, also the systematic error due to the non-gravitational perturbations mainly affecting the perigee of LAGEOS II was underestimated.
-The opportunities offered by the new generation of Earth gravity models from CHAMP and, especially, [[Gravity Recovery and Climate Experiment|GRACE]] allowed to discard the perigee of LAGEOS II, as pointed out by Ries et al. (2003a; 2003b). In 2003 Iorio put explicitly forth a suitable linear combination of the nodes of [[LAGEOS]] and LAGEOS II which cancels out the first even zonal harmonic <math>J_2</math> (Iorio and Morea 2004). Such an observable was used by Ciufolini and Pavlis in a test performed with the 2nd generation GRACE-only EIGEN-GRACE02S Earth gravity model over a time span of 11 years (Ciufolini and Pavlis 2004). The claimed total error budget is 5% at 1-sigma level and 10% at 3-sigma level. However, Iorio (2005a; 2005b; 2006a; 2006b) criticized such results because of the neglected impact of the secular variations of the uncancelled even zonal harmonics <math>\dot J_4,\ \dot J_6</math> which would amount to about 13%. This would yield a total error of <math>\sim</math> 20% at 1-sigma level. Moreover, the latest CHAMP/[[Gravity Recovery and Climate Experiment|GRACE]]-based Earth gravity models do not yet allow for a model-independent measurement. Indeed, the systematic error due to the static part of the even zonal harmonics amounts to 4% for EIGEN-GRACE02S, 6% for EIGEN-CG01C and 9% for GGM02S at 1-sigma level. Another potential source of additional systematic bias may be represented by the cross-coupling among   <math>J_2</math> and the residuals of the inclination <math>\delta i</math>, as pointed out by Iorio (2006b). Other papers on such a long-lasting, sometimes harsh, controversy are (Ciufolini and Pavlis 2005; Lucchesi 2005).
+The opportunities offered by the new generation of Earth gravity models from CHAMP and, especially, [[Gravity Recovery and Climate Experiment|GRACE]] allowed to discard the perigee of LAGEOS II, as pointed out by Ries et al. (2003a; 2003b). In 2003 Iorio, following the strategy put forth by Ciufolini (1996), put explicitly forth a suitable linear combination of the nodes of [[LAGEOS]] and LAGEOS II which cancels out the first even zonal harmonic <math>J_2</math> (Iorio and Morea 2004). Such an observable was used by Ciufolini and Pavlis in a test performed with the 2nd generation GRACE-only EIGEN-GRACE02S Earth gravity model over a time span of 11 years (Ciufolini and Pavlis 2004). The claimed total error budget is 5% at 1-sigma level and 10% at 3-sigma level. However, Iorio (2005a; 2005b; 2006a; 2006b) criticized such results because of the neglected impact of the secular variations of the uncancelled even zonal harmonics <math>\dot J_4,\ \dot J_6</math> which would amount to about 13%. This would yield a total error of <math>\sim</math> 20% at 1-sigma level. Moreover, the latest CHAMP/[[Gravity Recovery and Climate Experiment|GRACE]]-based Earth gravity models do not yet allow for a model-independent measurement. Indeed, the systematic error due to the static part of the even zonal harmonics amounts to 4% for EIGEN-GRACE02S, 6% for EIGEN-CG01C and 9% for GGM02S at 1-sigma level. Another potential source of additional systematic bias may be represented by the cross-coupling among   <math>J_2</math> and the residuals of the inclination <math>\delta i</math>, as pointed out by Iorio (2006b). Other papers on such a long-lasting, sometimes harsh, controversy are (Ciufolini and Pavlis 2005; Lucchesi 2005).
 By the way, such a controversial test has recently been superseded by an unexpected result in the gravitational field of Mars. Indeed, by suitably analyzing the residuals of the Mars Global Surveyor (MGS) spacecraft currently orbiting the red planet along a polar orbit, Iorio (2006c) reported a 6% measurement, on average, of the Lense-Thirring effect. A further analysis (Iorio 2007), based on a more detailed calculation, extended time span  and error analysis, has pushed the precision level of such a test below the 1% level.
@@ Line 36: / Line 36: @@
 Other preliminary tests of the Lense-Thirring effect induced by the Sun's gravitomagnetic field on the orbital motions of the inner planets of the Solar System can be found in (Iorio 2005c). The predictions of general relativity for the Lense-Thirring perihelion precessions are, in fact, in agreement with the latest determinations of the extra-perihelion advances of the inner planets (Pitjeva 2005a) obtained with the EPM2004 ephemerides (Pitjeva 2005b), but the errors are still large.
-A final remark to the interested readers: rather surprisingly (at least, for the non-experts in the field...), '''in previous versions of such an article all the references of a researcher in particular have systematically and completely been censored. They refer to the criticisms to the recent test performed with the LAGEOS satellites, and to the Mars and Sun tests.''' It is likely that the experts of this field are  sonorously laughing at this ridicolous, childish and really paranoid actions, but the non-expert readers may suffer a damage from such an unbelievable attitude which has nothing to do with the serious and genuine scientific practice and clearly. Thus, we strongly advise to carefully inspect all the relevant literature so to form an unbiased and autonomous judgement.
+A final remark to the interested readers: rather surprisingly (at least, for the non-experts in the field...), '''in previous versions of such an article all the references of a researcher in particular have systematically and completely been censored. They refer to the criticisms to the recent test performed with the LAGEOS satellites, and to the Mars and Sun tests.''' It is likely that the experts of this field are  sonorously laughing at this ridicolous, childish and really paranoid actions, but the non-expert readers may suffer a damage from such an unbelievable attitude which has nothing to do with the serious and genuine scientific practice. Thus, we strongly advise to carefully inspect all the relevant literature so to form an unbiased and autonomous judgement. '''In the meantime, we gladly note that the paranoid and ridicolous censor showed some encouraging signs of repentance; is it merit of some successful psychiatric therapy? We all hope he will continue along the route of the serious and fair scientific practice'''.
 == See also ==