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Albert Einstein's theory of general relativity predicts that rotating bodies drag spacetime 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 physicists Joseph Lense and Hans Thirring, and is also known as the 'Lense-Thirring effect'.^[1][2][3] 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, in order to detect it, you have to look at something very massive, or build an instrument that is incredibly sensitive. More generally, the subject of field effects caused by moving matter is known as gravitomagnetism.

Contents

Frame dragging effects

Experimental tests of frame-dragging

Astronomical evidence

See also

References

Mathematical derivation of frame-dragging

External links

Frame dragging effects

'Rotational frame-dragging' (the Lense-Thirring effect) appears in the general 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 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).^[4]

Experimental tests of frame-dragging

In 1976 Van Patten and Everitt^[5][6] 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^[7] 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^[8][9], 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^[10][11][12], 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^[13][14], but such a strategy has been shown to be unfeasible^[15][16][17].
Limiting 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^[18][19]. Tests have started to be effectively performed by using the LAGEOS and LAGEOS II satellites in 1996^[20], according to a strategy^[21] involving the use of a suitable combination of the nodes of both satellites and the perigee of LAGEOS II. The latest tests with the LAGEOS satellites have been performed in 2004-2006^[22][23] by discarding the perigee of LAGEOS II and using a linear combination^{[24][25][26][27]} 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^{[28][29][30][31][32]}. 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^[33]. Also such a test raised a debate^{[34][35][36][37][38][39][40]}. 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^[41]: the predictions of general relativity are compatible with the estimated corrections to the perihelia precessions^[42], although the errors are still large. The system of the Galilean satellites of Jupiter was investigated as well^[43], following the original suggestion by Lense and Thirring.
The Gravity Probe B experiment^[44][45] is currently under way to experimentally measure another gravitomagentic effect, i.e. the Schiff precession of a gyroscope^[46][47], 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^[48] 256-128%, with the hope of reaching about 13% in December 2007^[49].
A 1% measurement of the Lense-Thirring effect in the gravitational field of the Earth
could 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 pointed out in 2005 by Iorio^[50].
Recently, an indirect test of the gravitomagnetic interaction accurate to 0.1% has been reported by Murphy et al^[51] with the Lunar Laser Ranging (LLR) technique, but Kopeikin^[52] questioned the ability of LLR to be sensible to gravitomagnetism.

Astronomical evidence

Relativistic jets may provide evidence for the reality of frame-dragging. Gravitomagnetic forces produced by the Lense-Thirring effect (frame dragging) within the ergosphere of rotating black holes^[53][54] combined with the energy extraction mechanism by Sir Roger Penrose^[55][56] which describes the geometry of spacetime in the vicinity of a mass ''M'' rotating with angular momentum ''J''
:$$
c^{2} d au^{2} =
left( 1 - rac{r_{s} r}{
ho^{2}}
ight) c^{2} dt^{2}
- rac{
ho^{2}}{Lambda^{2}} dr^{2}
-
ho^{2} d heta^{2}
- left( r^{2} + a^{2} + rac{r_{s} r lpha^{2}}{
ho^{2}} sin^{2} heta
ight) sin^{2} heta dphi^{2}
+ rac{2r_{s} rlpha}{
ho^{2}} dphi dt

where ''r''_''s'' is the Schwarzschild radius
:$$
r_{s} = rac{2GM}{c^{2}}

and where the following shorthand variables have been introduced for brevity
:$$
lpha = rac{J}{Mc}

:$$
ho^{2} = r^{2} + lpha^{2} cos^{2} heta

:$$
Lambda^{2} = r^{2} - r_{s} r + lpha^{2}

In the non-relativistic limit where ''M'' (or, equivalently, ''r''_''s'') goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates
:$$
c^{2} d au^{2} =
c^{2} dt^{2}
- rac{
ho^{2}}{r^{2} + lpha^{2}} dr^{2}
-
ho^{2} d heta^{2}
- left( r^{2} + lpha^{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} - rac{g_{tphi}^{2}}{g_{phiphi}}
ight) dt^{2}
+ g_{rr} dr^{2} + g_{ heta heta} d heta^{2} +
g_{phiphi} left( dphi + rac{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 = -rac{g_{tphi}}{g_{phiphi}} = rac{r_{s} lpha r}{
ho^{2} left( r^{2} + lpha^{2}
ight) + r_{s} lpha^{2} r sin^{2} heta}

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 rotating black hole. The Kerr metric has two surfaces on which it appears to be singular. The inner surface corresponds to a spherical event horizon similar to that observed in the Schwarzschild metric; this occurs at
:$$
r_{inner} = rac{r_{s} + sqrt{r_{s}^{2} - 4lpha^{2}}}{2}

where the purely radial component ''g_rr'' of the metric goes to infinity. The outer surface is not a sphere, but an oblate 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} = rac{r_{s} + sqrt{r_{s}^{2} - 4lpha^{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 the ergosphere. A moving particle experiences a positive proper time along its worldline, its path through spacetime. 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.

See also

★ Kerr metric

★ Geodetic effect

★ Gravitomagnetism

★ Mach's principle

★ Broad Iron K line

★ Relativistic jet

References

1. 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]
2. 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"]
3. 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]
4. Einstein, A ''The Meaning of Relativity'' (contains transcripts of his 1921 Princeton lectures).
5. 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.
6. 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.
7. Ciufolini I., Measurement of Lense-Thirring Drag on High-Altitude Laser-Ranged Artificial Satellites, ''Phys. Rev. Lett.'', '56', 278-281, 1986.
8. 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.
9. Iorio, L., Lucchesi, D.M., and Ciufolini, I., The LARES Mission Revisited: An Alternative Scenario, ''Class. Quantum Grav.'', '19', 4311-4325, 2002.
10. 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.
11. Iorio, L., On a new observable for measuring the Lense-Thirring effect with Satellite Laser Ranging, ''Gen. Relativ. Gravit.'', '35', 1583-1595, 2003.
12. 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.
13. 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.
14. Ciufolini, I., On the orbit of the LARES satellite, (Preprint http://www.arxiv.org/abs/gr-qc/0609081), 2006.
15. 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.
16. Iorio, L., A critical approach to the concept of a polar, low-altitude LARES satellite, ''Class. Quantum Grav.'', '19', L175-L183, 2002.
17. Iorio, L., A comment on the paper "On the orbit of the LARES satellite", by I. Ciufolini, ''Planet. Space Sci.'', at press, (Preprint http://www.arxiv.org/abs/gr-qc/0609097) 2007.
18. 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.
19. Cugusi, L., Proverbio, E., Relativistic Effects on the Motion of Earth's Artificial Satellites, ''Astron. Astrophys'', '69', 321-325, 1978.
20. 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.
21. Ciufolini, I., On a new method to measure the gravitomagnetic field using two orbiting satellites., ''Il Nuovo Cimento A'', '109', 1709-1720, 1996.
22. Ciufolini, I., and Pavlis, E.C., A confirmation of the general relativistic prediction of the Lense-Thirring effect, ''Nature'', '431', 958-960, 2004
23. Ciufolini, 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.
24. 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.
25. 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).
26. 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.
27. 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).
28. 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.
29. 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.
30. 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.
31. 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.
32. 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.
33. Iorio, L., A note on the evidence of the gravitomagnetic field of Mars, ''Class. Quantum Grav.'', '23', 5451-5454, 2006.
34. Iorio, L., Testing frame-dragging with the Mars Global Surveyor spacecraft in the gravitational field of Mars, (Preprint http://www.arxiv.org/abs/gr-qc/0701042), 2007.
35. Krogh, K., Iorio's "high-precision measurement" of frame-dragging with the Mars Global Surveyor, (Preprint http://www.arxiv.org/abs/astro-ph/0701653), 2007.
36. Iorio, L., Reply to "Iorio's "high-precision measurement" of frame dragging with the Mars Global Surveyor", by Kris Krogh, (Preprint http://www.arxiv.org/abs/gr-qc/0701146), 2007.
37. Sindoni, G., Paris, C., Ialongo, P., On the Systematic Errors in the Detection of the Lense-Thirring Effect with a Mars Orbiter, (http://www.arxiv.org/abs/gr-qc/0701141), 2007.
38. Iorio, L., Reply to "On the Systematic Errors in the Detection of the Lense-Thirring Effect with a Mars Orbiter", by Giampiero Sindoni, Claudio Paris and Paolo Ialongo, (Preprint http://www.arxiv.org/abs/gr-qc/0701159), 2007.
39. Felici, G., The meaning of systematic errors, a comment to "Reply to On the Systematic Errors in the Detection of the Lense-Thirring Effect with a Mars Orbiter", by Lorenzo Iorio, (Preprint http://www.arxiv.org/abs/gr-qc/0703020), 2007.
40. Iorio, L., Reply to "The meaning of systematic errors, a comment to "Reply to On the Systematic Errors in the Detection of the Lense-Thirring Effect with a Mars Orbiter", by Lorenzo Iorio", by G. Felici, (Preprint http://www.arxiv.org/abs/gr-qc/0703042), 2007.
41. 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.'', doi:10.1016/j.pss.2007.04.001, 2007.
42. Pitjeva, E.V., Relativistic Effects and Solar Oblateness from Radar Observations of Planets and Spacecraft. ''Astron. Lett.'', '31', 340-349, 2005.
43. 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.
44. 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.
45. 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.
46. 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.
47. Schiff, L., On Experimental Tests of the General Theory of Relativity, ''Am. J. of Phys.'', '28', 340-343, 1960.
48. 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.
49. StanfordNews 4/14/07, downloadable at http://einstein.stanford.edu/
50. 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.
51. 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.

52. Kopeikin,Comment on "The gravitomagnetic influence on gyroscopes and on the lunar orbit", (Preprint http://arxiv.org/abs/gr-qc/0702120), 2007.
53. 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.
54. 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.
55. 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 of relativistic jets. The gravitomagnetic model developed by Reva Kay Williams predicts the observed high energy particles (~GeV) emitted by quasars and active 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, Gravitational field of a spinning mass as an example of algebraically special metrics, , RP, Kerr, Physical Review Letters, 1963
56. The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2), , LD, Landau, Pergamon Press, 1975,

External links

★ 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 misunderstandings of Iorio claims

★ Paper by G. Felici about the misunderstandings of Iorio claims

★ Paper by Kris Krogh about the misunderstandings of Iorio claims

★ Paper by Ignazio Ciufolini and Erricos Pavlis about the misunderstandings of Iorio claims

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