Speed of gravityFrom Wikipedia, the free encyclopedia

In classical theories of gravitation, the speed of gravity is the speed at which changes in a gravitationalfield propagate. This is the speed at which a change in the distribution of energy and momentum ofmatter results in subsequent alteration, at a distance, of the gravitational field which it produces. In amore physically correct sense, the "speed of gravity" refers to the speed of a gravitational wave, whichshould be the same speed as the speed of light (c).

Contents

1 Introduction

2 Static fields

3 Newtonian gravitation

4 Laplace

5 Electrodynamical analogies

5.1 Early theories

5.2 Lorentz

6 Lorentz covariant models

7 General relativity

7.1 Background

7.2 Aberration of field direction in general relativity, for a weakly accelerated observer

7.3 Formulaic conventions

7.4 Possible experimental measurements

8 References

9 External links

Introduction

The speed of gravitational waves in the general theory of relativity is equal to the speed of light invacuum, c.[1] Within the theory of special relativity, the constant c is not exclusively about light; insteadit is the highest possible speed for any interaction in nature. Formally, c is a conversion factor forchanging the unit of time to the unit of space.[2] This makes it the only speed which does not depend

either on the motion of an observer or a source of light and/or gravity. Thus, the speed of "light" is alsothe speed of gravitational waves and any other massless particle. Such particles include the gluon(carrier of the strong force), the photons that make up light, and the theoretical gravitons which make upthe associated field particles of gravity (however a theory of the graviton requires a theory of quantumgravity).

Static fields

The speed of physical changes in a gravitational or electromagnetic field should not be confused with"changes" in the behavior of static fields that are due to pure observer­effects. These changes in directionof a static field, because of relativistic considerations, are the same for an observer when a distant chargeis moving, as when an observer (instead) decides to move with respect to a distant charge. Thus,constant motion of an observer with regard to a static charge and its extended static field (either agravitational or electric field) does not change the field. For static fields, such as the electrostatic fieldconnected with electric charge, or the gravitational field connected to a massive object, the field extendsto infinity, and does not propagate. Motion of an observer does not cause the direction of such a field tochange, and by symmetrical considerations, changing the observer frame so that the charge appears to bemoving at a constant rate, also does not cause the direction of its field to change, but requires that itcontinue to "point" in the direction of the charge, at all distances from the charge.

The consequence of this is that static fields (either electric or gravitational) always point directly to theactual position of the bodies that they are connected to, without any delay that is due to any "signal"traveling (or propagating) from the charge, over a distance to an observer. This remains true if thecharged bodies and their observers are made to "move" (or not), by simply changing reference frames.This fact sometimes causes confusion about the "speed" of such static fields, which sometimes appear tochange infinitely quickly when the changes in the field are mere artifacts of the motion of the observer,or of observation.

In such cases, nothing actually changes infinitely quickly, save the point of view of an observer of thefield. For example, when an observer begins to move with respect to a static field that already extendsover light years, it appears as though "immediately" the entire field, along with its source, has begunmoving at the speed of the observer. This, of course, includes the extended parts of the field. However,this "change" in the apparent behavior of the field source, along with its distant field, does not representany sort of propagation that is faster than light.

Newtonian gravitation

Isaac Newton's formulation of a gravitational force law requires that each particle with mass respondinstantaneously to every other particle with mass irrespective of the distance between them. In modernterms, Newtonian gravitation is described by the Poisson equation, according to which, when the massdistribution of a system changes, its gravitational field instantaneously adjusts. Therefore the theoryassumes the speed of gravity to be infinite. This assumption was adequate to account for all phenomenawith the observational accuracy of that time. It was not until the 19th century that an anomaly inastronomical observations which could not be reconciled with the Newtonian gravitational model ofinstantaneous action was noted: the French astronomer Urbain Le Verrier determined in 1859 that theelliptical orbit of Mercury precesses at a significantly different rate from that predicted by Newtoniantheory.[3]

Laplace

The first attempt to combine a finite gravitational speed with Newton's theory was made by Laplace in1805. Based on Newton's force law he considered a model in which the gravitational field is defined as aradiation field or fluid. Changes in the motion of the attracting body are transmitted by some sort ofwaves.[4] Therefore, the movements of the celestial bodies should be modified in the order v/c, where vis the relative speed between the bodies and c is the speed of gravity. The effect of a finite speed ofgravity goes to zero as c goes to infinity, but not as 1/c2 as it does in modern theories. This led Laplaceto conclude that the speed of gravitational interactions is at least 7×106 times the speed of light. Thisvelocity was used by many in the 19th century to criticize any model based on a finite speed of gravity,like electrical or mechanical explanations of gravitation.

From a modern point of view, Laplace's analysis is incorrect. Not knowing about Lorentz' invariance ofstatic fields, Laplace assumed that when an object like the Earth is moving around the Sun, the attractionof the Earth would not be toward the instantaneous position of the Sun, but toward where the Sun hadbeen if its position was retarded using the relative velocity (this retardation actually does happen withthe optical position of the Sun, and is called annual solar aberration). Putting the Sun immobile at theorigin, when the Earth is moving in an orbit of radius R with velocity v presuming that the gravitationalinfluence moves with velocity c, moves the Sun's true position ahead of its optical position, by anamount equal to vR/c, which is the travel time of gravity from the sun to the Earth times the relativevelocity of the sun and the Earth. The pull of gravity (if it behaved like a wave, such as light) would thenbe always displaced in the direction of the Earth's velocity, so that the Earth would always be pulledtoward the optical position of the Sun, rather than its actual position. This would cause a pull ahead ofthe Earth, which would cause the orbit of the Earth to spiral outward. Such an outspiral would besuppressed by an amount v/c compared to the force which keeps the Earth in orbit; and since the Earth'sorbit is observed to be stable, Laplace's c must be very large. As is now known, it may be considered tobe infinite in the limit of straight­line motion, since as a static influence, it is instantaneous at distance,when seen by observers at constant transverse velocity. For orbits in which velocity (direction of speed)changes slowly, it is almost infinite.

The attraction toward an object moving with a steady velocity is towards its instantaneous position withno delay, for both gravity and electric charge. In a field equation consistent with special relativity (i.e., aLorentz invariant equation), the attraction between static charges moving with constant relative velocity,is always toward the instantaneous position of the charge (in this case, the "gravitational charge" of theSun), not the time­retarded position of the Sun. When an object is moving in orbit at a steady speed butchanging velocity v, the effect on the orbit is order v2/c2, and the effect preserves energy and angularmomentum, so that orbits do not decay.

Electrodynamical analogies

Early theories

At the end of the 19th century, many tried to combine Newton's force law with the established laws ofelectrodynamics, like those of Wilhelm Eduard Weber, Carl Friedrich Gauss, Bernhard Riemann andJames Clerk Maxwell. Those theories are not invalidated by Laplace's critique, because although theyare based on finite propagation speeds, they contain additional terms which maintain the stability of theplanetary system. Those models were used to explain the perihelion advance of Mercury, but they couldnot provide exact values. One exception was Maurice Lévy in 1890, who succeeded in doing so bycombining the laws of Weber and Riemann, whereby the speed of gravity is equal to the speed of light.So those hypotheses were rejected.[5][6]

However, a more important variation of those attempts was the theory of Paul Gerber, who derived in1898 the identical formula, which was also derived later by Einstein for the perihelion advance. Basedon that formula, Gerber calculated a propagation speed for gravity of 305 000 km/s, i.e. practically thespeed of light. But Gerber's derivation of the formula was faulty, i.e., his conclusions did not followfrom his premises, and therefore many (including Einstein) did not consider it to be a meaningfultheoretical effort. Additionally, the value it predicted for the deflection of light in the gravitational fieldof the sun was too high by the factor 3/2.[7][8][9]

Lorentz

In 1900 Hendrik Lorentz tried to explain gravity on the basis of his ether theory and the Maxwellequations. After proposing (and rejecting) a Le Sage type model, he assumed like Ottaviano FabrizioMossotti and Johann Karl Friedrich Zöllner that the attraction of opposite charged particles is strongerthan the repulsion of equal charged particles. The resulting net force is exactly what is known asuniversal gravitation, in which the speed of gravity is that of light. This leads to a conflict with the lawof gravitation by Isaac Newton, in which it was shown by Pierre Simon Laplace that a finite speed ofgravity leads to some sort of aberration and therefore makes the orbits unstable. However, Lorentzshowed that the theory is not concerned by Laplace's critique, because due to the structure of theMaxwell equations only effects in the order v2/c2 arise. But Lorentz calculated that the value for theperihelion advance of Mercury was much too low. He wrote:[10]

The special form of these terms may perhaps be modified. Yet, what has been said issufficient to show that gravitation may be attributed to actions which are propagated with nogreater velocity than that of light.

In 1908 Henri Poincaré examined the gravitational theory of Lorentz and classified it as compatible withthe relativity principle, but (like Lorentz) he criticized the inaccurate indication of the perihelionadvance of Mercury.[11]

Lorentz covariant models

Henri Poincaré argued in 1904 that a propagation speed of gravity which is greater than c wouldcontradict the concept of local time (based on synchronization by light signals) and the principle ofrelativity. He wrote:[12]

What would happen if we could communicate by signals other than those of light, thevelocity of propagation of which differed from that of light? If, after having regulated ourwatches by the optimal method, we wished to verify the result by means of these newsignals, we should observe discrepancies due to the common translatory motion of the twostations. And are such signals inconceivable, if we take the view of Laplace, that universalgravitation is transmitted with a velocity a million times as great as that of light?

However, in 1905 Poincaré calculated that changes in the gravitational field can propagate with thespeed of light if it is presupposed that such a theory is based on the Lorentz transformation. He wrote:[13]

Laplace showed in effect that the propagation is either instantaneous or much faster thanthat of light. However, Laplace examined the hypothesis of finite propagation velocityceteris non mutatis; here, on the contrary, this hypothesis is conjoined with many others,and it may be that between them a more or less perfect compensation takes place. Theapplication of the Lorentz transformation has already provided us with numerous examplesof this.

Similar models were also proposed by Hermann Minkowski (1907) and Arnold Sommerfeld (1910).However, those attempts were quickly superseded by Einstein's theory of general relativity.[14]Whitehead's theory of gravitation (1922) explains gravitational red shift, light bending, perihelion shiftand Shapiro delay.[15]

General relativity

Background

General relativity predicts that gravitational radiation should exist and propagate as a wave at lightspeed:a slowly evolving and weak gravitational field will produce, according to general relativity, effects likethose of Newtonian gravitation.

Suddenly displacing one of two gravitoelectrically interacting particles would, after a delaycorresponding to lightspeed, cause the other to feel the displaced particle's absence: accelerations due tothe change in quadrupole moment of star systems, like the Hulse–Taylor binary have removed muchenergy (almost 2% of the energy of our own Sun's output) as gravitational waves, which wouldtheoretically travel at the speed of light.

Two gravitoelectrically interacting particle ensembles, e.g., two planets or stars moving at constantvelocity with respect to each other, each feel a force toward the instantaneous position of the other bodywithout a speed­of­light delay because Lorentz invariance demands that what a moving body in a staticfield sees and what a moving body that emits that field sees be symmetrical.

A moving body's seeing no aberration in a static field emanating from a "motionless body" thereforecauses Lorentz invariance to require that in the previously moving body's reference frame the (nowmoving) emitting body's field lines must not at a distance be retarded or aberred. Moving charged bodies(including bodies that emit static gravitational fields) exhibit static field lines that bend not with distanceand show no speed of light delay effects, as seen from bodies moving with regard to them.

In other words, since the gravitoelectric field is, by definition, static and continuous, it does notpropagate. If such a source of a static field is accelerated (for example stopped) with regard to itsformerly constant velocity frame, its distant field continues to be updated as though the charged bodycontinued with constant velocity. This effect causes the distant fields of unaccelerated moving charges toappear to be "updated" instantly for their constant velocity motion, as seen from distant positions, in theframe where the source­object is moving at constant velocity. However, as discussed, this is an effectwhich can be removed at any time, by transitioning to a new reference frame in which the distantcharged body is now at rest.

The static and continuous gravitoelectric component of a gravitational field is not a gravitomagneticcomponent (gravitational radiation); see Petrov classification. The gravitoelectric field is a static fieldand therefore cannot superluminally transmit quantized (discrete) information, i.e., it could not constitute

a well­ordered series of impulses carrying a well­defined meaning (this is the same for gravity andelectromagnetism).

Aberration of field direction in general relativity, for a weakly accelerated observer

The finite speed of gravitational interaction in general relativity does not lead to the sorts of problemswith the aberration of gravity that Newton was originally concerned with, because there is no suchaberration in static field effects. Because the acceleration of the Earth with regard to the Sun is small(meaning, to a good approximation, the two bodies can be regarded as traveling in straight lines pasteach other with unchanging velocity) the orbital results calculated by general relativity are the same asthose of Newtonian gravity with instantaneous action at a distance, because they are modelled by thebehavior of a static field with constant­velocity relative motion, and no aberration for the forcesinvolved.[16] Although the calculations are considerably more complicated, one can show that a staticfield in general relativity does not suffer from aberration problems as seen by an unaccelerated observer(or a weakly accelerated observer, such as the Earth). Analogously, the "static term" in theelectromagnetic Liénard–Wiechert potential theory of the fields from a moving charge, does not sufferfrom either aberration or positional­retardation. Only the term corresponding to acceleration andelectromagnetic emission in the Liénard–Wiechert potential shows a direction toward the time­retardedposition of the emitter.

It is in fact not very easy to construct a self­consistent gravity theory in which gravitational interactionpropagates at a speed other than the speed of light, which complicates discussion of this possibility.[17]

Formulaic conventions

In general relativity the metric tensor symbolizes the gravitational potential, and Christoffel symbols ofthe spacetime manifold symbolize the gravitational force field. The tidal gravitational field is associatedwith the curvature of spacetime.

Possible experimental measurements

The speed of gravity (more correctly, the speed of gravitational waves) can be calculated fromobservations of the orbital decay rate of binary pulsars PSR 1913+16 (the Hulse–Taylor binary systemnoted above) and PSR B1534+12. The orbits of these binary pulsars are decaying due to loss of energyin the form of gravitational radiation. The rate of this energy loss ("gravitational damping") can bemeasured, and since it depends on the speed of gravity, comparing the measured values to theory showsthat the speed of gravity is equal to the speed of light to within 1%.[18] However, according to PPNformalism setting, measuring the speed of gravity by comparing theoretical results with experimentalresults will depend on the theory; use of a theory other than that of general relativity could in principleshow a different speed, although the existence of gravitational damping at all implies that the speedcannot be infinite.

In September 2002, Sergei Kopeikin and Edward Fomalont announced that they had made an indirectmeasurement of the speed of gravity, using their data from VLBI measurement of the retarded positionof Jupiter on its orbit during Jupiter's transit across the line­of­sight of the bright radio source quasarQSO J0842+1835. Kopeikin and Fomalont concluded that the speed of gravity is between 0.8 and 1.2times the speed of light, which would be fully consistent with the theoretical prediction of generalrelativity that the speed of gravity is exactly the same as the speed of light.[19]

Several physicists, including Clifford M. Will and Steve Carlip, have criticized these claims on thegrounds that they have allegedly misinterpreted the results of their measurements. Notably, prior to theactual transit, Hideki Asada in a paper to the Astrophysical Journal Letters theorized that the proposedexperiment was essentially a roundabout confirmation of the speed of light instead of the speed ofgravity.[20] However, Kopeikin and Fomalont continue to vigorously argue their case and the means ofpresenting their result at the press­conference of AAS that was offered after the peer review of theresults of the Jovian experiment had been done by the experts of the AAS scientific organizingcommittee. In later publication by Kopeikin and Fomalont, which uses a bi­metric formalism that splitsthe space­time null cone in two – one for gravity and another one for light, the authors claimed thatAsada's claim was theoretically unsound.[21] The two null cones overlap in general relativity, whichmakes tracking the speed­of­gravity effects difficult and requires a special mathematical technique ofgravitational retarded potentials, which was worked out by Kopeikin and co­authors[22][23] but was neverproperly employed by Asada and/or the other critics.

Stuart Samuel also suggested that the experiment did not actually measure the speed of gravity becausethe effects were too small to have been measured.[24] A response by Kopeikin and Fomalont challengesthis opinion.[25]

It is important to understand that none of the participants in this controversy are claiming that generalrelativity is "wrong". Rather, the debate concerns whether or not Kopeikin and Fomalont have reallyprovided yet another verification of one of its fundamental predictions. A comprehensive review of thedefinition of the speed of gravity and its measurement with high­precision astrometric and othertechniques appears in the textbook Relativistic Celestial Mechanics in the Solar System.[26]

References1. Hartle, JB (2003). Gravity: An Introduction to Einstein's General Relativity. Addison­Wesley. p. 332.

ISBN 981­02­2749­3.2. Taylor, Edwin F. and Wheeler, John Archibald, Spacetime Physics, 2nd edition, 1991, p. 12.3. U. Le Verrier, Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhéliede cette planète (http://www.archive.org/stream/comptesrendusheb49acad#page/379), C. R. Acad. Sci. 49(1859), 379–383.

4. Laplace, P.S.: (1805) "A Treatise in Celestial Mechanics", Volume IV, Book X, Chapter VII, translated byN. Bowditch (Chelsea, New York, 1966)

5. Zenneck, J. (1903). "Gravitation". Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrerAnwendungen (in German) 5: 25–67. doi:10.1007/978­3­663­16016­8_2.

6. Roseveare, N. T (1982). Mercury's perihelion, from Leverrier to Einstein. Oxford: University Press. ISBN 0­19­858174­2.

7. Gerber, P. (1898). "Die räumliche und zeitliche Ausbreitung der Gravitation". Zeitschrift für mathematischePhysik (in German) 43: 93–104.

8. Zenneck, pp. 49–519. "Gerber's Gravity". Mathpages. Retrieved 2 Dec 2010.10. Lorentz, H.A. (1900). "Considerations on Gravitation". Proc. Acad. Amsterdam 2: 559–574.11. Poincaré, H. (1908). "La dynamique de l'électron" (PDF). Revue générale des sciences pures et appliquées 19:

386–402. Reprinted in Poincaré, Oeuvres, tome IX, S. 551–586 and in "Science and Method" (1908)12. Poincaré, Henri (1904). "L'état actuel et l'avenir de la physique mathématique". Bulletin des Sciences

Mathématiques 28 (2): 302–324.. English translation in Poincaré, Henri (1905). "The Principles ofMathematical Physics". In Rogers, Howard J. Congress of arts and science, universal exposition, St. Louis,1904 1. Boston and New York: Houghton, Mifflin and Company. pp. 604–622. Reprinted in "The value ofscience", Ch. 7–9.

13. Poincaré, H. (1906). "Sur la dynamique de l'électron" (PDF). Rendiconti del Circolo Matematico di Palermo(in French) 21 (1): 129–176. doi:10.1007/BF03013466. See also the English Translation

(http://henripoincarepapers.univ­lorraine.fr/chp/hp­pdf/hp1906rpen.pdf).14. Walter, Scott (2007). Renn, J., ed. "Breaking in the 4­vectors: the four­dimensional movement in gravitation,

1905–1910" (PDF). The Genesis of General Relativity (Berlin: Springer) 3: 193–252.15. Will, Clifford & Gibbons, Gary. "On the Multiple Deaths of Whitehead's Theory of Gravity

(http://arxiv.org/abs/gr­qc/0611006)", to be submitted to Studies In History And Philosophy Of ModernPhysics (2006).

16. Carlip, S. (2000). "Aberration and the Speed of Gravity". Phys. Lett. A 267 (2–3): 81–87. arXiv:gr­qc/9909087. Bibcode:2000PhLA..267...81C. doi:10.1016/S0375­9601(00)00101­8.

17. * Carlip, S. (2004). "Model­Dependence of Shapiro Time Delay and the "Speed of Gravity/Speed of Light"Controversy". Class. Quant. Grav. 21: 3803–3812. arXiv:gr­qc/0403060.

18. C. Will (2001). "The confrontation between general relativity and experiment". Living Rev. Relativity 4: 4.arXiv:gr­qc/0103036. Bibcode:2001LRR.....4....4W.

19. Ed Fomalont & Sergei Kopeikin (2003). "The Measurement of the Light Deflection from Jupiter:Experimental Results". The Astrophysical Journal 598 (1): 704–711. arXiv:astro­ph/0302294.Bibcode:2003ApJ...598..704F. doi:10.1086/378785.

20. Hideki Asada (2002). "Light Cone Effect and the Shapiro Time Delay". The Astrophysical Journal Letters574 (1): L69. arXiv:astro­ph/0206266. Bibcode:2002ApJ...574L..69A. doi:10.1086/342369.

21. Kopeikin S.M. & Fomalont E.B. (2006). "Aberration and the Fundamental Speed of Gravity in the JovianDeflection Experiment". Foundations of Physics 36 (8): 1244–1285. arXiv:astro­ph/0311063.Bibcode:2006FoPh...36.1244K. doi:10.1007/s10701­006­9059­7.

22. Kopeikin S.M. & Schaefer G. (1999). "Lorentz covariant theory of light propagation in gravitational fields ofarbitrary­moving bodies". Physical Review D 60 (12): id. 124002 [44 pages]. arXiv:gr­qc/9902030.Bibcode:1999PhRvD..60l4002K. doi:10.1103/PhysRevD.60.124002.

23. Kopeikin S.M. & Mashhoon B. (2002). "Gravitomagnetic effects in the propagation of electromagnetic wavesin variable gravitational fields of arbitrary­moving and spinning bodies". Physical Review D 65 (6): id.064025 [20 pages]. arXiv:gr­qc/0110101. Bibcode:2002PhRvD..65f4025K.doi:10.1103/PhysRevD.65.064025.

24. http://www.lbl.gov/Science­Articles/Archive/Phys­speed­of­gravity.html25. Kopeikin, Sergei & Fomalont, Edward (2006). "On the speed of gravity and relativistic v/c corrections to the

Shapiro time delay". Physics Letters A 355 (3): 163–166. arXiv:gr­qc/0310065.Bibcode:2006PhLA..355..163K. doi:10.1016/j.physleta.2006.02.028.

26. S. Kopeikin, M. Efroimsky and G. Kaplan [1] (http://www.wiley­vch.de/publish/en/books/forthcomingTitles/PH00/3­527­40856­8/authorinformation/?sID=p2qlnooj68su7htl8qrrc2qjt3) Relativistic Celestial Mechanics in the Solar System, Wiley­VCH, 2011.XXXII, 860 Pages, 65 Fig., 6 Tab.

Kopeikin, Sergei M. (2001). "Testing Relativistic Effect of Propagation of Gravity by Very­Long BaselineInterferometry". Astrophys. J. 556 (1): L1–L6. arXiv:gr­qc/0105060. Bibcode:2001ApJ...556L...1K.doi:10.1086/322872.

Asada, Hidecki (2002). "The Light­cone Effect on the Shapiro Time Delay". Astrophys. J. 574 (1): L69.arXiv:astro­ph/0206266. Bibcode:2002ApJ...574L..69A. doi:10.1086/342369.

Will, Clifford M. (2003). "Propagation Speed of Gravity and the Relativistic Time Delay". Astrophys. J. 590(2): 683–690. arXiv:astro­ph/0301145. Bibcode:2003ApJ...590..683W. doi:10.1086/375164.

Fomalont, E. B. & Kopeikin, Sergei M. (2003). "The Measurement of the Light Deflection from Jupiter:Experimental Results". Astrophys. J. 598 (1): 704–711. arXiv:astro­ph/0302294.Bibcode:2003ApJ...598..704F. doi:10.1086/378785.

Kopeikin, Sergei M. (Feb 21, 2003). "The Measurement of the Light Deflection from Jupiter: TheoreticalInterpretation". arXiv:astro­ph/0302462.

Kopeikin, Sergei M. (2003). "The Post­Newtonian Treatment of the VLBI Experiment on September 8,2002". Phys. Lett. A 312 (3–4): 147–157. arXiv:gr­qc/0212121. Bibcode:2003PhLA..312..147K.doi:10.1016/S0375­9601(03)00613­3.

Faber, Joshua A. (Mar 14, 2003). "The speed of gravity has not been measured from time delays".

arXiv:astro­ph/0303346.

Kopeikin, Sergei M. (2004). "The Speed of Gravity in General Relativity and Theoretical Interpretation of theJovian Deflection Experiment". Classical and Quantum Gravity 21 (13): 3251–3286. arXiv:gr­qc/0310059.Bibcode:2004CQGra..21.3251K. doi:10.1088/0264­9381/21/13/010.

Samuel, Stuart (2003). "On the Speed of Gravity and the v/c Corrections to the Shapiro Time Delay". Phys.Rev. Lett. 90 (23): 231101. arXiv:astro­ph/0304006. Bibcode:2003PhRvL..90w1101S.doi:10.1103/PhysRevLett.90.231101. PMID 12857246.

Kopeikin, Sergei & Fomalont, Edward (2006). "On the speed of gravity and relativistic v/c corrections to theShapiro time delay". Physics Letters A 355 (3): 163–166. arXiv:gr­qc/0310065.Bibcode:2006PhLA..355..163K. doi:10.1016/j.physleta.2006.02.028.

Hideki, Asada (Aug 20, 2003). "Comments on "Measuring the Gravity Speed by VLBI" ". arXiv:astro­ph/0308343.

Kopeikin, Sergei & Fomalont, Edward (2006). "Aberration and the Fundamental Speed of Gravity in theJovian Deflection Experiment". Foundations of Physics 36 (8): 1244–1285. arXiv:astro­ph/0311063.Bibcode:2006FoPh...36.1244K. doi:10.1007/s10701­006­9059­7.

Carlip, Steven (2004). "Model­Dependence of Shapiro Time Delay and the "Speed of Gravity/Speed of Light"Controversy". Class. Quant. Grav. 21 (15): 3803–3812. arXiv:gr­qc/0403060.Bibcode:2004CQGra..21.3803C. doi:10.1088/0264­9381/21/15/011.

Kopeikin, Sergei M. (2005). "Comment on 'Model­dependence of Shapiro time delay and the "speed ofgravity/speed of light" controversy". Class. Quant. Grav. 22 (23): 5181–5186. arXiv:gr­qc/0510048.Bibcode:2005CQGra..22.5181K. doi:10.1088/0264­9381/22/23/N01.

Pascual­Sánchez, J.­F. (2004). "Speed of gravity and gravitomagnetism". Int. J. Mod. Phys. D 13 (10): 2345–2350. arXiv:gr­qc/0405123. Bibcode:2004IJMPD..13.2345P. doi:10.1142/S0218271804006425.

Kopeikin, Sergei (2006). "Gravitomagnetism and the speed of gravity". Int. J. Mod. Phys. D 15 (3): 305–320.arXiv:gr­qc/0507001. Bibcode:2006IJMPD..15..305K. doi:10.1142/S0218271806007663.

Samuel, Stuart (2004). "On the Speed of Gravity and the Jupiter/Quasar Measurement". Int. J. Mod. Phys. D13 (9): 1753–1770. arXiv:astro­ph/0412401. Bibcode:2004IJMPD..13.1753S.doi:10.1142/S0218271804005900.

Kopeikin, Sergei (2006). "Comments on the paper by S. Samuel "On the speed of gravity and theJupiter/Quasar measurement" ". Int. J. Mod. Phys. D 15 (2): 273–288. arXiv:gr­qc/0501001.Bibcode:2006IJMPD..15..273K. doi:10.1142/S021827180600853X.

Kopeikin, Sergei & Fomalont, Edward (2007). "Gravimagnetism, Causality, and Aberration of Gravity in theGravitational Light­Ray Deflection Experiments". General Relativity and Gravitation 39 (10): 1583–1624.arXiv:gr­qc/0510077. Bibcode:2007GReGr..39.1583K. doi:10.1007/s10714­007­0483­6.

Kopeikin, Sergei & Fomalont, Edward (2008). "Radio interferometric tests of general relativity". "A GiantStep: from Milli­ to Micro­arcsecond Astrometry", Proceedings of the International Astronomical Union,IAU Symposium 248 (S248): 383–386. Bibcode:2008IAUS..248..383F. doi:10.1017/S1743921308019613.

Zhu, Yin (2011). "Measurement of the Speed of Gravity". arXiv:1108.3761.

External links

Does Gravity Travel at the Speed of Light?(http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html) in The Physics FAQ (alsohere (http://www.faqs.org/faqs/astronomy/faq/part4/section­6.html)).

Measuring the Speed of Gravity (http://www.mathpages.com/home/kmath451/kmath451.htm) atMathPages

Hazel Muir, First speed of gravity measurement revealed(http://www.newscientist.com/article/dn3232­first­speed­of­gravity­measurement­revealed.html),a New Scientist article on Kopeikin's original announcement.

Clifford M. Will, Has the Speed of Gravity Been Measured?(http://wugrav.wustl.edu/people/CMW/SpeedofGravity.html).

Kevin Carlson, MU physicist defends Einstein's theory and 'speed of gravity' measurement(http://www.spaceref.com/news/viewpr.html?pid=23705 –).

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