Speed of gravity

Speed of gravity

In the context of classical theories of gravitation, the speed of gravity refers to the speed at which a gravitational field propagates. This is the speed at which changes in the distribution of energy and momentum result in noticeable changes in the gravitational field which they produce.

Where no other theory is specified, discussion of the speed of gravity is normally in reference to general relativity, which predicts it to equal "c".

Newtonian gravitation

Isaac Newton's formulation of a gravitational force law requires that each particle respond instantaneously to every other massive particle irrespective of the distance between them. In modern terms, Newtonian gravitation is described by the Poisson equation, according to which, when the mass distribution of a system changes, its gravitational field instantaneously adjusts. Therefore the theory requires the speed of gravity to be infinite.

Newton was troubled by this aspect of his theory. He felt that the gravitational effect should propagate at some finite speed. He experimented with introducing such a finite propagation speed, but found that it would destroy the remarkable agreement between his original theory and the astronomical observations available at the time. It was not until the 19th century, long after Newton's death, that discrepancies between the Newtonian gravitational model and astronomical observation were noted.


The one who first tried to combine a finite gravitational speed with Newton's theory was Laplace in 1805. Based on Newton's force law he considered a model in which the gravitational field is defined as a radiation field or fluid. Changes in the motion of the attracting body are transmitted by some sort of waves. [Laplace, P.S.: (1805) "A Treatise in Celestial Mechanics", Volume IV, Book X, Chapter VII, translated by N. Bowditch (Chelsea, New York, 1966)] Therefore, the movements of the celestial bodies should be modified in the order "v/c", where "v" is the relative speed between the bodies and "c" is the wave velocity. This argument draws an analogy to the aberration of light, which causes the Sun to appear in a position slightly displaced from its actual position. So, introducing a speed of light time delay into Newtonian gravitation would result in unstable planetary orbits.

According to Laplace, the stability of orbits can only be maintained by introducing a velocity for gravitational interactions of at least 7×106 times the speed of light. This fantastic velocity 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.

Field theories

;Early theoriesAt the end of the 19th century, many tried to combine Newton's force law with the established laws of electrodynamics, like those of Wilhelm Eduard Weber, Carl Friedrich Gauß, Bernhard Riemann and James Clerk Maxwell. Those theories are not concerned by Laplace's critique, because although they are based on finite propagation speeds, they contain additional terms which maintain the stability of the planetary system. Those models were used to explain the perihelion advance of Mercury, but they could not provide exact values. One exception was Lévy in 1890, who succeeded in doing so by combining the laws of Weber and Riemann, whereby the speed of gravity is equal to the speed of light. [cite journal | author=Zenneck, J. | authorlink=Jonathan Zenneck | title=Gravitation | journal=Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen | year=1903 | volume=5 | pages=25–67 | url=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D189514 | language=German] However, a more important variation of those attempts was the theory of Paul Gerber, who derived in 1898 the identical formula, which was later also derived by Einstein for the perihelion advance. Based on that formula, Gerber calculated a propagation speed for gravity of 305 000 km/s, i.e. practically the speed of light. [cite journal | author=Gerber, P. | authorlink=Paul Gerber | title=Die räumliche und zeitliche Ausbreitung der Gravitation | journal=Zeitschrift für mathematische Physik | year=1898 | volume=43 | pages=93–104 | url=http://www.mahag.com/fremd/gerber.htm | language=German] [Zenneck, pp. 49-51] But Gerber's derivation of the formula was faulty, i.e., his conclusions did not follow from his premises, and therefore many (including Einstein) did not consider it to be a meaningful theoretical effort. Additionally, the value it predicted for the deflection of light in the gravitational field of the sun was too high by the factor 3/2. [Mathpages: [http://www.mathpages.com/home/kmath527/kmath527.htm Gerber's Gravity] ] So those hypotheses, including Lévi's and Gerber's, were rejected.

;LorentzIn 1900 Hendrik Lorentz tried to explain gravity on the basis of his ether theory and the Maxwell equations. After proposing (and rejecting) a Le Sage type model, he assumed like Ottaviano Fabrizio Mossotti and Johann Karl Friedrich Zöllner that the attraction of opposite charged particles is stronger than the repulsion of equal charged particles. The resulting net force is exactly what is known as universal gravitation, in which the speed of gravity is that of light. This leads to a conflict with the law of gravitation by Isaac Newton, in which it was shown by Pierre Simon Laplace that a finite speed of gravity leads to some sort of aberration and therefore makes the orbits unstable. However, Lorentz showed that the theory is not concerned by Laplace's critique, because due to the structure of the Maxwell equations only effects in the order "v²/c²" arise. But Lorentz calculated that the value for the perihelion advance of Mercury was much too low. He wrote: [cite journal | author=Lorentz, H.A. | authorlink = Hendrik Lorentz | title=Considerations on Gravitation | journal=Proc. Acad. Amsterdam | year=1900 | volume=2 | pages=559–574 | url=http://www.historyofscience.nl/search/detail.cfm?pubid=260&view=image&startrow=1] "The special form of these terms may perhaps be modified. Yet, what has been said is sufficient to show that gravitation may be attributed to actions which are propagated with no greater velocity than that of light."

Henri Poincaré argued in 1904 that a propagation speed of gravity which is greater than c would contradict the concept of local time (based on synchronization by light signals) and the principle of relativity. He wrote: Citation
author=Poincaré, Henri
title=L'état actuel et l'avenir de la physique mathématique
journal=Bulletin des sciences mathématiques
. English translation in Citation
author=Poincaré, Henri
chapter=The Principles of Mathematical Physics
editor=Rogers, Howard J.
title=Congress of arts and science, universal exposition, St. Louis, 1904
publisher=Houghton, Mifflin and Company
place=Boston and New York
Reprinted in "The value of science", Ch. 7-9.] "What would happen if we could communicate by signals other than those of light, the velocity of propagation of which differed from that of light? If, after having regulated our watches by the optimal method, we wished to verify the result by means of these new signals, we should observe discrepancies due to the common translatory motion of the two stations. And are such signals inconceivable, if we take the view of Laplace, that universal gravitation 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 the speed of light if it is presupposed that such a theory is based on the Lorentz transformation. He wrote: [cite journal | author=Poincaré, H. | authorlink = Henri Poincaré | title=Sur la dynamique de l'électron | journal=Rendiconti del Circolo matematico di Palermo | year=1906 | volume=21 | pages=129–176 | url=http://www.soso.ch/wissen/hist/SRT/P-1905.pdf | language=French See also the [http://www.univ-nancy2.fr/poincare/bhp/pdf/hp1906rpen.pdf English Translation] .] "Laplace showed in effect that the propagation is either instantaneous or much faster than that of light. However, Laplace examined the hypothesis of finite propagation velocity ceteris 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. The application of the Lorentz transformation has already provided us with numerous examples of this." In 1908 he examined the gravitational theory of Lorentz and classified it as compatible with the relativity principle, but (like Lorentz) he criticized the inaccurate indication of the perihelion advance of Mercury. [Citation | author=Poincaré, H. | year=1908a | title= [http://www.univ-nancy2.fr/Poincaré/bhp/pdf/hp1908rg.pdf La dynamique de l'électron] | journal=Revue générale des sciences pures et appliquées | volume =19 | pages =386-402 Reprinted in Poincaré, Oeuvres, tome IX, S. 551-586 and in "Science and Method" (1908)]

General relativity


In general relativity, the gravitational potential is identified with the metric tensor and the gravitational force field with the Christoffel symbols of the space-time manifold. Tidal gravitational field is associated with the curvature of space-time. General relativity predicts that gravitational radiation should exist and propagate as a wave at the speed of light. To avoid confusion, we should point out that a "slowly evolving" source for a weak gravitational field will produce, according to general relativity, similar effects to those we might expect from Newtonian gravitation. In particular, a slowly evolving "Coulomb component" of a gravitational field should not be confused with a possible additional "radiation component"; see Petrov classification. Nonetheless, any of the Petrov-type gravitational field obeys the principle of causality, so that the slowly evolving "Coulomb component" of the gravitational field can not transfer information about position of the source of the gravitational field faster than the speed of light.

Aberration in general relativity

The finite speed of gravitational interaction in general relativity may at first seem to lead to exactly the same sorts of problems with the aberration of gravity that Newton was originally concerned with. In general relativity, however, (similar to the field theories above), gravitomagnetism effects cancel out the effects of aberration.clarifyme As shown by Carlip, in the weak stationary field limit, the orbital results calculated by general relativity are the same as those of Newtonian gravity (with instantaneous action at a distance), despite the fact that the full theory gives a speed of gravity of "c". [cite journal | author=Carlip, S. | title=Aberration and the Speed of Gravity | journal= Phys. Lett. A | year=2000 | volume=267 | pages=81–87 | id= arxiv|archive=gr-qc|id=9909087 | doi=10.1016/S0375-9601(00)00101-8 ] Although the calculations are considerably more complicated, one can show that general relativity does not suffer from aberration problems just as electromagnetic retarded Liénard–Wiechert potential theory does not. It is not very easy to construct a self-consistent gravity theory in which gravitational interaction propagates at a speed other than the speed of light, which complicates discussion of this possibility.

Following Laplace, on the other hand, Van Flandern claims that in general relativity the speed of gravity must be at least 20 billion times that of light. [cite journal | author=Van Flandern, T.; and Vigier, J.P. | title=Experimental Repeal of the Speed Limit for Gravitational, Electrodynamic, and Quantum Field Interactions | journal=Found. Phys. | year=2002 | volume=32 | pages=1031–1068 | doi=10.1023/A:1016530625645] These claims are generally dismissed as erroneous by relativity experts. [ Citation | url = http://www.cosmosmagazine.com/node/1162 | title = Was Einstein a fake? | first = John | last = Farrell | journal = Cosmos | year = 2006 | month = Oct | volume = 11 . This article includes some blunt assessments from relativity experts. Steve Carlip says: "As far as I can tell, Van Flandern simply doesn't understand the Einstein field equations." Flandern does not accept this assessment.] [ * Carlip, S. (2004). "Model-Dependence of Shapiro Time Delay and the "Speed of Gravity/Speed of Light" Controversy". Class. Quant. Grav. 21: 3803-3812. arXiv:gr-qc/0403060. ]

Experimental measurement?

The speed of gravity can be calculated from observations of the orbital decay rate of binary pulsars PSR 1913+16 and PSR B1534+12. The orbits of these pulsars around each other is decaying due to loss of energy in the form of gravitational radiation. The rate of this energy loss ("gravitational damping") can be measured, and since it depends on the speed of gravity, comparing the measured values to theory shows that the speed of gravity is equal to the speed of light to within 1%. [ cite journal | author = C. Will | title = The confrontation between general relativity and experiment | year = 2001 | journal = Living Rev. Relativity | volume = 4 | url = http://www.livingreviews.org/lrr-2001-4 | pages = 4] (However, measuring the speed of gravity by comparing theoretical results with experimental results will depend on the theory; use of a theory other than that of general relativity could in principle show a different speed, although the existence of gravitational damping at all implies that the speed cannot be infinite.)

In September 2002, Sergei Kopeikin and Edward Fomalont announced that they had made an indirect measurement of the speed of gravity, using their data from VLBI measurement of the retarded position of Jupiter on its orbit during Jupiter's transit across the line-of-sight of the bright radio source quasar QSO J0842+1835. Kopeikin and Fomalont concluded that the speed of gravity is between 0.8 and 1.2 times the speed of light, which would be fully consistent with the theoretical prediction of general relativity that the speed of gravity is "exactly" the same as the speed of light.

Several physicists, including Clifford M. Will and Steve Carlip, have criticized these claims on the grounds that they have allegedly misinterpreted the results of their measurements. However, Kopeikin and Fomalont continue to vigorously argue their case. (See the citations below for the details of the arguments pro and con.)

It is important to understand that none of the participants in this controversy are claiming that general relativity is "wrong". Rather, the debate concerns whether or not Kopeikin and Fomalont have really provided yet another verification of one of its fundamental predictions.


*cite journal | author=Kopeikin, Sergei M. | title=Testing Relativistic Effect of Propagation of Gravity by Very-Long Baseline Interferometry | journal= Astrophys. J. | year=2001 | volume=556 | pages=L1–L6 | id= arxiv|archive=gr-qc|id=0105060 | doi=10.1086/322872

*cite journal | author=Asada, Hidecki | title=The Light-cone Effect on the Shapiro Time Delay |journal= Astrophys. J. | year=2002 | volume=574 | pages=L69 | id= arxiv|archive=astro-ph|id=0206266 | doi=10.1086/342369

*cite journal | author=Will, Clifford M. | title=Propagation Speed of Gravity and the Relativistic Time Delay | journal=Astrophys. J.| year=2003 | volume=590 | pages=683–690 | id= arxiv|archive=astro-ph|id=0301145 | doi=10.1086/375164

*cite journal | author=Fomalont, E. B.; and Kopeikin, Sergei M. | title=The Measurement of the Light Deflection from Jupiter: Experimental Results | journal= Astrophys. J.| year=2003 | volume=598 | pages=704–711 | id= arxiv|archive=astro-ph|id=0302294 | doi=10.1086/378785

*cite arXiv | author=Kopeikin, Sergei M. | title=The Measurement of the Light Deflection from Jupiter: Theoretical Interpretation | year = 2003 | version=Feb 21, 2003 | eprint=astro-ph/0302462

*cite journal | author=Kopeikin, Sergei M. | title=The Post-Newtonian Treatment of the VLBI Experiment on September 8, 2002 | journal= Phys. Lett. A| year = 2003 |volume=312 | pages=147–157 | id = arxiv|archive=gr-qc|id=0212121 | doi=10.1016/S0375-9601(03)00613-3

*cite arXiv | author=Faber, Joshua A. | title=The speed of gravity has not been measured from time delays | year=2003 | version=Mar 14, 2003 | eprint=astro-ph/0303346

*cite journal | author=Kopeikin, Sergei M. | title= The Speed of Gravity in General Relativity and Theoretical Interpretation of the Jovian Deflection Experiment|journal= Classical and Quantum Gravity | year = 2004 |volume= 21| pages=3251–3286 | id = arxiv|archive=gr-qc|id=0310059 | doi= 10.1088/0264-9381/21/13/010

*cite journal | author=Samuel, Stuart | title=On the Speed of Gravity and the v/c Corrections to the Shapiro Time Delay | journal= Phys. Rev. Lett. | year=2003 | volume= 90 | pages=231101 | id= arxiv|archive=astro-ph|id=0304006 | doi=10.1103/PhysRevLett.90.231101

*cite journal | author=Kopeikin, Sergei and Fomalont, Edward | title=On the speed of gravity and relativistic v/c corrections to the Shapiro time delay | journal= Physics Letters A | year=2006|volume=355 |pages= 163–166| id = arxiv|archive=gr-qc|id=0310065 | doi=10.1016/j.physleta.2006.02.028

*cite arXiv | author=Asada, Hideki | title=Comments on "Measuring the Gravity Speed by VLBI" | year = 2003 | version=Aug 20, 2003 | eprint=astro-ph/0308343

*cite journal | author=Kopeikin, Sergei and Fomalont, Edward |title=Aberration and the Fundamental Speed of Gravity in the Jovian Deflection Experiment| journal= Foundations of Physics| year=2006 |volume=36| pages= 1244–1285| id=arxiv|archive=astro-ph|id=0311063 |doi=10.1007/s10701-006-9059-7

*cite journal | author=Carlip, S. | title=Model-Dependence of Shapiro Time Delay and the "Speed of Gravity/Speed of Light" Controversy | journal= Class. Quant. Grav. | year=2004 | volume=21 | pages=3803–3812 | id= arxiv|archive=gr-qc|id=0403060 | doi=10.1088/0264-9381/21/15/011

*cite journal | author=Kopeikin, Sergei M. | title=Comment on 'Model-dependence of Shapiro time delay and the "speed of gravity/speed of light" controversy | journal= Class. Quant. Grav. | year=2005 | volume=22 | pages=5181–5186 | id= arxiv|archive=gr-qc|id=0510048 | doi=10.1088/0264-9381/22/23/N01

*cite journal | author=Pascual-Sánchez, J.-F. | title=Speed of gravity and gravitomagnetism | journal= Int. J. Mod. Phys. D | year=2004 | volume=13 | pages=2345–2350 | id= arxiv|archive=gr-qc|id=0405123 | doi=10.1142/S0218271804006425

*cite journal | author=Kopeikin, Sergei | title=Gravitomagnetism and the speed of gravity | journal= Int. J. Mod. Phys. D | year=2006|volume=15 |pages= 305–320 | id = arxiv|archive=gr-qc|id=0507001 | doi=10.1142/S0218271806007663

*cite journal | author=Samuel, Stuart | title=On the Speed of Gravity and the Jupiter/Quasar Measurement | journal=Int. J. Mod. Phys. D | year=2004 | volume=13 | pages=1753–1770 | id= arxiv|archive=astro-ph|id=0412401 | doi=10.1142/S0218271804005900

*cite journal | author=Kopeikin, Sergei | title=Comments on the paper by S. Samuel "On the speed of gravity and the Jupiter/Quasar measurement" | journal= Int. J. Mod. Phys. D | year=2006|volume=15 |pages= 273–288 | id = arxiv|archive=gr-qc|id=0501001 | doi=10.1142/S021827180600853X

*cite journal | author=Kopeikin, Sergei and Fomalont, Edward | title=Gravimagnetism, Causality, and Aberration of Gravity in the Gravitational Light-Ray Deflection Experiments | journal= General Relativity and Gravitation | year = 2007 |volume= 39 | pages= 1583–1624 | id=arxiv|archive=gr-qc|id=0510077 | doi=10.1007/s10714-007-0483-6

External links

* [http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html Does Gravity Travel at the Speed of Light?] in "The Physics FAQ" (also [http://www.faqs.org/faqs/astronomy/faq/part4/section-6.html here] ).
* [http://www.mathpages.com/home/kmath451/kmath451.htm Measuring the Speed of Gravity] at MathPages

*Hazel Muir, [http://www.newscientist.com/news/news.jsp?id=ns99993232 First speed of gravity measurement revealed] , a "New Scientist" article on Kopeikin's original announcement.

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

*Tom Van Flandern, [http://metaresearch.org/cosmology/speed_of_gravity.asp The Speed of Gravity – What the Experiments Say] and [http://metaresearch.org/media%20and%20links/press/SOG-Kopeikin.asp Kopeikin and the Speed of Gravity] .

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

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