Speed of gravity

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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.

Contents

[edit] 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 mode and astronomical observation were noted.

[edit] Laplace

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.[1] 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.

[edit] Field theories

At the end of the 19th century, many tried to combine Newton's force law with the established laws of electrodynamics, like those of Weber, Gauß, Riemann and 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. [2]

Lorentz

In 1900 Hendrik Lorentz tried to explain gravity on the basis of his Lorentz 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:[3]

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.

Poincaré

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:[4]

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:[5]

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.[6]

Gerber

In 1898 Gerber derived the correct value for the perihelion advance and based on that formula, he calculated a propagation speed for gravity of 305 000 km/s, i.e. practically the speed of light.[7][8] But Gerber's derivation of the formula was faulty and therefore many (including Einstein) rejected Gerber's proposal. Additionally, the value for the deflection of light in the gravitational field of the sun was too high by the factor 3/2.[9]

[edit] General relativity

[edit] Background

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 with the speed faster than the speed of light.

[edit] 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. 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.[10] 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.[11] These claims are generally dismissed as erroneous by relativity experts.[12][13]

[edit] 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%. [14] (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.

[edit] References

  1. ^ Laplace, P.S.: (1805) "A Treatise in Celestial Mechanics", Volume IV, Book X, Chapter VII, translated by N. Bowditch (Chelsea, New York, 1966)
  2. ^ Zenneck, J. (1903). "Gravitation" (in German). Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen 5: 25–67. 
  3. ^ Lorentz, H.A. (1900). "Considerations on Gravitation". Proc. Acad. Amsterdam 2: 559–574. 
  4. ^ Poincaré, H. (1904/2000), “The present and the future of mathematical physics”, Bull. Amer. Math. Soc. 37: 25-38 
  5. ^ Poincaré, H. (1906). "Sur la dynamique de l'électron" (in French). Rendiconti del Circolo matematico di Palermo 21: 129–176.  See also the English Translation.
  6. ^ Poincaré, H. (1908a), “La dynamique de l'électron”, 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)
  7. ^ Gerber, P. (1898). "Die räumliche und zeitliche Ausbreitung der Gravitation" (in German). Zeitschrift für mathematische Physik 43: 93–104. 
  8. ^ Zenneck, pp. 49-51
  9. ^ Mathpages: Gerber's Gravity
  10. ^ Carlip, S. (2000). "Aberration and the Speed of Gravity". Phys. Lett. A 267: 81–87. doi:10.1016/S0375-9601(00)00101-8. arXiv:gr-qc/9909087. 
  11. ^ Van Flandern, T.; and Vigier, J.P. (2002). "Experimental Repeal of the Speed Limit for Gravitational, Electrodynamic, and Quantum Field Interactions". Found. Phys. 32: 1031–1068. doi:10.1023/A:1016530625645. 
  12. ^ Farrell, John (2006), “Was Einstein a fake?”, Cosmos 11, <http://www.cosmosmagazine.com/node/1162> . 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.
  13. ^ * 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.
  14. ^ C. Will (2001). "The confrontation between general relativity and experiment". Living Rev. Relativity 4: 4. 
  • Kopeikin, Sergei M. (2001). "Testing Relativistic Effect of Propagation of Gravity by Very-Long Baseline Interferometry". Astrophys. J. 556: L1–L6. doi:10.1086/322872. arXiv:gr-qc/0105060. 
  • Fomalont, E. B.; and Kopeikin, Sergei M. (2003). "The Measurement of the Light Deflection from Jupiter: Experimental Results". Astrophys. J. 598: 704–711. doi:10.1086/378785. arXiv:astro-ph/0302294. 
  • Kopeikin, Sergei M. (2004). "The Speed of Gravity in General Relativity and Theoretical Interpretation of the Jovian Deflection Experiment". Classical and Quantum Gravity 21: 3251–3286. doi:10.1088/0264-9381/21/13/010. arXiv:gr-qc/0310059. 
  • Kopeikin, Sergei and Fomalont, Edward (2007). "Gravimagnetism, Causality, and Aberration of Gravity in the Gravitational Light-Ray Deflection Experiments". General Relativity and Gravitation 39: 1583–1624. doi:10.1007/s10714-007-0483-6. arXiv:gr-qc/0510077. 

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