Isaac Newton's formulation of a gravitational force law requires that each particle with mass respond instantaneously to every other particle with mass 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 assumes the speed of gravity to be infinite. This assumption was adequate to account for all phenomena with the observational accuracy of that time. It was not until the 19th century that an anomaly in astronomical observations which could not be reconciled with the Newtonian gravitational model of instantaneous action was noted: the French astronomer Leverier determined in 1847 that the elliptical orbit of Mercury precesses at a significantly different rate than is predicted by Newtonian theory.
Laplace
The first attempt to combine a finite gravitational speed with Newton's theory was made by 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 speed of gravity. The effect of a finite speed of gravity goes to zero as c goes to infinity, but not like 1/c2 as it does in modern theories. This led Laplace to conclude that the speed of gravitational interactions is 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.
From a modern point of view, Laplace's analysis is incorrect. Not knowing about relativistic field equations, Laplace assumed that when an object like the Earth is moving around the sun, the attraction of the Earth would not be toward the position of the sun in the sun's rest frame, but toward where the sun would be now if its position were extrapolated linearly using the relative velocity. Putting the sun immobile at the origin, when the Earth is moving in an orbit of radius R with velocity v and gravity moves with velocity c, extrapolating the position of the sun from its apparent position relative to the Earth moves the sun over by an amount equal to vR/c, which is the travel time of gravity from the sun to the Earth times the relative velocity of the sun and the earth. The pull of gravity is always displaced opposite the direction of the Earth's velocity, so that the Earth is always being pulled backwards, causing a drag which leads the orbit to decay. The decay is only suppressed by an amount v/c compared to the force which keeps the Earth in orbit, and since the Earth's orbit is stable, c must be very very big.
In a field equation consistent with special relativity, the attraction is always toward the instantaneous position of the sun, not the extrapolated position. When an object is moving at a steady speed, the effect on the orbit is order v2/c2, and the effect preserves energy, and orbits don't decay, they precess. The attraction toward an object moving with a steady velocity is towards its instantaneous position.
[edit] Electrodynamical analogies
Early 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 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 Maurice 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. So those hypotheses were rejected.[2][3]
However, a more important variation of those attempts was the theory of Paul Gerber, who derived in 1898 the identical formula, which was also derived later 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. 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.[4][5][6]
Lorentz
In 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:[7]
"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."
In 1908 Henri Poincaré 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.[8]
Lorentz covariant models
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:[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:[10]
"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."
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.[11]
General relativity
Background
In general relativity, the gravitational potential is identified with the metric tensor and the gravitational force field with the Christoffel symbols of the spacetime manifold. Tidal gravitational field is associated with the curvature of spacetime. 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.[clarification needed] 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.[12] 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.[13]
Possible Experimental measurements
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. Notably, prior to the actual transit, Hideki Asada in a paper to the Astrophysical Journal Letters theorized that the proposed experiment was essentially a roundabout confirmation of the speed of light instead of the speed of gravity. [15] Further, there has been criticism of the means by which these results were presented, in that they were announced at a meeting of the AAS instead of being submitted for peer review. [16] However, Kopeikin and Fomalont continue to vigorously argue their case and the means of presenting their result at the press-conference of AAS that was offered after the peer review of the results of the Jovian experiment had been done by the experts of the AAS scientific organizing committee. Asada's claim was found theoretically unsound and disproved in later publication by Kopeikin and Fomalont [17], which operates with a bi-metric formalism that splits the space-time null cone in two - one for gravity and another one for light. The two null cones overlap in general relativity, which makes tracking the speed-of-gravity effects difficult and requires a special mathematical technique of gravitational retarded potentials, which was worked out by Kopeikin and co-authors [18][19] but was never properly employed by Asada and/or the other critics.
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.
