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Speed of gravity

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General relativity subsumes the Newtonian instantaneous gravitation in the form of the gravitoelectric field.[1][2][3] There is also a special case of nonuniformly accelerating masses, which generate gravitational waves propagating at the speed of light. However, the instantaneously acting Newtonian component of gravitation usually provides the dominant contribution.[4]

Newtonian gravitation

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 Urbain Le Verrier determined in 1859 that the elliptical orbit of Mercury precesses at a significantly different rate than is predicted by Newtonian theory.[5]

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.[6] 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 as 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 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.

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.[7][8]

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.[9][10][11]

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:[12]

"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.[13]

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:[14]

"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:[15]

"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.[16]

General relativity

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 and weak gravitational field (whose intensity is below the minimum quantum of action—the Planck constant) will produce, according to general relativity, similar effects to those we might expect from Newtonian gravitation. In particular, the gravitoelectric (static and continuous) component of a gravitational field should not be confused with a possible additional gravitoelectromagnetic component (quanta of gravitational radiation); see Petrov classification.[2][3][17]

Since the intensity of the gravitoelectric field is below the Planck constant, it cannot be used for superluminal transmission of quantized (discrete) information, i.e., it could not constitute a well-ordered series of impulses carrying a well-defined meaning. If an elementary particle were to suddenly be displaced from its position, the whole universe would instantaneously receive a corresponding gravitoelectric signal, but this signal would be scrambled by its constructive and destructive interference with the gravitoelectric signals of all the other particles of the universe.

However, in the case of two gravitoelectrically interacting particle ensembles, such as two planets or stars, the quasi-random quantum fluctuations in the instantaneous gravitoelectric signals of individual particles become averaged out, so that if one of the two interacting celestial bodies becomes displaced, the other will be instantaneously and precisely informed about the new position of its gravitoelectric counterpart (the precision will be proportional to the number of the particles constituting the two gravitoelectrically interacting celestial bodies).

On the scale of the whole universe, the instantaneous gravitoelectric signals of individual elementary particles experience mutual constructive and destructive interference and form an interference pattern of gravitoelectric peaks and troughs, so that at every Planckian moment of time, the universe becomes precisely informed which of its particles' possible next configurations is characterized by the lowest total gravitational potential energy and should be chosen. This provides the arrow of time and eliminates the necessity for the many-worlds interpretation ("God does not play dice").

Measuring the speed of gravity

In relativistic quantum theory, a system cannot be localized to a precision better than its Compton wavelength,[18] expressed as λ = hc/E = c/f.

At f = 1 Hz, a quantum's energy (E = fh) is equal to h (the breakeven point between wave-likeness and particle-likeness), while the quantum's Compton wavelength (the radius of nonlocality, instantaneous propagation) is 1 light-second (300 thousand kilometres). According to the theory of relativity, superluminal propagation is propagation into the past. Therefore:

Conclusion:

  1. Gravitoelectromagnetic waves, whose frequencies are below 1 Hz, are undetectable.[20][21]
  2. Gravitoelectromagnetic waves, whose frequencies are above 1 Hz, do not exist.

References

  1. ^ Yau, Shing-Tung ♦ Seminar on differential geometry Institute for Advanced Study (Princeton, N.J.), p. 554
  2. ^ a b Grøn, Øyvind; Hervik, Sigbjørn ♦ Einstein's general theory of relativity: with modern applications in cosmology Springer, 2007, p. 203 ♦ "The gravitoelectric field is the Newtonian part of the gravitational field, while the gravitomagnetic field is the non-Newtonian part."
  3. ^ a b Hawking, Stephen W.; Israel, W. ♦ Three hundred years of gravitation Cambridge University Press, 1989, p. 283
  4. ^ a b "McGraw-Hill Dictionary of Scientific and Technical Terms". Retrieved 2 Dec 2010.
  5. ^ U. Le Verrier, Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète, C. R. Acad. Sci. 49 (1859), 379–383.
  6. ^ Laplace, P.S.: (1805) "A Treatise in Celestial Mechanics", Volume IV, Book X, Chapter VII, translated by N. Bowditch (Chelsea, New York, 1966)
  7. ^ Zenneck, J. (1903). "Gravitation". Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (in German). 5: 25–67.
  8. ^ Roseveare, N. T (1982). Mercury's perihelion, from Leverrier to Einstein. Oxford: University Press. ISBN 0198581742.
  9. ^ Gerber, P. (1898). "Die räumliche und zeitliche Ausbreitung der Gravitation" . Zeitschrift für mathematische Physik (in German). 43: 93–104.
  10. ^ Zenneck, pp. 49-51
  11. ^ "Gerber's Gravity". Mathpages. Retrieved 2 Dec 2010. {{cite web}}: Cite has empty unknown parameter: |1= (help)
  12. ^ Lorentz, H.A. (1900). "Considerations on Gravitation" . Proc. Acad. Amsterdam. 2: 559–574.
  13. ^ Poincaré, H. (1908). "La dynamique de l'électron". Revue générale des sciences pures et appliquées. 19: 386–402. {{cite journal}}: External link in |title= (help) Reprinted in Poincaré, Oeuvres, tome IX, S. 551-586 and in "Science and Method" (1908)
  14. ^ 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 of Mathematical Physics". In Rogers, Howard J. (ed.). Congress of arts and science, universal exposition, St. Louis, 1904. Vol. 1. Boston and New York: Houghton, Mifflin and Company. pp. 604–622. {{cite book}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help) Reprinted in "The value of science", Ch. 7-9.
  15. ^ Poincaré, H. (1906). "Sur la dynamique de l'électron" (PDF). Rendiconti del Circolo matematico di Palermo (in French). 21: 129–176. doi:10.1007/BF03013466. See also the English Translation.
  16. ^ Walter, Scott (2007). Renn, J. (ed.). The Genesis of General Relativity. 3. Berlin: Springer: 193–252. {{cite journal}}: |contribution= ignored (help); Missing or empty |title= (help)
  17. ^ On page 94 of his book Space, Time and Gravitation: an Outline of the General Relativity Theory, Sir Arthur Eddington draws an analogy between the gravitational field and the attraction exerted by an electric charge. In Note 6 (p. 94), he cites a formula for the scalar Liénard–Wiechert potential, showing how the action of a slowly moving electric charge reduces to the instantaneous action of its static potential.
  18. ^ Ji, Xiangdong (2004). "Viewing the proton through "color" filters". In Boffi, S.; Ciofi degli Atti, C.; Giannini, M. M. (ed.). Perspectives in hadronic physics: 4th international conference held at ICTP, Trieste, Italy, 12-16 May 2003. p. 24. {{cite book}}: |access-date= requires |url= (help); External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)CS1 maint: multiple names: editors list (link)
  19. ^ Corning, Peter A. ♦ Holistic Darwinism: synergy, cybernetics, and the bioeconomics of evolution University of Chicago Press, 2005, p. 340 (Since the gravitational field propagates into the past, its entropy decreases with time.)
  20. ^ Davies, P. C. W. ♦ The search for gravity waves CUP Archive, 1980, p. 129
  21. ^ When the average Compton wavelength of the gravitons is 300 thousand kilometres, it does not matter whether one detects the velocity of a single graviton or the group velocity of a myriad of gravitons—in both cases, the signal will arrive to both arms of the interferometer simultaneously.
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