Speed of gravity

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In classical theories of gravitation, the changes in a gravitational field propagate. A change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational field which it produces. In a more physically correct sense, the "speed of gravity" refers to the speed of a gravitational wave, which is the same speed[citation needed] as the speed of light (c) as predicted by General Relativity and confirmed by observation of the GW170817 neutron star merger.

Introduction[edit]

The speed of gravitational waves in the general theory of relativity is equal to the speed of light in a vacuum, c.[1] Within the theory of special relativity, the constant c is not exclusively about light; instead it is the highest possible speed for any interaction in nature. Formally, c is a conversion factor for changing 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 also the speed of gravitational waves and any massless particle. Such particles include the gluon (carrier of the strong force), the photons that make up light, and the hypothetical gravitons which make up the associated field particles of gravity (however a theory of the graviton requires a theory of quantum gravity).

Static fields[edit]

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 observer-effects. These changes in direction of a static field, because of relativistic considerations, are the same for an observer when a distant charge is 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 a gravitational or electric field) does not change the field. For static fields, such as the electrostatic field connected with electric charge, or the gravitational field connected to a massive object, the field extends to infinity, and does not propagate. Motion of an observer does not cause the direction of such a field to change, and by symmetrical considerations, changing the observer frame so that the charge appears to be moving at a constant rate, also does not cause the direction of its field to change, but requires that it continue to "point" in the direction of the charge, at all distances from the charge. Challenges to Self-Acceleration in Modified Gravity from Gravitational Waves and Large-Scale Structure The consequence of this is that static fields (either electric or gravitational) always point directly to the actual 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 the charged 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 to change 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 the field. For example, when an observer begins to move with respect to a static field that already extends over light years, it appears as though "immediately" the entire field, along with its source, has begun moving 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 represent any sort of propagation that is faster than light.

Newtonian gravitation[edit]

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 from that predicted by Newtonian theory.[3]

Laplace[edit]

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.[4] 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 Lorentz' invariance of static fields, Laplace assumed that when an object like the Earth is moving around the Sun, the attraction of the Earth would not be toward the instantaneous position of the Sun, but toward where the Sun had been if its position was retarded using the relative velocity (this retardation actually does happen with the optical position of the Sun, and is called annual solar aberration). Putting the Sun immobile at the origin, when the Earth is moving in an orbit of radius R with velocity v presuming that the gravitational influence moves with velocity c, moves the Sun's true position ahead of its optical position, 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 (if it behaved like a wave, such as light) would then be always displaced in the direction of the Earth's velocity, so that the Earth would always be pulled toward the optical position of the Sun, rather than its actual position. This would cause a pull ahead of the Earth, which would cause the orbit of the Earth to spiral outward. Such an outspiral would be suppressed by an amount v/c compared to the force which keeps the Earth in orbit; and since the Earth's orbit is observed to be stable, Laplace's c must be very large. As is now known, it may be considered to be infinite in the limit of straight-line 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 with no delay, for both gravity and electric charge. In a field equation consistent with special relativity (i.e., a Lorentz 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 the Sun), not the time-retarded position of the Sun. When an object is moving in orbit at a steady speed but changing velocity v, the effect on the orbit is order v2/c2, and the effect preserves energy and angular momentum, so that orbits do not decay.

Electrodynamical analogies[edit]

Early theories[edit]

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 invalidated 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.[5][6]

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.[7][8][9]

Lorentz[edit]

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:[10]

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.[11]

Lorentz covariant models[edit]

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:[12]

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:[13]

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.[14] Whitehead's theory of gravitation (1922) explains gravitational red shift, light bending, perihelion shift and Shapiro delay.[15]

General relativity[edit]

Background[edit]

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 like those of Newtonian gravitation.

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

Two gravitoelectrically interacting particle ensembles, e.g., two planets or stars moving at constant velocity with respect to each other, each feel a force toward the instantaneous position of the other body without a speed-of-light delay because Lorentz invariance demands that what a moving body in a static field 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" therefore causes Lorentz invariance to require that in the previously moving body's reference frame the (now moving) 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 distance and 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 not propagate. If such a source of a static field is accelerated (for example stopped) with regard to its formerly constant velocity frame, its distant field continues to be updated as though the charged body continued with constant velocity. This effect causes the distant fields of unaccelerated moving charges to appear to be "updated" instantly for their constant velocity motion, as seen from distant positions, in the frame where the source-object is moving at constant velocity. However, as discussed, this is an effect which can be removed at any time, by transitioning to a new reference frame in which the distant charged body is now at rest.

The static and continuous gravitoelectric component of a gravitational field is not a gravitomagnetic component (gravitational radiation); see Petrov classification. The gravitoelectric field is a static field and therefore cannot superluminally transmit quantized (discrete) information, i.e., it could not constitute a well-ordered series of impulses carrying a well-defined meaning (this is the same for gravity and electromagnetism).

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

The finite speed of gravitational interaction in general relativity does not lead to the sorts of problems with the aberration of gravity that Newton was originally concerned with, because there is no such aberration 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 past each other with unchanging velocity) the orbital results calculated by general relativity are the same as those of Newtonian gravity with instantaneous action at a distance, because they are modelled by the behavior of a static field with constant-velocity relative motion, and no aberration for the forces involved.[16] Although the calculations are considerably more complicated, one can show that a static field 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 the electromagnetic Liénard–Wiechert potential theory of the fields from a moving charge, does not suffer from either aberration or positional-retardation. Only the term corresponding to acceleration and electromagnetic emission in the Liénard–Wiechert potential shows a direction toward the time-retarded position of the emitter.

It is in fact 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.[17]

Formulaic conventions[edit]

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

Measurements[edit]

The detection of the neutron star inspiral GW170817 in 2017, detected through both gravitational waves and gamma rays, provides the so far by far best limit on the difference between the speed of light and that of gravity. Photons were detected 1.7 seconds after peak gravitational wave emission; assuming a delay of zero to ten seconds, the difference between the speeds of gravitational and electromagnetic waves, vGW − vEM, is constrained to between −3×10−15 and +7×10−16 times the speed of light[18].

This also excluded some alternatives to general relativity, including variants of scalar–tensor theory[19][20][21][22] and Hořava–Lifshitz gravity[23][24][25].

Notes[edit]

References[edit]

  1. ^ Hartle, JB (2003). Gravity: An Introduction to Einstein's General Relativity. Addison-Wesley. p. 332. ISBN 981-02-2749-3. 
  2. ^ Taylor, Edwin F. and Wheeler, John Archibald, Spacetime Physics, 2nd edition, 1991, p. 12.
  3. ^ Verrier U. Le (1859). "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: 379–383. 
  4. ^ Laplace, P.S.: (1805) "A Treatise in Celestial Mechanics", Volume IV, Book X, Chapter VII, translated by N. Bowditch (Chelsea, New York, 1966)
  5. ^ Zenneck, J. (1903). "Gravitation". Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (in German). 5: 25–67. doi:10.1007/978-3-663-16016-8_2. 
  6. ^ Roseveare, N. T (1982). Mercury's perihelion, from Leverrier to Einstein. Oxford: University Press. ISBN 0-19-858174-2. 
  7. ^ Gerber, P. (1898). "Die räumliche und zeitliche Ausbreitung der Gravitation". Zeitschrift für mathematische Physik (in German). 43: 93–104. 
  8. ^ Zenneck, pp. 49–51
  9. ^ "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 of Mathematical 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 of science", 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.
  14. ^ Walter, Scott A. (2007). Renn, J.; Schemmel, M., eds. "Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910". 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", to be submitted to Studies In History And Philosophy Of Modern Physics (2006).
  16. ^ Carlip, S. (2000). "Aberration and the Speed of Gravity". Phys. Lett. A. 267 (2–3): 81–87. arXiv:gr-qc/9909087Freely accessible. Bibcode:2000PhLA..267...81C. doi:10.1016/S0375-9601(00)00101-8. 
  17. ^ * 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/0403060Freely accessible. Bibcode:2004CQGra..21.3803C. doi:10.1088/0264-9381/21/15/011. 
  18. ^ Abbott, B. P.; et al. (2017). "Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A". The Astrophysical Journal Letters. 848 (2): L13. arXiv:1710.05834Freely accessible [astro-ph.HE]. doi:10.3847/2041-8213/aa920cFreely accessible. 
  19. ^ Lombriser, Lucas; Taylor, Andy (28 September 2015). "Breaking a Dark Degeneracy with Gravitational Waves". arXiv:1509.08458Freely accessible [astro-ph.CO]. 
  20. ^ Lombriser, Lucas; Lima, Nelson (24 February 2016). "Challenges to Self-Acceleration in Modified Gravity from Gravitational Waves and Large-Scale Structure". arXiv:1602.07670Freely accessible [astro-ph.CO]. 
  21. ^ "Quest to settle riddle over Einstein's theory may soon be over". phys.org. 2017-02-10. Retrieved 2017-02-10. 
  22. ^ "Theoretical battle: Dark energy vs. modified gravity". arstechnica.co.uk. 2017-02-25. Retrieved 2017-10-27. 
  23. ^ Creminelli, Paolo; Vernizzi, Filippo (16 October 2017). "Dark Energy after GW170817". arXiv:1710.05877Freely accessible [astro-ph.CO]. 
  24. ^ Sakstein, Jeremy; Jain, Bhuvnesh (16 October 2017). "Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories". arXiv:1710.05893Freely accessible [astro-ph.CO]. 
  25. ^ Ezquiaga, Jose María; Zumalacárregui, Miguel (16 October 2017). "Dark Energy after GW170817". arXiv:1710.05901Freely accessible [astro-ph.CO]. 

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