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Kopeikin and the Speed of Gravity

Good judgment comes from experience. Experience comes from bad judgment.

Meta Research Press Release

"The speed of gravity"

Abstract. New findings were announced on 2003/01/08 by S. Kopeikin, claiming to have measured the "speed of gravity" and finding it essentially equal to the speed of light. These findings are invalid by both experimental and theoretical standards because the quantity measured was already known to propagate at the speed of light. The hyped claims therefore do a disservice to science in general and the advancement of physics in particular because the announced findings do not represent the meaning of the actual experimental results and cannot possibly represent the physical quantity heretofore called "the speed of gravity", which has already been proved by six experiments to propagate much faster than light, perhaps billions of times faster. Several mainstream relativists have also stated their disagreement that the experiment really measured what it claimed to measure.


In 2001, S. Kopeikin proposed an experiment to test the speed of gravity [[1]]. However, his result as described would have been a hybrid of near-instantaneous effects and lightspeed-delayed effects. The physical interpretation in his proposal (but not the math or the experiment itself) was objected to by T. Van Flandern [[2]] and independently by H. Asada [[3]]. The experiment was then funded and carried out in 2002 September, with results initially expected last October. When no results were forthcoming yet in December, a rumor began circulating in USENET newsgroup sci.physics that the results were not coming out in accord with expectations and were being scrutinized. On December 30, Kopeikin posted a new paper on the internet containing new algorithms and formulas for the analysis [[4]], incompatible with his own pre-observation published formulas. On 2003 January 8, he gave an oral talk at the Seattle meeting of the American Astronomical Society (AAS) announcing his numerical results, continuing to claim that they measured "the speed of gravity": cg = (0.95+/-0.25)c, where c is the speed of light. 

The physical meaning of "the speed of gravity"

            Although gravitation and relativity are technical subjects, the mistake made by Kopeikin is not unlike measuring the speed of a falling apple and claiming that is the speed of gravity. The following section provides the background to understand what actually was done wrongly. 

            All gravitational phenomena unique to Einstein's general relativity (GR), such as light bending, gravitational redshift, perihelion advance, and Shapiro delay of radio or radar signals, arise in a static or near-static gravitational potential field, also sometimes called in various contexts by the names "the gravitational field", "the space-time medium", "the light-carrying medium", "aether", or "elysium". Disturbances of this potential field or medium are called "gravitational waves". According to GR, such waves propagate at the speed of light, as do all other phenomena associated with the potential field that propagate at all. This gravitational wave speed has been confirmed directly by measures of the aforementioned phenomena unique to GR, and indirectly by binary pulsar observations. There is no current dispute about this, and no expectation of any other result for the propagation speed of gravitational waves. However, the name notwithstanding, "gravitational waves" have nothing to do with gravitational force. They are ultra-weak disturbances of the potential field or space-time medium due to the acceleration of bodies. So far, they have proved too weak to detect directly in any laboratory or astrophysical experiment. They are certainly far too weak to have any observable influence on any macroscopic body in their path. 

            By contrast, gravitational forces are large, readily detected, and control the dynamics of most of the visible universe. Gravimeters easily detect the gravitational force from, and motion of, a person entering a room, for example. The propagation speed of gravitational force is bounded by six experiments to be much faster than the speed of light [[5]]. For example:

n     In 1825, Laplace determined that the minimum speed of gravity consistent with observations was at least 10 million times the speed of light, c.

n     Modern, high-precision solar system observations show that the direction from which the Sun's light comes, and the direction toward which the Sun's gravity pulls us, are not the same. The former is retarded by the time it takes light to travel from Sun to Earth, 8.3 minutes; and the latter is not retarded by any detectible amount.

n     Eclipses of the Sun by the Moon occur about 40 seconds before the time of the Sun's maximum gravitational pull on the Moon. The delay indicates that light and gravity do not have the same propagation speed.

n     A 1997 laboratory experiment by Walker & Dual showed that gravitational signals propagated much faster than light signals.

n     Binary pulsars (with large masses and speeds) show that the speed of gravity must be at least 20 billion times the speed of light. 

            In classical Newtonian gravitation, that propagation speed is infinite [[6]]. But instead of getting closer to GR if propagation in Newtonian gravity is reduced to the speed of light, the gravitational model disintegrates and fails to represent reality in a gross way. Nature insists that gravitational forces must propagate without any delays nearly as large as light-speed delays. 

            The transition from equations governing the potential field (Einstein equations) to equations governing the force or 3-space acceleration of gravity (equations of motion) involves forming the gradient of the potential (the slope of the potential well). The issue of the "speed of gravity" (meaning propagation speed of gravitational force) revolves around whether one uses an instantaneous or retarded gradient. If nearly instantaneous, that corresponds with approximately infinite propagation speed for gravitational force. If the gradient is retarded by light-propagation speed, orbits become growing spirals that quickly depart from any observed motion. The physical significance of this difference – that gravity appears to act instantly, and light appears to act with delay – is what has been the subject of discussion and debate for the last decade. 

Propagation delay is not important for gravitational potential because the potential field is already present near any source mass, and does not need to propagate to get there. Its shape is described by the Einstein equations, or for some purposes by retarded potential equations (analogous to the Lienard-Wiechert potential in electrodynamics). Propagation delay effects depend on the propagation speed to the first power. By contrast, relativistic phenomena depend on c2, where c (the speed of light) then plays the role of the permeability or permittivity of space-time (its stiffness or resistance to curving) or the index of refraction of the space-time medium (in flat space-time interpretations of GR). So any use of terms containing c2 would have nothing to do with propagation speed. If we let vg represent the physical propagation speed of gravitational force (not the same thing as Kopeikin's cg!), propagation delay would be proportional to vg to the first power. 

Einstein's GR, while not addressing the speed of gravity directly, always adopts infinite speed for it when deriving equations of motion [[7]], even in Einstein's own work [[8]], by dropping any retardation for the gradient of the gravitational potential field. That is, of course, the primary manifestation for any propagation delay for gravitational force; and dropping it effectively sets that propagation delay to zero, or its propagation speed to infinity. 

A common thought experiment asks: "What would happen to the Earth's orbit if the Sun suddenly ceased to exist?" The answer is now clear. The usual relationship "force is the gradient of the potential" would instantly end. The Sun's potential field would then begin to dissipate, taking 8.3 minutes to dissipate out to the distance of the Earth's orbit; so effects such as light-bending and clock-slowing would persist for that long. But the Newtonian component of gravitational force, the force that keeps Earth in its orbit, would cease almost instantly, and Earth would fly off along a straight line like a weight on a spinning merry-go-round that broke free from its moorings. 

Kopeikin's experiment

Kopeikin's latest paper on the internet, giving the basis for his findings announced at the AAS meeting, contains some egregious errors. The following claims appear therein: "… a moving gravitating body deflects light not instantaneously but with retardation caused by the finite speed of gravity propagating from the body to the light ray. … We calculated this correction for Jupiter by making use of the post-Minkowskian approximation based on the retarded Lienard-Wiechert solutions of the Einstein equations. … Speed of gravity cg must enter the left side of the Einstein equations (2) … This will lead to the wave operator depending explicitly on the speed of gravity cg." 

None of these statements is correct even in GR, provided only that "the speed of gravity" retains its classical meaning for the past two centuries of force propagation speed. The Einstein equations require the potential field of all bodies to act from the body's instantaneous direction, not its retarded direction, because they set propagation delay for the gradient to zero. But Kopeikin adopts the Sun acting from its instantaneous position and Jupiter acting from its retarded position, which is inconsistent. In fact, although the Sun moves 1000 times more slowly than Jupiter, it is 1000 times more massive, making any hypothetical retardation effects comparably important. The Lienard-Wiechert equations consider retardation in mutual distance, but not in direction – the latter being a much larger effect of propagation delay. And the parameter on the left side of the Einstein equations is c2, and therefore has nothing to do with the speed of gravity, as we noted above. This does not prevent Kopeikin from calling it "cg" and solving for this parameter as if it were the speed of gravity, which is what he has done. 

            Sadly, Kopeikin here ignores both the existence of a long-standing controversy about the speed of gravity (defined as the propagation speed of gravitational force) [5] and the aforementioned arguments raised against his original interpretation by others. Kopeikin used the notion that this experiment might determine "the speed of gravity" to aggrandize the experiment, and perhaps also to justify funding for doing it. Yet the cg parameter measured is more closely related to the speed of light per se than anything else. 

To clarify, it is well known to physicists that electromagnetic signals (whether light passing the Sun or quasar radio signals passing Jupiter) are not bent or slowed by the force of gravity, but by passage through a gravitational potential field. A potential field slows the rates at which clocks tick, produces gravitational redshift, bends light, and retards radar and radio signals. Gravitational force, by contrast, has no such effects even in fields as strong as 1019 g, where g = acceleration of gravity at Earth's surface [[9]]. Gravitational force simply produces the 3-space (Newtonian) acceleration of bodies. Kopeikin has not cleanly separated potential-change propagation effects from force propagation effects, despite an attempt to do so in his latest paper that was absent from the original paper. 

Kopeikin makes another claim in his new paper: "The spectrum of plausible values of cg ranges from cg = c in general relativity to cg = infinity as advocated by Van Flandern (1998)." This is also false. Van Flandern has long maintained in USENET discussions and on the Meta Research web site [2] that Kopeikin's cg parameter is essentially equal to the speed of light. So this statement by Kopeikin is again an attempt to falsely claim that his experiment bears on the subject of the speed of propagation of gravitational force, which it does not. 

            However, the misrepresentation in this new paper and announcement is more serious than mixing speed-of-light and speed-of-gravity parameters. Kopeikin's new paper has modified the equations to be used in determining the speed of gravity in a fundamental way. His own formalism now rules out the possibility of cg = infinity or cg >> c in his results even before the experiment is performed. Here is why. Kopeikin now defines a new time tau = (c/cg) t to replace the coordinate time t in the Einstein equation. However, because (c/cg) is obviously forced to become very small or zero for large or infinite cg, the role of the time coordinate is diminished or suppressed altogether by this substitution, which effectively eliminates many relativistic effects already verified in other experiments. So even if the speed of "gravitational waves" had been much faster than the speed of light, Kopeikin's experiment is incapable of showing that with his present method of analysis. More than that, Kopeikin has violated scientific protocol by changing the equations to be used for the analysis after the results are in, thereby presumably avoiding the embarrassment of having to announce an unexpected result. We were also unable to verify one of his key references in the December 30 paper, "E. Fomalont & S. Kopeikin (2002)" which says simply "submitted to Science". But as of January 6, Science magazine has no record of such a submission. 

            The basic point here about the physical meaning of the speed of gravitational force as it appears in relativity theory has been brought to Kopeikin's attention by at least two authors of published technical papers, yet is still being ignored. Now Kopeikin has altered the analysis equations after the results were in. This raises serious questions about whether Kopeikin has maintained his scientific objectivity after using the promise of a measurement of the speed of gravity to justify funding for his experiment. Almost certainly, his erroneous announcement has damaged scientific inquiry into an important and worthy matter, the speed of propagation of gravitational force. 

The speed of gravity is the subject of a recent definitive paper concluding that the real physical parameter vg must be much greater than c [[10]]. Because this paper is the third in this series to appear in mainstream journals, because both its authors are senior and widely published, because this paper was rigorously peer-reviewed (as appropriate for controversial subject matter), and because it addresses every objection raised by anyone over the past decade in a way that was satisfactory to neutral parties, including the journal editors, there is no scientifically valid excuse for ignoring or riding roughshod over these results by creating the false impression that Kopeikin's experiment supercedes these already published findings. Moreover, because the viewpoint that the speed of gravity vg must be >> c is in good standing by the aforementioned criteria, there is no good reason why Kopeikin should refuse to debate this matter in a suitable forum. He is hereby challenged to do so. 

          Note added 2003/01/10: Noted relativist C. Will has now joined those who agree that Kopeikin's result measured only the speed of light and not the speed of gravity [[11]]. 

Note added 2003/01/18: C. Will's objections are detailed; and Peter van Nieuwenhuizen, a physicist at Stony Brook University in New York, calls the interpretation of the results by Fomalont and Kopeikin "complete nonsense" [[12]]. 

          Note added 2003/02/09: Retired physicist K. Nordtvedt, instrumental in proposing a key test of relativity theory, is quoted in Nature magazine of 16 January (p. 198): "The experiment is wonderful, but it has nothing to do with the speed of gravity." 

          Note added 2003/03/20: J.A. Faber, Northwestern U. has calculated the expected experimental results using an infinite speed of gravity and found no difference [[13]]. He also concludes that only the speed of light, not the speed of gravity, was measured.

          Note added 2003/05/26: Clifford Will made a presentation to the American Physical Society explaining why Kopeikin and Fomalont's interpretation is incorrect [[14]].

          Note added 2003/06/22: C.M. Will argues that the Jupiter-passing-quasar experiment is not sensitive to the speed of propagation of gravity [[15]]. In a related paper, S. Samuel at Lawrence Berkeley National Laboratory makes a similar argument, and concludes that Kopeikin measured the speed of light, not gravity [[16]].

Note added 2003/08/04: Stuart Samuel showed that the real effects of the speed of gravity were at least 100 times too small to have been measured in the Kopeikin experiment [[17]].

          Note added 2004/01/14: The Kopeikin-Fomalont paper on the experimental results, previously rejected, is restyled as a measurement of the deflection of a light signal by Jupiter and published [[18]].


n     S. Kopeikin misquotes Van Flandern as predicting that cg (Kopeikin’s “speed of gravity”) will be infinity. Van Flandern and Vigier are in print showing that six experiments better than Kopeikin’s already show that the speed of gravity is >> c (c = speed of light). But in posted discussions with Kopeikin and in USENET newsgroups, Van Flandern clearly states that Kopeikin’s “cg” parameter cannot be the speed of gravity and will certainly come out near the value c, as it did.

n     Asada followed up Kopeikin’s Astrophysical Journal paper with his own paper in the same journal showing that Kopeikin was simply measuring a quantity that propagated at the speed of light, and was definitely not measuring the speed of gravity.

n     Kopeikin has mostly ignored these well-founded corrections. We note that he justified funding for his experiment by the claim that it would measure the speed of gravity.

n     Kopeikin new paper at the Los Alamos archive revises his protocol, equations, and methodology. Scientific method forbids changing the protocol after the results are in, especially when it is done to avoid an unwanted or unexpected result.

n     Kopeikin’s new equations introduce a new factor, c/cg, for time in the Einstein equations. This factor drives time intervals to zero for large values of cg, thereby making large cg results *impossible* for any experimental data regardless of reality.

n     Kopeikin’s now-forced results do a great disservice to science in general and the advancement of physics in particular because they no longer represent what his own experiment showed, much less the speed of gravity. 

Significance to the public supporting the research

            Contrary to Kopeikin's announced result, reference [10] shows that the speed of light is no longer a universal speed limit. Travel and communication at unlimited speeds are now possible. These take place in forward time, creating no paradoxes. (E.g., you can't go back in time and kill your own grandfather when he was still a child.) Nothing at all about the mathematical theory of relativity is altered. However, the experimental interpretation of special relativity now favors Lorentz's version over Einstein's. And the experimental interpretation of general relativity now favors the force interpretation (as preferred by Einstein, Dirac, and Feynman, among others) over the geometric interpretation ("curved space-time"). 

Tom Van Flandern

Meta Research



[last updated: 2004/01/14]

[1] S. Kopeikin (2001), "Testing the relativistic effect of the propagation of gravity by very long baseline interferometry", Astrophys.J. 556, L1-L5.

[3] H. Asada  (2002), Astrophys.J. 574, L69-L70.

[5] T. Van Flandern (1998) , “The speed of gravity – What the experiments say”, Phys.Lett.A 250, 1-11.

[6] C.W. Misner, K.S. Thorne & J.A. Wheeler (1973), Gravitation, W.H. Freeman & Co., San Francisco, 177.

[7] C.W. Misner et al. (1973), op.cit., 1095.

[8] A. Einstein, L. Infeld & B. Hoffmann (1938), “The gravitational equations and the problem of motion”, Ann.Math. 39, 65-100.

[9] J. Bailey, K. Borer, F. Combley, et al. (1997), “Measurements of relativistic time dilation for positive and negative muons in a circular orbit”, Nature 268, 301-305.

[10] T. Van Flandern and J.P. Vigier (2002), “Experimental Repeal of the Speed Limit for Gravitational, Electrodynamic, and Quantum Field Interactions”, Found.Phys. 32, 1031-1068.

[11] C.M. Will (2003), "Propagation speed of gravity and the relativistic time delay", <>.

[12] R.R. Britt (2003), "Speed of gravity results 'incorrect', physicist says", <>.

[13] J.A. Faber (2003), "The speed of gravity has not been measured from time delays", <>.

[14] APS News Online (2003), "Physicist disputes speed of gravity claim", <>.

[15] C.M. Will (2003), "Propagation speed of gravity and the relativistic time delay", ApJ 590, 683-690.

[16] S. Samuel (2003), "On the speed of gravity and the v/c corrections to the Shapiro time delay", <>.

[17] S. Samuel (2003), "On the speed of gravity and the v/c corrections to the Shapiro time delay", Phys.Rev.Lett. June; also at <>.

[18] E.B. Fomalont and S.M. Kopeikin (2003), "The measurement of light deflection from Jupiter: experimental results", ApJ 598, 704-711.