"The speed of
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.
2001, S. Kopeikin proposed an experiment to test the speed of gravity [].
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 []
and independently by H. Asada [].
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 [],
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
The physical meaning of "the speed of gravity"
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.
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.
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 [].
In 1825, Laplace determined that the
minimum speed of gravity consistent with observations was at least 10 million
times the speed of light, c.
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.
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
A 1997 laboratory experiment by Walker
& Dual showed that gravitational signals propagated much faster than light
Binary pulsars (with large masses and
speeds) show that the speed of gravity must be at least 20 billion times the
speed of light.
Newtonian gravitation, that propagation speed is infinite [].
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
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
Einstein's GR, while not addressing the speed of gravity directly, always
adopts infinite speed for it when deriving equations of motion [],
even in Einstein's own work [],
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.
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 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."
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.
here ignores both the existence of a long-standing controversy about the speed
of gravity (defined as the propagation speed of gravitational force)  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.
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
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  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.
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.
speed of gravity is the subject of a recent definitive paper concluding that
the real physical parameter vg must be much greater than c [].
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
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"
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
He also concludes that only the speed of light, not the speed of gravity, was
Note added 2003/05/26: Clifford Will made a presentation to the American
Physical Society explaining why Kopeikin and Fomalont's interpretation is
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 [].
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
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 [].
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 [].
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.
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.
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.
Kopeikin new paper at the
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.
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.
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.
the public supporting the research
Kopeikin's announced result, reference  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
S. Kopeikin (2001), "Testing the relativistic
effect of the propagation of gravity by very long baseline
interferometry", Astrophys.J. 556, L1-L5.
H. Asada (2002), Astrophys.J. 574,
T. Van Flandern (1998) , “The speed of gravity –
What the experiments say”, Phys.Lett.A 250, 1-11.
C.W. Misner, K.S. Thorne & J.A. Wheeler (1973),
Gravitation, W.H. Freeman & Co., San Francisco, 177.
C.W. Misner et al. (1973), op.cit., 1095.
A. Einstein, L. Infeld & B. Hoffmann (1938),
“The gravitational equations and the problem of motion”, Ann.Math.
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.
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.