"$d_web"'My sports & clubs/natural- CNPS/Flandern 1998 Speed of light Meta Research Bulletin of 6_15_94.txt'
The Speed of Gravity – What the Experiments Say
Tom Van Flandern
Meta Research
>
[as published in Physics Letters A 250:1-11 (1998)]
/Abstract./Standard experimental techniques exist to determine the
propagation speed of forces. When we apply these techniques to gravity,
they all yield propagation speeds too great to measure, substantially
faster than lightspeed. This is because gravity, in contrast to light,
has no detectable aberration or propagation delay for its action, even
for cases (such as binary pulsars) where sources of gravity accelerate
significantly during the light time from source to target. By contrast,
the finite propagation speed of light causes radiation pressure forces
to have a non-radial component causing orbits to decay (the
�Poynting-Robertson effect�); but gravity has no counterpart force
proportional to to first order. General relativity (GR) explains these
features by suggesting that gravitation (unlike electromagnetic forces)
is a pure geometric effect of curved space-time, not a force of nature
that propagates. Gravitational radiation, which surely does propagate at
lightspeed but is a fifth order effect in , is too small to play a role
in explaining this difference in behavior between gravity and ordinary
forces of nature. Problems with the causality principle also exist for
GR in this connection, such as explaining how the external fields
between binary black holes manage to continually update without benefit
of communication with the masses hidden behind event horizons. These
causality problems would be solved without any change to the
mathematical formalism of GR, but only to its interpretation, if gravity
is once again taken to be a propagating force of nature in flat
space-time with the propagation speed indicated by observational
evidence and experiments: not less than 2x10^10 c. Such a change of
perspective requires no change in the assumed character of gravitational
radiation or its lightspeed propagation. Although faster-than-light
force propagation speeds do violate Einstein special relativity (SR),
they are in accord with Lorentzian relativity, which has never been
experimentally distinguished from SR—at least, not in favor of SR.
Indeed, far from upsetting much of current physics, the main changes
induced by this new perspective are beneficial to areas where physics
has been struggling, such as explaining experimental evidence for
non-locality in quantum physics, the dark matter issue in cosmology, and
the possible unification of forces. Recognition of a
faster-than-lightspeed propagation of gravity, as indicated by all
existing experimental evidence, may be the key to taking conventional
physics to the next plateau.
*Introduction***
The most amazing thing I was taught as a graduate student of celestial
mechanics at Yale in the 1960s was that all gravitational interactions
between bodies in all dynamical systems had to be taken as
instantaneous. This seemed unacceptable on two counts. In the first
place, it seemed to be a form of �action at a distance�. Perhaps no one
has so elegantly expressed the objection to such a concept better than
Sir Isaac Newton: �That one body may act upon another at a distance
through a vacuum, without the mediation of any thing else, by and
through which their action and force may be conveyed from one to the
other, is to me so great an absurdity, that I believe no man who has in
philosophical matters a competent faculty of thinking, can ever fall
into it.� (See Hoffman, 1983.) But mediation requires propagation, and
finite bodies should be incapable of propagation at infinite speeds
since that would require infinite energy. So instantaneous gravity
seemed to have an element of magic to it.
The second objection was that we had all been taught that Einstein�s
special relativity (SR), an experimentally well-established theory,
proved that nothing could propagate in forward time at a speed greater
than that of light in a vacuum. Indeed, as astronomers we were taught to
calculate orbits using instantaneous forces; then extract the position
of some body along its orbit at a time of interest, and calculate where
that position would appear as seen from Earth by allowing for the finite
propagation speed of light from there to here. It seemed incongruous to
allow for the finite speed of light from the body to the Earth, but to
take the effect of Earth�s gravity on that same body as propagating from
here to there instantaneously. Yet that was the required procedure to
get the correct answers.
These objections were certainly not new when I raised them. They have
been raised and answered thousands of times in dozens of different ways
over the years since general relativity (GR) was set forth in 1916. Even
today in discussions of gravity in USENET newsgroups on the Internet,
the most frequently asked question and debated topic is �What is the
speed of gravity?� It is only heard less often in the classroom because
many teachers and most textbooks head off the question by hastily
assuring students that gravitational waves propagate at the speed of
light, leaving the firm impression, whether intended or not, that the
question of gravity�s propagation speed has already been answered.
Text Box: Figure 1. Orbits are unstable if forces propagate with finite
speed.Yet, anyone with a computer and orbit computation or numerical
integration software can verify the consequences of introducing a delay
into gravitational interactions. The effect on computed orbits is
usually disastrous because conservation of angular momentum is
destroyed. Expressed less technically by Sir Arthur Eddington, this
means: �If the Sun attracts Jupiter towards its present position S, and
Jupiter attracts the Sun towards its present position J, the two forces
are in the same line and balance. But if the Sun attracts Jupiter toward
its previous position S�, and Jupiter attracts the Sun towards its
previous position J�, when the force of attraction started out to cross
the gulf, then the two forces give a couple. This couple will tend to
increase the angular momentum of the system, and, acting cumulatively,
will soon cause an appreciable change of period, disagreeing with
observations if the speed is at all comparable with that of light.�
(Eddington, 1920, p. 94) See Figure 1.
Indeed, it is widely accepted, even if less widely known, that the speed
of gravity in Newton�s Universal Law is unconditionally infinite. (E.g.,
Misner et al., 1973, p. 177) This is usually not mentioned in proximity
to the statement that GR reduces to Newtonian gravity in the
low-velocity, weak-field limit because of the obvious question it begs
about how that can be true if the propagation speed in one model is the
speed of light, and in the other model it is infinite.
The same dilemma comes up in many guises: Why do photons from the Sun
travel in directions that are not parallel to the direction of Earth�s
gravitational acceleration toward the Sun? Why do total eclipses of the
Sun by the Moon reach maximum eclipse about 40 seconds before the Sun
and Moon�s gravitational forces align? How do binary pulsars anticipate
each other�s future position, velocity, and acceleration faster than the
light time between them would allow? How can black holes have gravity
when nothing can get out because escape speed is greater than the speed
of light?
Herein we will examine the experimental evidence bearing on the issue of
the speed of propagation of gravity. By gravity, we mean the
gravitational �force� from some source body. By force, we mean that
which gives rise to the acceleration of target bodies through space.
[Note: Orbiting bodies do accelerate through space even if gravity is
geometry and not a true force. For example, one spacecraft following
another in the same orbit can stretch a tether between the two. The taut
tether then describes a straight line, and the path of both spacecraft
will be curved with respect to it.] We will examine the explanations
offered by GR for these phenomena. And we will confront the dilemma that
remains when we are through: whether to give up our existing
interpretation of GR, or the principle of causality.
*Propagation Delay versus Aberration***
Text Box: Figure 2. Top shows transit delay: source fixed, target moves.
Bottom shows aberration: source moves, target fixed. See animation #4 at
.
To understand how propagation speeds of phenomena are normally measured,
it will be useful to discuss propagation or transit delay and
aberration, and the distinction between them. The points in this section
are illustrated in Figure 2.
In the top half of the figure, we consider the view from the source. A
fixed source body on the left (for example, the Sun) sends a projectile
(the arrow, which could also be a photon) toward a moving target (for
example, the Earth). Infinitely far to the right are shown a bright
(large, 5-pointed) star and a faint (small, 4-pointed) star, present to
define directions in space. Because of transit delay, in order to hit
the target, the source body must send the projectile when it is seen in
the direction of the faint star, but send it toward the direction of the
bright star, leading the target. The tangent of the lead angle (the
angle between the two stars) is the ratio of the tangential target speed
to the radial projectile speed. For small angles, this ratio equals the
lead angle in radians.
In the bottom half of the figure, we consider the view from the target,
which will consider itself at rest and the source moving. By the
principle of relativity, this view is just as valid since no experiment
can determine which of two bodies in uniform, linear relative motion is
�really moving� and which is not. The projectile will be seen to
approach from the retarded position of the source, which is the spatial
direction headed toward the faint star. The angle between the true and
retarded positions of the source, which equals the angle between the two
stars, is called �aberration�. It will readily be recognized as the same
angle defined in the first view due to transit delay.
Indeed, that is generally true: The initial and final positions of the
target as viewed from the source differ by the motion of the target
during the transit delay of the projectile. The same difference between
initial and final positions of the source as viewed from the target is
called the angle of aberration. Expressed in angular form, both are
equal, and are manifestations of the finite propagation speed of the
projectile as viewed from different frames. So the most basic way to
measure the speed of propagation of any entity, whether particle or wave
or dual entity or neither, is to measure transit delay, or equivalently,
the angle of aberration.
*Fact: Gravity Has No Aberration***
1. The effect of aberration on orbits is not seen
As viewed from the Earth�s frame, light from the Sun has aberration.
Light requires about 8.3 minutes to arrive from the Sun, during which
time the Sun seems to move through an angle of 20 arc seconds. The
arriving sunlight shows us where the Sun was 8.3 minutes ago. The true,
instantaneous position of the Sun is about 20 arc seconds east of its
visible position, and we will see the Sun in its true present position
about 8.3 minutes into the future. In the same way, star positions are
displaced from their yearly average position by up to 20 arc seconds,
depending on the relative direction of the Earth�s motion around the
Sun. This well-known phenomenon is classical aberration, and was
discovered by the astronomer Bradley in 1728.
Orbit computations must use true, instantaneous positions of all masses
when computing accelerations due to gravity for the reason given by
Eddington. When orbits are complete, the visible position of any mass
can be computed by allowing for the delay of light traveling from that
mass to Earth. This difference between true and apparent positions of
bodies is not merely an optical illusion, but is a physical difference
due to transit delay that can alter an observer�s momentum. For example,
small bodies such as dust particles in circular orbit around the Sun
experience a mostly radial force due to the radiation pressure of
sunlight. But because of the finite speed of light, a portion of that
radial force acts in a transverse direction, like a drag, slowing the
orbital speed of the dust particles and causing them to eventually
spiral into the Sun. This phenomenon is known as the Poynting-Robertson
effect.
If gravity were a simple force that propagated outward from the Sun at
the speed of light, as radiation pressure does, its mostly radial effect
would also have a small transverse component because of the motion of
the target. Analogous to the Poynting-Robertson effect, the magnitude of
that tangential force acting on the Earth would be 0.0001 of the Sun�s
radial force, which is the ratio of the Earth�s orbital speed (30 km/s)
to the speed of this hypothetical force of gravity moving at light-speed
(300,000 km/s). It would act continuously, but would tend to speed the
Earth up rather than slow it down because gravity is attractive and
radiation pressure is repulsive. Nonetheless, the net effect of such a
force would be to double the Earth�s distance from the Sun in 1200
years. There can be no doubt from astronomical observations that no such
force is acting. The computation using the instantaneous positions of
Sun and Earth is the correct one. The computation using retarded
positions is in conflict with observations. From the absence of such an
effect, Laplace set a lower limit to the speed of propagation of
classical gravity of about 10^8 /c/, where /c/is the speed of light.
(Laplace, 1825, pp. 642-645 of translation)
In the general case, let be the speed of propagation of gravitational
force, and let be the initial semi-major axis at time of an orbiting
body in a system where the product of the gravitational constant and the
total system mass is . Then the following formula, derived from the
ordinary perturbation formulas of celestial mechanics (e.g., Danby,
1988, p. 327), allows us to compute the semi-major axis at any other time :
[1]
We will use this formula later to set limits on .
2. Gravity and light do not act in parallel directions
There is no cause to doubt that photons arriving now from the Sun left
8.3 minutes ago, and arrive at Earth from the direction against the sky
that the Sun occupied that long ago. But the analogous situation for
gravity is less obvious, and we must always be careful not to mix in the
consequences of light propagation delays. Another way (besides
aberration) to represent what gravity is doing is to measure the
acceleration vector for the Earth�s motion, and ask if it is parallel to
the direction of the arriving photons. If it is, that would argue that
gravity propagated to Earth with the same speed as light; and conversely.
Such measurements of Earth�s acceleration through space are now easy to
make using precise timing data from stable pulsars in various directions
on the sky. Any movement of the Earth in any direction is immediately
reflected in a decreased delay in the time of arrival of pulses toward
that direction, and an increased delay toward the opposite direction. In
principle, Earth�s orbit could be determined from pulsar timings alone.
In practice, the orbit determined from planetary radar ranging data is
checked with pulsar timing data and found consistent with it to very
high precision.
How then does the direction of Earth�s acceleration compare with the
direction of the visible Sun? By direct calculation from geometric
ephemerides fitted to such observations, such as those published by the
U.S. Naval Observatory or the Development Ephemerides of the Jet
Propulsion Laboratory, the Earth accelerates toward a point 20 arc
seconds in front of the visible Sun, where the Sun will appear to be in
8.3 minutes. In other words, the acceleration now is toward the true,
instantaneous direction of the Sun now, and is not parallel to the
direction of the arriving solar photons now. This is additional evidence
that forces from electromagnetic radiation pressure and from gravity do
not have the same propagation speed.
3. The solar eclipse test
Yet another manifestation of the difference between the propagation
speeds of gravity and light can be seen in the case of solar eclipses
(Van Flandern, 1993, pp. 49-50). The Moon, being relatively nearby and
sharing the Earth�s 30 km/s orbital motion around the Sun, has
relatively little aberration (0.7 arc seconds, due to the Moon�s 1 km/s
orbital speed around Earth). The Sun, as mentioned earlier, has an
aberration of just over 20 arc seconds. It takes the Moon about 38
seconds of time to move 20 arc seconds on the sky relative to the Sun.
Since the observed times of eclipses of the Sun by the Moon agree with
predicted times to within a couple of seconds, we can use the orbits of
the Sun and the Moon near times of maximum solar eclipse to compare the
time of predicted gravitational maximum with the time of visible maximum
eclipse.
In practice, the maximum gravitational perturbation by the Sun on the
orbit of the Moon near eclipses may be taken as the time when the lunar
and solar longitudes are equal. Details of the procedure are provided in
the reference cited. We find that maximum eclipse occurs roughly 38�1.9
seconds of time, on average, before the time of gravity maximum. If
gravity is a propagating force, this 3-body (Sun-Moon-Earth) test
implies that gravity propagates at least 20 times faster than light.
*Electromagnetic Analogies and Gravitational Radiation***
1. Myth: Gravity from an accelerating source experiences
light-time delay
Text Box: Figure 3. Comparison of a star's true position, At, with its
linearly extrapolated retarded position, Ae.In electromagnetism, it is
said that moving charges anticipate each other�s linear motion, but not
acceleration, and that acceleration causes the emission of photons. If
gravity behaved in an analogous way, moving masses would anticipate each
other�s linear motion, but not acceleration, and accelerating masses
would emit gravitational radiation. Indeed, the orbit of binary pulsar
PSR1913+16 is observed to slowly decay at a rate close to that predicted
by GR from the emission of gravitational radiation. Could that be
evidence for changes in gravity propagating at lightspeed?
First, we will calculate the acceleration predicted for any two stars if
each star responds to the linearly extrapolated retarded position and
velocity, but not acceleration, of its companion over one light time
between the stars. This would be consistent with the electromagnetic
analogy. In Figure 3, we will consider the orbit of component A relative
to component B during the light time between the two stars. We will then
consider three positions of component A: its true, instantaneous
position, A_t; its retarded position one light time ago, A_r; and its
linearly extrapolated position one light time ahead from its retarded
position, A_e . As before, let the product of the gravitational constant
and the total system mass be , and the radius of A�s circular orbit
around B be . Also let the speed of light be , and A�s orbital period be
. Finally, is the angle at B through which A moves during the light
time , and is the angle at B between A_e and A_t . By construction, the
linear distance from A_r to A_e is equal to the length of the arc from
A_r to A_t , and both are equal to .
The difference in the distance of A_e and A_t from B causes only small,
non-cumulative effects on the orbit. However, the angle causes the
extrapolated retarded position to feel a transverse force component that
continually increases the orbital period . From the triangles in the
figure we see that . Since is normally a very small angle, we can
expand the arctangent into a series and retain only significant terms.
The result is . However, is times the light time, or . So the
transverse perturbing acceleration , which is times the radial orbital
acceleration , can be found from . Finally, from (Danby, 1988, p. 327)
and with some minor change of variables and simplification, we arrive at:
[2]
Now we are ready to compare this prediction for binary pulsars
PSR1913+16 and PSR1534+12 with the measured values of in the two
best-observed cases. Orbital quantities are taken from (Taylor et al.,
1992) – see Text Box: PSR1913+16 PSR1534+12 (sec) 2.342 3.729 (sec)
27,907 36,352 -observed -2.42x10-12 �0.6x10-12 -predicted +921x10-12
+1682x10-12 Table I. Observed and predicted period change rate for two
binary pulsars. Table I. The period change rate for PSR1534+12 is not
yet seen, so the table shows the observational error of the measurement.
At a glance, we see there is no possible match. The predicted period
changes that would result if gravity propagated at the speed of light in
a manner analogous to electromagnetic forces are orders of magnitude
larger than the observed period changes. For PSR1913+16, they have the
opposite sign as well. From PSR1534+12, we can set a lower limit to the
speed of gravity as an electromagnetic-type propagating force: 2800.
We could have seen the essence of this result at the outset. Binary
pulsars decay as they radiate away angular momentum, presumably in the
form of gravitational radiation. However, a finite speed of propagation
of gravitational force must add angular momentum to orbits. This is
because the retarded position of any source of gravity must lie in the
same direction relative to its true position as the tangential motion of
the target body. Therefore, any delay in gravity will always pull the
target in a direction that will increase its instantaneous orbital speed
– the opposite of the effect of gravitational radiation.
In concluding this section, we should also note that, even in the solar
system, the Sun moves around the barycenter in a path that often takes
the barycenter a million kilometers or so from the Sun. So the idea that
the Sun�s field can be treated as �static� and unchanging is not a good
approximation even for our own planetary system. The Sun�s motion during
the light time to the planets is appreciable, yet its gravity field is
continually updated without apparent delay.
2. Myth: Gravitational waves contribute to gravitational force
Few subjects in physics are in such a state of confusion as is the
subject of gravitational waves. Normally, this term is synonymous with
gravitational radiation, a hypothetical, ultra-weak disturbance of
space-time induced by a certain type of asymmetric change in the
distribution of matter called a quadrupole moment. It is supposed to be
analogous to accelerating charges emitting photons. This form of
radiation is predicted by GR. The acceleration of binary pulsar
PSR1913+16 is said to be in accord with the predicted amount of
gravitational radiation, and therefore to provide an indirect
confirmation of the prediction. However, attempts to detect
gravitational waves in the laboratory from any source have yet to yield
events that have convinced a consensus of their reality. The LIGO
experiment is being designed to provide definitive detections, assuming
these waves exist.
When gravitational waves were predicted, it was natural to associate
them with supernova explosions, since no known event in nature
redistributes mass in space more rapidly. However, the explosion must be
asymmetric to produce gravitational waves. Because the gravitational
field of the supernova is changing rapidly during the explosion, it is
natural to associate the production of gravitational waves with changes
in gravitational fields. So far, so good.
However, many physicists do more than associate the two concepts, and
consider that changes in gravitational fields /are/gravitational waves.
The heart of this confusion is illustrated by the following passage from
(Synge, 1960): �Suppose that a man, standing on the earth, holds in his
hand a heavy club. At first the club hangs down toward the ground, but
at a certain moment the man raises it quickly over his head. Any theory
of gravitation recognizes that the club produces a gravitational field,
however minute it may be, and that the action of the man changes that
field, not only in his neighborhood, but throughout the whole universe.
According to Newtonian theory, the effect is instantaneously felt on the
moon, on the sun and in every remote nebula. Since we are not concerned
with Newtonian theory, we do not have to discuss the absurdity of this.
As relativists, familiar with the idea that no causal effect can travel
faster than light, ..., we would guess that the change in the
gravitational field of the moving club travels out into space with the
speed of light. And we would call this moving disturbance a
/gravitational wave/. Thus, on a very general basis, we must regard the
physical existence of gravitational waves, so understood, as self-evident.�
The sudden displacement of the club may cause a disturbance of
space-time, which would be a form of gravitational radiation.
Separately, if gravitation is itself some sort of wave phenomenon,
changes in gravitational fields will propagate away from a source as
waves. Now there is no doubt that changes in gravitational fields exist,
or that they can be detected in the laboratory. Therefore, this
phenomenon cannot be the same thing as gravitational radiation, since
the latter has not yet been reliably detected, and its existence still
remains unverified. However, both phenomena are called �gravitational
waves� without further distinction. For the former type, we must look to
ultra-small accelerations of distant, massive pulsars for some hint of
their existence. For the latter type, we see indirect evidence of
changes in the gravitational fields of Sun and Moon every day in the
tides, or can measure them directly with a gravimeter. We can even
measure gravitational field changes using small masses in a purely
laboratory setting.
The consequences of this distinction become clearer when we are careful
to distinguish sources and targets of gravity. Ordinary gravitational
acceleration of a target results from some form of communication from a
source of gravity that may or may not be carried from source to target
in wave form. Separately, the acceleration of a target body must change
the nearby space-time, and such changes seem likely to be propagated
outward in wave form away from the target. If possible waves associated
with sources of gravity (those that may induce acceleration in other
bodies), and other possible waves induced by targets of gravity (those
that result from acceleration), are not distinguished, we are certain to
have massive confusion over the meaning of the very concept of �the
speed of gravity�.
In a binary pulsar, where both masses are comparable, both stars may
emit gravitational radiation. But each would do so as a consequence of
its acceleration induced by the other, not as a consequence of its own
gravity. Moreover, as we noted earlier, gravitational waves in the sense
of gravitational radiation cause orbiting bodies to lose angular
momentum; whereas gravitational aberration that must accompany any
finite speed of propagation of gravity from a source to a target would
cause orbits to gain angular momentum.
Therefore, it seems fairly certain that, if gravitational radiation
exists, its waves will propagate at the speed of light. In what way this
type of disturbance of space-time may differ from very-long-wavelength
electromagnetic disturbances of space-time, if indeed it does differ,
remains to be seen.
By contrast, the speed of propagation of gravitational fields and of
changes in those fields, whatever the nature of the propagating agents,
are different matters, and pose a question we hope to answer in this paper.
*Space-Time Curvature and Retarded Potentials***
1. Is gravity caused by a curvature of space and time?
A common way to explain why gravity can appear to act instantaneously,
yet still propagate with a delay, is the rubber sheet analogy. (See
Figure 4.) A large mass sitting on a rubber sheet would make a large
indentation, and that indentation would induce smaller nearby masses to
role toward the indentation. This is an analogy for curved space-time,
which is likewise supposed to be the cause of bodies accelerating toward
large masses. The reasoning in the analogy further suggests that target
bodies simply respond instantly to the local curvature of the underlying
space-time medium (like the rubber sheet). Therefore, any delay
associated with altering that local curvature would not produce
aberration, and the target body would appear to respond instantaneously
to the source unless the source suddenly changed its motion.
The rubber sheet analogy is represented as a way of visualizing why
bodies attract one another. However, in that regard, it is highly
defective. A target body sitting on the side of an indentation would
stay in place, with no tendency to roll downhill, unless there were
already a force such as gravity underneath the rubber sheet pulling
everything downhill. And this failure of the analogy helps us identify
the precise problem with the curved space-time description of gravity –
the lack of causality. Without consideration of why a target body is
induced to accelerate through space, and how quickly it receives updates
of information about how to accelerate through space, neither the
space-time curvature explanation nor the rubber sheet analogy can help
us understand why gravity appears to act so much faster than light.
Moreover, contrary to what the rubber sheet analogy implies, an orbiting
body such as a spacecraft orbiting the Earth is not following the
curvature of space near the Earth. As we remarked earlier, two
spacecraft some distance apart in the same orbit could stretch a tether
between them and pull it taut, thereby describing a straight line
through space different from their orbital path. In more mathematical
terms, the supposed curvature of space-time produced by a gravitational
field is an effect proportional to the local gravitational potential ,
the variable part of which is in turn proportional to , where is
orbital speed. Yet, orbital curvature through space, like stellar
aberration, is proportional to , a much larger effect. For example, for
the Earth orbiting the Sun, is of order 10^-4 , and is of order 10^-8
. So we see that almost all of the acceleration of bodies through space
is not a consequence of the curvature of space. In the GR explanation,
the acceleration through space is due to the curvature of �space-time�,
a mathematical entity not to be confused with the combined separate
concepts of space and time.
While relativists have always been partial to the curved space-time
explanation of gravity, it is not an essential feature of GR. Eddington
(1920, p. 109) was already aware of the mostly equivalent �refracting
medium� explanation for GR features, which retains Euclidean space and
time in the same mathematical formalism. In essence, the bending of
light, gravitational redshift, Mercury perihelion advance, and radar
time delay can all be consequences of electromagnetic wave motion
through an underlying refracting medium that is made denser in
proportion to the nearness of a source of gravity. (Van Flandern, 1993,
pp. 62-67 and Van Flandern, 1994) And it is now known that even ordinary
matter has certain electromagnetic-wave-like characteristics. The
principal objection to this conceptually simpler refraction
interpretation of GR is that a faster-than-light propagation speed for
gravity itself is required. In the context of this paper, that cannot be
considered as a fatal objection.
Lastly, we note experimental evidence from neutron interferometers that
purports to demonstrate a failure of the geometric weak equivalence
principle, that gravity is due to a curvature of space-time.
(Greenberger & Overhauser, 1980) This experiment confirmed the strong
equivalence principle (local equivalence of a uniform acceleration and a
gravitational field), but its results are incompatible with the
geometrical weak equivalence principle because interference effects in
quantum mechanics depend on the mass. This is because the wave nature of
the neutron depends on the momentum of the neutron, which is mass times
velocity. So all phase-dependent phenomena depend on the mass through
the wavelength, a feature intrinsic to quantum mechanics.
Since the experiment confirms the applicability of quantum mechanics
even in the presence of gravity, including this non-geometrical mass
dependence, the experiment seems to be a step in the undermining of the
purely geometrical point of view, and �tends to bother theorists who
prefer to think of gravity as being intrinsically related to geometry�,
according to the authors.
2. Does GR really reduce to Newtonian gravity in low-velocity,
weak-field limit?
As we have already noted, Newtonian gravity propagates with
unconditionally infinite speed. How, then, can GR reduce to Newtonian
gravity in the weak-field, low-velocity limit? The answer is that
conservation of angular momentum is implicit in the assumptions on which
GR rests. However, as we have already seen, finite propagation speeds
and conservation of angular momentum are incompatible. Therefore, GR was
forced to claim that gravity is not a force that propagates in any
classical sense, and that aberration does not apply.
In practice, this suppression of aberration is done through so-called
�retarded potentials�. In electromagnetism, these are called
�Lienard-Wiechert potentials�. For examples of the use of retarded
potentials, see (Misner et al., 1973, p. 1080) or (Feynman, 1963, p.
21-4). Suppose we let be the gravitational potential at a field point
and time , be the gravitational constant, be an element of volume in
the source of the potential, be the coordinates of that volume element
in the source, be the matter density at point and time , , be the
distance from the source volume element at time to the field point at
time , and be the relative velocity between the field point and the
source. Then two different forms of retarded potentials in common use
for gravitation are these:
[3]
[4]
In [3], we have used as the retarded time. Then the triple integral
evaluates the density one light time ago in place of the present
density, as might be useful if a non-spherically symmetric source body
were rotating. In [4], the mutual distance is taken to depend on the
scalar distance of the source one light time ago.
However, in neither form of retarded potential is any consideration
given to the transverse motion between source and target during the
light time; i.e., the aberration. Ignoring aberration is logically
equivalent to adopting an infinite propagation speed for gravitational
force. That point is glossed over by emphasizing that the density
distribution or the mutual distance is being taken at its retarded
position, as if a finite propagation speed for gravity were being
adopted. Nevertheless, the only practical consequence of a finite
propagation speed that matters in most applications is missing from
these potentials. And that clever trick then allows a theory with
�gravity propagating at the speed of light� to be equivalent to a theory
with infinite propagation speed in the weak-field, low velocity limit.
In short, both GR and Newtonian gravity use infinite propagation speeds
with aberration equal to zero. In Newton�s laws, that fact is explicitly
recognized even though aberration and delay terms do not appear because
of an infinity in their denominator. In GR, much effort has been
expended in disguising the continued absence of the same delay terms by
including retardation effects in ways that are presently unobservable
and ignoring aberration. Every physicist and physics student should be
at least annoyed at having been tricked by this sleight of hand, and
should demand that the neglect of aberration be clearly justified by
those who propose to do so.
*Does a Gravitational Field Continuously Regenerate, or is it �Frozen�?***
In attempts to describe how GR can affect distant bodies seemingly
without delay, relativists often speak of the field of a body as if it
were a rigid extension of the body itself. If such a �static� field has
no moving parts, it then would have no need of a propagation speed
unless something changes. The objection to this picture is that it is
acausal. Somehow, momentum is transferred from a source body to a target
body. It seems impossible to conceive of a static field with literally
no moving parts as capable of transferring momentum. This is the dilemma
of the �rubber sheet� analogy again. Just because a rubber sheet or
space-time is curved, why should a stationary target body on the slope
of such a curve begin moving toward the source? What is the source of
the momentum change?
To retain causality, we must distinguish two distinct meanings of the
term �static�. One meaning is unchanging in the sense of no moving
parts. The other meaning is sameness from moment to moment by continual
replacement of all moving parts. We can visualize this difference by
thinking of a waterfall. A frozen waterfall is static in the first
sense, and a flowing waterfall is static in the second sense. Both are
essentially the same at every moment, yet the latter has moving parts
capable of transferring momentum, and is made of entities that propagate.
As this applies to gravitational fields for a fixed source, if the field
were static in the first sense, there would be no need of aberration,
but also no apparent causality link between source and target. If the
field were static in the second sense, then the propagation speed of the
entities carrying momentum would give rise to aberration; and the
observed absence of aberration demands a propagation speed far greater
than lightspeed.
So are gravitational fields for a rigid, stationary source frozen, or
continually regenerated? Causality seems to require the latter. If such
fields are frozen, then what is the mechanism for updating them as the
source moves, even linearly? Even a �rigid� bar pushed at one end would
not move at the other end until a pressure wave had propagated its
entire length. Moreover, we seem to need two mechanisms – one to curve
space-time when a mass approaches, and another to unbend it when the
mass recedes. This is because, once a curve is �frozen� into space-time,
it will not necessarily �melt� back to its original condition when the
cause is removed. Yet, there is no available cause for either process to
result from a field with no moving parts.
We can also deduce the consequences for a source in continual
acceleration, such as the Sun in our solar system. The Sun�s path around
the solar system barycenter induced by planetary perturbations causes
excursions of over a million kilometers, and the barycenter is sometimes
outside the physical body of the Sun. So the Sun�s field must be
continually updated at all distances to infinity. Surely, this updating
requires the propagation of causal agents from the source. And since the
source is continually accelerating, the regeneration of the distant
field must likewise be a continuous process, requiring propagation.
However, propagation involves delays, and even in the solar system, we
have observationally ruled out delays as great as lightspeed propagation
would produce. For example, the solar eclipse experiment is sensitive to
delays in the continual updating of the Earth�s field by the Sun as they
both affect the Moon, and update speeds of at least 20are required.
The binary pulsar experiment provides another, more direct demonstration
that even changes in gravitational fields must propagate faster than
light. Ultimately, GR proposes that such changes appear to act
instantaneously in the �near field�, but eventually show their true,
light-speed-delayed character in the �far field�, which is conveniently
beyond our present ability to observe. The necessity of this dual
behavior is to prevent the logical need for changes to continue to
appear to act instantaneously at ever increasing distances, even to
infinity.
Text Box: Figure 5. How can binary black holes update their external
fields as they interact, when the masses are hidden behind event
horizons? However, this only prevents certain types of paradoxes from
arising. When the subject of �black holes� first comes up in physics
classes, a frequently asked question is �If nothing can escape the event
horizon because nothing can propagate faster than light, how does
gravity get out of a black hole?� The answer usually provided is that
the field around a black hole was frozen into the surrounding space-time
prior to the collapse of the parent star behind an event horizon, and
has remained in that state ever since. By implication, there is no need
for continual regeneration of the external field by causal agents from
the source.
However, let us suppose we have a binary black hole, with the two
collapsed stars in elliptical orbits around one another. See Figure 5.
Then each field must be continually updated by a changing contribution
from the orbiting field of the other. How does each field know what it
is supposed to do if it is no longer in communication with its source
mass hidden behind an event horizon? If the curvature of space-time at a
point near black hole A becomes zero because black hole B is equally
distant, what makes it non-zero again once black hole B recedes?
Indeed, if each source mass is forced to accelerate, why should each
field point with a certain curvature undergo exactly the same
acceleration as the source, making the whole field (to infinity?) appear
frozen rigidly to the parent black hole? Perturbations by the other star
are different at every different field point, so each such space-time
field point should experience a different acceleration. With no
communication, how can the whole system remain intact and coherent?
We conclude that the concept of frozen gravitational fields is acausal
and paradoxical. Gravitational fields must continually regenerate, like
a flowing waterfall. In doing so, they must consist of entities that
propagate. And the speed of propagation of those entities must greatly
exceed the speed of light.
*Conclusion: The Speed of Gravity is **³**2x10^10 **^**^*
We conclude that gravitational fields, even �static� ones, continually
regenerate through entities that must propagate at some very high speed,
. We call this the speed of gravity. Equation [1] then tells us how
orbits will expand in response to this large but finite propagation
speed, since the field itself, and not merely changes in the field, will
transfer momentum to orbiting target bodies. Rewriting equation [1] in a
form suitable for comparisons with observations, we derive:
[5]
For the Earth�s orbit, = 1 year, = 10^-4 , and we take as an upper
limit to the value 2.4x10^-12 /year (derived from ½ ) in solutions
using radar ranging and spacecraft data (Pitjeva, 1993). Substituting
these values, we get from Earth-orbit data that ³10^9.
Using the same equation with binary pulsar PSR1534+12 and the parameters
in Table I, we can place the most stringent limit yet from the observed
uncertainty in : ³2x10^10.
A direct experimental verification in the laboratory that gravity
propagates faster than light may now be possible. The protocol and
preliminary results were reported in (Walker, 1997).
It might be tempting to conclude that the speed of gravity is infinite.
But these limits on are still a long way from infinite velocity, and
Newton�s statement, quoted at the beginning of this paper, still seems
applicable. Infinite speeds, too, are acausal.
*Consistency with Special Relativity***
Einstein special relativity (SR) is able to prove based on its premises
that nothing can propagate faster than the speed of light in forward
time. Is our result for the speed of gravity an experimental
falsification of SR? The correct answer must be a qualified �yes and
no�. Strictly, the minor new interpretation of SR needed for consistency
with our result is no more a falsification of SR than GR was a
falsification of Newtonian gravity. In both cases, the earlier theory
was incomplete rather than wrong. We will now examine exactly what must
change about SR for full consistency with all existing experimental
evidence and this new result as well.
Text Box: Experiment Description Year Bradley Discovery of aberration
of light 1728 Fresnel Light suffers drag from the local medium 1817 Airy
Aberration is independent of the local medium 1871 Michelson-Morley
Speed of light is independent of Earth's orbital motion 1881 De Sitter
Speed of light is independent of speed of source 1913 Sagnac Speed of
light depends on speed of a rotating platform 1913 Kennedy-Thorndike
Measured time as well as length is affected by motion 1932 Ives-Stilwell
Ions radiate at frequencies affected by their motion 1941 Frisch-Smith
Radioactive decay of mesons is slowed by their motion 1963
Hafele-Keating Atomic clock changes depend on Earth's rotation 1972 GPS
(Various -- see text) 1997 Table II. Independent experiments bearing on
special relativity.
A brief overview of the history of relativity will provide useful
background for this section, since everything proposed here has been
proposed before. The �principle of relativity�, that the laws of physics
should be the same as viewed from any inertial frame, dates to the 19^th
century, well before it was popularized by H. Poincare. The well known
�Lorentz transformations� embody that principle, but were not original
when Lorentz adopted them for his own theory of relativity, first
published in 1904 in an �aether� context. Einstein�s main contribution
with his famous 1905 paper, then, was the addition of a second
postulate, that the speed of light will be locally the same for all
observers regardless of their own state of motion. This did away with
the need for an aether, or more generally, with a preferred frame of
reference.
The ensuing years saw much discussion of whether nature was more like
Einstein�s SR or Lorentzian relativity (LR). The experiments relevant to
testing relativity are listed in Table II. The discovery of Fresnel drag
had seemed at first to demand the existence of an aether, but
relativists eventually found ways to explain it using SR too. The Airy
water-filled telescope experiment showed that the aberration of
starlight was unchanged by passing through a water medium even though
that medium slowed the speed of light by about 30%. This too seemed to
favor the existence of a preferred frame because the local speed of
light did not affect aberration, showing that aberration was determined
outside the telescope rather than by the conditions most local to the
observer. However, Einstein supporters could also explain this result
using SR, albeit with somewhat more complexity.
The Michelson-Morley experiment is the first (and only) observation that
seemed to strongly favor SR over LR, although Michelson himself never
accepted that. The expected aether-drift speed did not put in an
appearance in the test results, and the speed of light did indeed seem
to be the same in all directions, as SR postulated, even though the
observer was obviously moving at high speed in some direction through
space. It was not until the last decade that serious consideration was
given to the possibility that the local gravity field may always
constitute a preferred frame. This idea was popularized in (Beckmann,
1987) and then widely discussed in the journals /Galilean
Electrodynamics/ and
/Apeiron/, and occasionally in the
/Meta Research Bulletin/>.
It is now well-established that LR is fully compatible with the
Michelson-Morley experiment, and in general with the expectation that
the speed of light will seem to be the same even when the observer is
moving provided that certain conditions are met, although not under all
circumstances. That the speed of light is independent of the speed of
its source is unremarkable, since that is a property of all wave motion.
However, being independent of the speed of the observer is special.
Choosing to synchronize clocks using the Einstein convention
automatically makes one-way speed of light independent of the speed of
the observer because that assumption is built into the Einstein
synchronization method. If some other convention were used to
synchronize clocks, such as synchronizing them to an underlying common
inertial frame (as is done for the Global Positioning System satellites,
or when astronomers synchronize phenomena to a barycentric frame using
time provided by distant pulsars), then the one-way speed of light would
be different in each direction when measured by observers moving with
respect to that special frame. The round-trip speed of light uses a
single clock to measure elapsed time, and so does not depend on
synchronization. But if the rate of an ordinary clock is affected by its
speed in a Lorentzian way, which we now know to be the case, then the
measured speed of light will appear to be an invariant in all
directions. Using a clock whose rate is not affected by its
translational speed, for example pulses in the strength of the
gravitational field from a compact, massive binary star, would
apparently allow the speed of the observer relative to the local mean
gravity field to be detected.
Following the publication of Einstein�s SR paper, two new experimental
results were published in 1913, both favoring LR over SR. Indeed, Sagnac
claimed a falsification of SR on the grounds that the local speed of
light was affected by observer velocity if the observer was attached to
a rotating platform. He showed that the Michelson-Morley experiment
performed in such a rotating frame did show fringe shifts, and concluded
that, even if linear motion was relative, rotational motion was
absolute. DeSitter noted that stellar aberration was the same for both
components of distant binary stars, even though the relative velocity of
each with respect to the observer was quite different. Therefore
velocity in some special frame (we might now say velocity in the local
gravity field relative to the distant gravity field) rather than
relative velocity between source and observer determines aberration.
Both of these experiments were blows to SR�s contention that all motion
was relative. Nonetheless, SR supporters came up with explanations of
these phenomena too in an SR context, and these fairly non-trivial
explanations are the subjects of textbooks on relativity today.
The Michelson-Gale experiment of 1925 involving the same Michelson as in
the Michelson-Morley experiment again claimed a contradiction of SR – a
theory that Michelson never found acceptable. History has concluded that
this experiment is essentially another demonstration of the Sagnac
effect, and no longer cites it as a significant independent experiment;
so it is omitted from our table. Ives and Stilwell (1938) drew
conclusions similar to those of Michelson, and specifically argued that
their own experiment confirmed LR (which they called the Larmor-Lorentz
theory) over SR. Yet today, it is simply added to the list of
SR-confirming experiments.
When the muon lifetime experiments were performed in the 1960s, LR had
been all but forgotten. Questions were raised briefly about whether the
situation was reciprocal – whether high-speed muons would really see
laboratory muons live longer. SR offered assurance that they would, but
no test was then possible. By the time the Hafele-Keating experiment
compared traveling atomic clocks sent around the world in opposite
directions with a stay-at-home clock, an experiment later improved upon
by C.O. Alley at the Univ. of Maryland, it was no longer considered
remarkable that the velocity effects on clocks had to be based on speeds
in the underlying inertial frame instead of the relative velocities of
the clocks.
Finally, the Global Positioning System (GPS) showed the remarkable fact
that all atomic clocks on board orbiting satellites moving at high
speeds in different directions could be simultaneously and continuously
synchronized with each other and with all ground clocks. No �relativity
of simultaneity� corrections, as required by SR, were needed. This too
seemed initially to falsify SR. But on further inspection, continually
changing synchronization corrections for each clock exist such that the
predictions of SR are fulfilled for any local co-moving frame. To avoid
the embarrassment of that complexity, GPS analysis is now done
exclusively in the Earth-centered inertial frame (the local gravity
field). And the pre-launch adjustment of clock rates to compensate for
relativistic effects then hides the fact that all orbiting satellite
clocks would be seen to tick slower than ground clocks if not
rate-compensated for their orbital motion, and that no reciprocity would
exist when satellites view ground clocks.
Why then did SR win out over LR? Three circumstances conspired to make
SR appear to be the better solution to describing nature in the early
years of the 20^th century. (1) Classical thinking about the aether
almost always involved a universal field rather than a local field. No
one took seriously that each local gravity field might serve as a
preferred frame for local observers. Yet that now seems the case. (2)
The wave nature of matter had not yet been discovered by deBroglie.
Before that happened, there was no logical reason to expect that clocks
based ultimately on atomic oscillations would have their rates affected
by observer motion in the same way that the speed of light would be
affected by observer motion, rendering observer motion undetectable in
experiments. However, that also now seems to be true (Van Flandern,
1993, p. 72-77). (3) The success of GR in predicting the light-bending
effect at the 1918 solar eclipse gained great credibility for GR, and SR
benefited from this success because it was widely believed that GR was
based on SR. But GR is usually implemented using a preferred frame
closely coinciding with the local gravity field, with the consequence
that only the features that SR and LR have in common were integrated
into GR. The reciprocity of time dilation between two inertial frames, a
key way in which SR differs from LR, plays no role in GR.
The principal differences between the two relativity theories stem from
the equivalence of all inertial frames in SR, and the existence of a
preferred frame in LR. Otherwise, SR�s time dilation is equivalent to
LR�s clock slowing; SR�s space contraction is equivalent to LR�s
meter-stick shrinkage; and SR�s change in the momentum of moving bodies
is equivalent to LR�s. However, LR recognizes a �universal time� apart
from the time kept by electromagnetic-based clocks affected by motion.
And the law of addition of velocities between two frames, neither of
which is the preferred frame, is different in LR than in SR. For a
derivation of this law and the revised form of the Lorentz
transformations for Lorentzian universal time, see (Mansouri & Sexl,
1977). For our purposes here, we simply note that the proof that nothing
can propagate faster than the speed of light in forward time does not
apply to LR.
Near the end of his career, Lorentz is quoted as having graciously
conceded the contest: �My theory can obtain all the same results as
special relativity, but perhaps not with a comparable simplicity.�
(private communication from C.O. Alley) Today, with hindsight, we might
make a somewhat different assessment: �Special relativity can explain
all the experimental results in Table II that Lorentzian relativity can,
but perhaps not with a comparable simplicity.� Even so, SR cannot
explain the faster-than-light propagation of gravity, although LR
readily can.
We conclude that the speed of gravity may provide the new insight
physics has been awaiting to lead the way to unification of the
fundamental forces. As shown in (Van Flandern, 1993, pp. 80-85 and Van
Flandern, 1996), it may also be connected with the explanation of the
dark matter problem in cosmology. Moreover, the modest switch from SR to
LR may correct the �wrong turn� physics must have made to get into the
dilemma presented by quantum mechanics, that there appears to be no
�deep reality� to the world around us. Quantum phenomena that violate
the locality criterion may now be welcomed into conventional physics.
*Acknowledgments***
The author is indebted to numerous correspondents who have challenged
the conclusions of this paper in so many different ways, especially in
USENET discussion groups such as sci.physics, sci.physics.relativity and
sci.astro. Each of these challenges has forced a new and deeper
investigation, without all of which the present paper could never have
hoped to pass peer review. One relativist in particular, Steve Carlip of
UC Davis, had the patience to stay with this issue over a span of
several years, defending the GR interpretation to the fullest extent
possible. Between us we have written enough prose, created enough
analogies, pondered enough equations, and consulted enough references to
fill a book.
The author further thanks Jeffery Kooistra for his key role. His
/Analog/article (Kooistra, 1997) flushed this subject to the forefront
once again, and his inquiries to both Steve Carlip and to the author
forced us to explain our positions in layman�s language, and thereby
diverted us from talking past one another. Discussions with colleagues
too numerous to mention must likewise be acknowledged. But Jean-Pierre
Vigier, in addition to several penetrating questions, encouraged the
author to stop talking and start writing, promising a fair peer review
process at the conclusion. Without such encouragement, this paper would
certainly not have come into existence.
*[Final version published: _Physics Letters A_ 250:1-11 (1998); also:
/Infinite Energy/**5 #27:50-58 (1999).*
*See published comment: G.E. Marsch, C. Nissim-Sabat, �Comments on �The
speed of gravity��, _Phys.Lett.A_ 262:103-106 (1999).*
*See response: T. Van Flandern, �Reply to comments on �The speed of
gravity��, _Phys.Lett.A_ 262:261-263 (1999).*
*See additional comment: S. Carlip, �Aberration and the speed of
gravity�, _Phys.Lett.A_ 267:81-87 (2000).*
*See response to above and all other comments: �Experimental Repeal of
the Speed Limit for Gravitational, Electrodynamic, and Quantum Field
Interactions�, T. Van Flandern and J.P. Vigier, _Foundations of Physics_
32:1031-1068 (2002).*
*As of 2006/02/04, no further comment or criticism has appeared.]*
* *
*Bibliography***
Beckmann, P., /Einstein Plus Two/, Golem Press (1987).
Danby, J.M.A., /Fundamentals of Celestial Mechanics/, Willmann-Bell,
Richmond, VA (1988).
Eddington, A.E., /Space, Time and Gravitation/, original printed in
1920, reprinted by Cambridge Univ. Press, Cambridge (1987).
Feynman, R.P., Leighton, R.B. and Sands, M., The Feynman Lectures on
Physics, Vol. II, Addison-Wesley, Reading, Mass. (1963).
Greenberger, D.M. and Overhauser, A.W., �The role of gravity in quantum
theory�, _Sci.Amer._ 242 (May):66-76 (1980).
Hoffman, B., /Relativity and its Roots/, Freeman, New York, NY (1983).
Kooistra, J.D., �Paradigm shifty things�, _Analog_ CXVII #6:59-69 (1997).
Ives, H.E. and Stilwell, G.R., �An experimental study of the rate of a
moving atomic clock�, _J.Opt.Soc.Amer._ 28#7:215-226 (1938).
Laplace, P., /_Mechanique Celeste_/, volumes published from 1799-1825,
English translation reprinted by Chelsea Publ., New York (1966).
Mansouri, R. and Sexl, R.U., �A test theory of special relativity: I.
Simultaneity and clock synchronization�, _Gen.Rel.&Grav._ 8:497-513 (1977).
Misner, C.W., K.S. Thorne & J.A. Wheeler, Gravitation, W.H. Freeman &
Co., San Francisco, CA (1973).
Pitjeva, E.V., �Experimental testing of relativity effects, variability
of the gravitational constant and topography of Mercury surface from
radar observations 1964-1989�, _Cel.Mech.&Dyn.Astron._ 55:313-321 (1993).
Synge, J.L., /Relativity/, North-Holland Publishing Co., Amsterdam, Ch.
IX (1960).
Taylor, J.H., Wolszczan, A., Damour, T. & Weisberg, J.M., �Experimental
constraints on strong-field relativistic gravity�, _Nature_ 355:132-136
(1992).
Van Flandern, T., /Dark Matter, Missing Planets and New Comets/, North
Atlantic Books, Berkeley, CA (1993).
Van Flandern, T., �Relativity with Flat Spacetime�, _MetaRes.Bull._
3:9-13 [see >] (1994).
Van Flandern, T., �Possible new properties of gravity�, Parts I & II,
_MetaRes.Bull._ 5:23-29 & 38-50 [see >] (1996).
Walker, W.D., �Superluminal propagation speed of longitudinally
oscillating electrical fields�, abstract in /Causality and Locality in
Modern Physics and Astronomy: Open Questions and Possible Solutions/, S.
Jeffers, ed., York University, North York, Ontario, #72 (1997).