AOH :: GRVSPEED.TXT What is the speed of gravity?
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At What Speed Does Gravity Propagate?

Notation:
DM = "Dark Matter, Missing Planets and New Comets", T. Van
Flandern, North Atlantic Books, Berkeley (1993)
MTW = "Gravitation", C.W. Misner, K.S. Thorne & J.A.
Wheeler, W.H. Freeman & Co., San Francisco (1973)
SR = special relativity
v = orbital velocity relative to a primary such as the Sun
c = speed of light
cg = speed of propagation of gravity
G = gravitational constant
M = mass of primary or reference body, usually the Sun
r = distance between two bodies
r-vect = vector distance between two bodies, which has a
magnitude and a direction

The speed of propagation of Newtonian gravity is infinite
(MTW, p. 177; DM, p. 43).  Forces are computed using only the
true, instantaneous positions of moving bodies.  Newtonian
gravity gives an excellent approximation of the motions of bodies
in the solar system.

General relativity reduces to Newtonian gravity in the weak-
field, low-velocity limit, which is why Newtonain gravity works
so well in the solar system.  To avoid the implication of
infinite propagation speed that goes with this reduction, general
relativity postulates that bodies appear to accelerate because
spacetime is curved by the presence of nearby mass; and the
nearest equivalent of a straight line through curved spacetime is
the geodesic path.  So general relativity suggests that no
propagating force is needed, since moving bodies are simply
following geodesic paths (straight lines through curved
spacetime) without the help of forces.

However, that explanation avoids the question of how quickly
distant spacetime responds to changes in the position of the
nearby mass acting as the cause of the spacetime curvature.
There must in principle be communication between a mass and the
spacetime around it at some finite speed.  If there were not, we
would have a case of true "action at a distance", about which
Isaac Newton said, "That one body may act upon another at a
distance through a vacuum without the mediation of anything else
... is to me so great an absurdity, that I believe no man, who
has in philosophical matters a competent faculty for thinking,
can ever fall into."

Put another way, Einstein's equivalence principle tells us
that a gravitational force and a uniform acceleration are
equivalent.  So our question about the speed of propagation of
gravity is equivalent to a question about the direction in which
moving bodies accelerate relative to the body causing that
acceleration.

In the following, we will first show why gravity is
different from electromagnetism.  We will then eliminate several
possible red herring arguments.  Finally, we will consider the
experimental evidence bearing on the answer to our title
question.

Similarities between gravity and electromagnetism:
(1) The force of both has inverse square behavior.

Dissimilarities between gravity and electromagnetism:
(1) Electromagnetic forces are both attractive and
repulsive, gravity is attractive only.
(2) A body's own charge affects its motion in the presence
of other charges, but its own mass does not affect its
motion in the presence of other masses.
(3) There is no Equivalence Principle for charges, and no
analog of magnetism or Maxwell's equations for gravity.
(4) Massive bodies are free to move continuously; but
electrons are confined to discrete energy levels.
(5) The relative strengths of gravity and electromagnetism
differ by 40 orders of magnitude at comparable
distances.

We will consider the question of the speed of propagation of
the force of gravity in a moment.  In this connection, the
following items are not relevant:

(1) Gravitational radiation, or gravity waves.  These
hypothetical ripples in spacetime should result from
rapid mass changes or accelerations, but are too weak
by many orders of magnitude to affect ordinary gravity
to the precision that matters here.  Even for binary
pulsars, the time scale for significant orbital change
is 10^8 years.  And there is still no certainty that
gravity waves do exist.

(2) Special relativity.  Velocities in the solar system are
on the order of 10^-4 c.  So the Lorentz factor for
masses, distances, and time intervals is of order
10^-8.  For our purposes here, mass increase, distance
contraction, and time dilation can all be ignored.
Only one frame of reference is needed: that with fixed
origin at the center of mass.

(3) Electromagnetic analogs of gravity.  From the preceding
list of dissimilarities, any such analogy is suspect.
But specifically, the Lienard-Wiechert "retarded
potentials" used in electromagnetism do not bear on the
question of the speed of gravity (or for that matter,
on the speed of virtual photons either) because they do
not consider retarded positions, only potentials.
These are integrals over the space distribution of the
charge and current densities evaluated at the time of
emission, but using instantaneous positions.  The same
is true of the equivalent formulas for gravity (MTW, p.
996 & p. 1080): the "retarded integrals" consider only
delays of stress energy density, but not delays of
position.  Only delays of effective position are
relevant to the speed of propagation issue for
communication between a gravitational source body and
the position of some moving test particle.  The
integrals in question determine the effective
distribution of mass, but not its effective position,
which is assumed to be the instantaneous one.

(4) "Rubber sheet" analogies.  Sometimes, these are invoked
to visualize the effect of a mass on spacetime.  In
such analogies, it appears at first glance that nothing
need propagate continuously from a mass for its
gravitational effect to continue.  However, if a body
finds itself on the side of a "spacetime hill", it has
no tendency to roll downhill unless some force, such as
gravity, acts dynamically on it.  To that extent the
analogy is faulty.  Besides, real masses such as the
Sun are in a continual state of motion around the solar
system barycenter, which occasionally is found outside
the Sun's physical body.  So whatever the Sun may do to
spacetime must be continually updated to reflect such
motions.

We consider the results of six experiments sensitive to the
speed of propagation of gravity.

(1) Experiment 1: A static Sun orbited by a relatively
massless Earth.  [Even in a static field, propagation
delay should manifest itself.]

Celestial mechanics experiments, such as radar
measurements of the direction of the radius of
curvature of the Earth's orbit, or the directions of
accelerations in the equations of motion, show that
gravity accelerates all bodies toward the true,
instantaneous position of the gravity source.

The radius of curvature may be computed from solar
system numerical integrations that have been fitted to
observations of high precision, such as those of JPL or
the Astronomical Almanac.  All such numerical
integrations use true, instantaneous positions of all
masses in a frame centered on the system center of
mass, because the results do not resemble the
observations if that is not done.

Full relativistic equations of motion, such as the
Einstein-Infeld-Hoffman equations (MTW, p. 1095),
compute the accelerations of each body using the true,
instantaneous positions of each other body.  No delayed
positions are present, nor were any used in the
derivation of these equations that later cancelled out
(see, e.g., Robertson & Noonan, 1938).  For example,
most of the acceleration is contained in a GM/r^3 r-
vect term.  The delayed value of r-vect = r-vect -
(r/cg) d(r-vect)/dt, where r/cg is the gravity
propagation delay time.  No such terms appear in the
equations, directly or hidden.  This is because cg has
been taken as infinite.

All of the post-Newtonian terms contain either the
potential, GM/r, or the velocity squared, for one or
more bodies.  In either case, a factor of c^2 appears
as a divisor.  These c^2 factors arise from the Lorentz
transformations of SR, not from propagation
considerations.  All such terms are negligible in size
compared to the motion of the bodies during one
lighttime interval, which is proportional to v/c.  For
the Earth, v/c is about 10^-4, and both (v/c)^2 and

It is argued that, since the Sun's field is
static, continual dynamic regeneration is not needed.
But this is a play on the meaning of the word "static",
which does not mean "cemented in place", but rather
"unchanging in character from moment to moment"; e.g.,
a steady rainfall.  If there were literally nothing
continually pulling or pushing the Earth toward the
Sun, or continually reshaping spacetime near the Earth,
the Earth would not appear to accelerate toward the
Sun.

By way of contrast with gravity, light and
radiation pressure from the Sun strike orbiting planets
from a forward angle equal to the ratio of the
transverse speed of the orbiting body to the
propagation speed of light (v/c).  For the Earth, this
angle (called "aberration") is 20 arc seconds.  In the
case of light, radiation pressure actually retards the
forward motion of planets, but by a negligible amount
since light has so little momentum.  For orbiting dust,
the effect is sensible, and the corresponding
deceleration is called the "Poynting-Robertson effect".

Anything that propagates from Sun to Earth will
act with aberration in the manner of a steady, vertical
rainfall.  As seen from a moving train, the raindrops
appear to streak across the windows at an angle that
depends on the ratio of the train's speed to the
raindrop speed, and the drops oppose the train's
forward motion.  Likewise, an arrow shot radially out
of the Sun toward an orbiting spacecraft with open
windows will be moving directly away from the Sun at
every instant; yet it will cross the moving spacecraft
cabin at an angle that depends on the spacecraft's
forward speed, and appear to the astronauts to come
from a somewhat forward direction.  If the arrow
strikes the spacecraft, part of its momentum opposes
the spacecraft's motion.

When momentum is transferred by a gravitational
pulse from the Sun to a test particle in orbit, the
momentum of the pulse adds vectorially to the momentum
of the particle to determine the net impulse applied.
The integrated effect of a continuous source of such
pulses produces continuous impulses on the test
particle, which become its acceleration.  Obviously,
the vector momentum of the gravitational pulse depends
on the instantaneous (not future) direction of its
source; and the vector momentum of the test particle
depends on the particle's velocity.  So the net
acceleration must depend on both the direction of the
source and the affected particle velocity.  In general,
all physical forces act along the effective direction
of their approach.  And that effective direction of
approach must be influenced by the affected body's
velocity.  The fact that this does not appear to happen
in the case of gravity must be attributed to gravity's
relatively very high propagation speed.

Sunlight and the radiation pressure of sunlight
display this aberration effect and retard the motions
of orbiting bodies.  But gravity appears to have zero
aberration.  So the directions of arrival of forces
from gravity and from solar radiation pressure differ
by the aberration angle, showing that their respective
propagation speeds are not equal.  Laplace and Poincare
used the absence of observable gravitational aberration
to set a lower limit for the speed of gravitational
propagation of about 10^7 to 10^8 times the speed of
light.  Modern data would improve this lower limit to
10^10 c.

(2) Experiment 2: The Earth fixed at the origin, orbited by
the Sun.  [Propagation delay affects motion independent
of what frame of reference is chosen]

The principle of relativity tells us that no
experiment enables us to determine which of two
relatively moving bodies is actually moving, and which
is not.  So we can consider the Earth as origin with
the Sun moving around us, as is frequently done in
astronomy for convenience.  Even if any ambiguity about
which body is moving can be resolved over a revolution
by examining the aberration of the distant stars, there
is no way to make such a distinction in an interval as
short as the Sun-Earth lighttime of 500 seconds, during
which the relative motion is essentially linear and
uniform, and the relativity principle applies.  But if
the Sun's "static field" gave it any special status,
e.g., if it were the cause of gravitational aberration
being zero, we would then have a means of detecting
absolute motion of the Earth through the Sun's static
field, in violation of the relativity principle.

In the Earth-centered frame, the direction from
which the Sun's gravitational force emanates relative
to background stars is toward the Sun's true,
instantaneous position rather than toward the apparent
solar disk we see now.  We know that from simple
computer modeling, which shows that if we use delayed
positions, the periods of orbiting bodies continually
increase as unphysical angular momentum gets pumped
into the system.  We will not be able to see the Sun in
the direction from which its gravity now acts until 500
seconds in the future -- the lighttime from Sun to
Earth.  This again shows that gravity acts faster than
light.  If gravity acted with delay as light does, the
Earth would accelerate out of orbit.

Eddington summarized the reason best:  "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."

This point may be further emphasized with a
thought experiment in which the Sun's mass is
increased, and the Earth's velocity increased
commensurately so as to maintain a circular orbit.
When the Earth's velocity is near the speed of light,
the apparent Sun always leads the Earth in orbit.  If
gravity came from that same direction, the Earth would
always accelerate its orbital speed.  So gravity must
be communicated from Sun to Earth much faster than
light.

(3) Experiment 3: The Sun-Earth-Moon system.  [Propagation
delay should also appear in the 3-body problem.]

It is sometimes said that the Sun's field curves
space, and that the curved space then mimics the Sun's
accelerations around the solar system barycenter and
through the galaxy in the manner of a rigid body;
hence, without delay.  There are two problems with that
scenario.  Even rigid bars do not respond to an
acceleration instantly; instead, a pressure wave
propagates from one end to the other at less than the
speed of light.  Secondly, the field of the Sun would
have to act in this "rigid body" manner out to
essentially infinite distances from the Sun to avoid
delayed accelerations occuring at great distances,
contrary to observations.

As a further test, I considered the Earth-Moon
system as a two-body problem perturbed by the Sun.  The
effects of all other smaller perturbations were
isolated and eliminated.  The Moon's orbit around the
Earth was taken as an ellipse with a "bulge" toward a
third perturbing body.  I then used lunar occultation
timings to solve for the direction of the "bulge" of
the Moon's orbit away from elliptical.  That direction
turned out to be toward the Sun's true, instantaneous
direction, and not the direction from which its light
arrives (20 arc seconds away), to within an uncertainty
of one arc second.  This 3-body experiment sets a lower
limit to the speed of action of gravity of not less
than 20 times faster than light.

(4) Experiment 4: Binary star systems.  [Propagation delay
affects orbits even if the source of a force is
linearly extrapolated into the future.]

It is sometimes suggested that gravity acts from
the delayed position of a source body extrapolated
forward linearly for one lighttime.  This would be very
close to its true, instantaneous position.  But
computer experiments with close binary star systems
show that even this small difference adds angular
momentum to the orbiting pair at an observationally
unacceptable rate.  The rate of change of period with
time from this effect is 2 pi (v/c)^3.

It should be noted that close binary stars are
generally on precessing elliptical orbits that never
precisely repeat their paths through space, so no
steady-state condition can be set up in spacetime.  Any
spacetime curvature must be continually regenerated.

Binary pulsars provide an even more critical test,
showing that any computational assumption other than
true, instantaneous positions for the sources of
gravitation for two massive orbiting bodies lead to
unacceptable deviations from real orbits.

(5) Experiment 5: Equal masses, S and E, in linear motion
past one another, with S deflected at some point.
[Even if forces were extrapolated ahead on geodesic

It is sometimes postulated that, nonetheless,
massive orbiting bodies know each other's future
geodesic paths, so no new information is communicated
by the apparently instantaneous action of gravity.
Consider the following thought experiment which assumes

Consider two bodies of equal mass, S and E, moving
on linear paths from infinity, and passing one another
at some minimum distance d at time t (the same in both
frames, since SR effects are negligible).  The
lighttime between the two bodies is then d/c.  Now
suppose that body S carries a charge, so that it may be
deflected from its normal path by an electromagnet
without disturbing E.  Let S be deflected onto a new
trajectory at time t (closest approach).

At time t, E is accelerating toward the true,
instantaneous position of S.  By postulate here, it
must continue to do so for the lighttime d/c after time
t, since it cannot know of S's deflection during that
interval.  Paradox (1) is that, between time t and t +
d/c, E is accelerating toward positions in space that S
never occupied -- an apparent effect without a cause.

Eventually, the lightime elapses and E learns that
S is now on a new trajectory.  By postulate, E must now
begin accelerating toward the new true, instantaneous
position of S at time t + d/c.  Paradox (2) is that
apparently S could have gone anywhere during the
lighttime d/c without any effect whatever on E -- a
gravational cause without any effect.

It has been suggested that there might be some
"transient behavior" period following such a deflection
during which E accelerates toward the retarded position
of S, gradually resuming its acceleration toward S's
true, instantaneous position.  Paradox (3) is that d
can be as small or as large as we please, so that no
possible "transient behavior" interval can be
satisfactory for all possible values of d.

All of these paradoxes arise because the
accelerations of bodies must be toward the true,
instantaneous position of the source to be in accord
with observation and experiment.  In this example, that
is assumed to happen as the result of a geodesic
extrapolation of a source body to some hypothetical
future position during the transit time, instead of by
the price that must be paid if one wishes to assume
gravity propagation at lightspeed or slower.  If
gravity propagates at a speed too fast to measure, no

(6) Experiment 6: Black holes.  [The difference in
propagation speed between gravity and light is
essential, not disposable.]

The escape velocity from a black hole exceeds the
speed of light, which is the basic reason why no light
escapes.  It follows that anything communicating
between a gravity source inside a black hole and space
outside the hole must propagate faster than light to
escape.

We conclude that a reasonable interpretation of the
experimental evidence is that gravity propagates without
detectable delay; i.e., faster than light.  Reconciliations of
this conclusion with the two postulates and ten basic experiments
supporting SR, and with cosmology, may be found in the book DM.

Tom Van Flandern
1993/06/06

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