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
          GM/(rc^2) are about 10^-8.

               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
          paths, paradoxes would arise.]

               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
          this is true, and leads to a paradox.

               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
          near-instantaneous propagation.  These paradoxes are
          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
          such logical paradoxes arise.

     (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