An Analysis of the Speed of Gravity Measurement by Jupiter Scientific including a suggested resolution to the criticism concerning the experiment.
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Analysis of the Speed of Gravity Measurement

Problems with the Concept of a Speed of Gravity

     The staff of Jupiter Scientific believes that it understands the source of the confusion by the scientific community over the speed of gravity measurement. Consider the following animation:

propagating gravity
Figure 1. Propagation of Gravity at Various Speeds.
Top Box: cg < c. Middle Box: cg = c. Bottom Box: cg = infinity.
Because the speed of gravity is infinite for the bottom box,
one does not see gravity propagating; the effect is instantaneous.

     The black disk represents Jupiter. The open circle O is a point of observation. The important questions are: what is the strength of gravity at O at a particular time due to Jupiter when it is moving toward O, and how does it depend on the speed of propagation of gravity? Recall that Jupiter is creating gravity by causing a curvature of spacetime in its vicinity. As Jupiter moves, its effect on spacetime propagates at cg, which is c in Einstein's theory. The upper box shows what happens when cg is less than c. The position of Jupiter as sensed by O is farther away because more time is needed for the signal to propagate to O than the situation for which, cg = c, which is the case shown in the middle box. The relevant distance for the "slow case" cg < c is rs, which is larger than the cg = c case for which the relevant distance is rc. The lower box shows the situation for which the speed of gravity is infinite; the relevant distance ri in this case is smaller than rc. Since the strength of the gravitational force weakens with distance, it seems that the larger cg is, the stronger the gravity of Jupiter is at O.
     Let v denote the speed of Jupiter. Then the differences between the various distances rs, rc, and ri vanish as v/cg and v/c for small v; meaning that these differences are proportional to v for low speeds and vanish if the speed of gravity and the speed of light are made infinite.
     Suppose, however, that a new reference frame is chosen that moves along with Jupiter. In other words, suppose we, as observers, just happen to be moving at the same velocity as Jupiter so that Jupiter does not appear to us to be moving at all. Since, for short times, Jupiter is almost moving in a straight line with constant speed, its acceleration is small, and such a reference frame is, according to Einstein's special theory of relativity, an equally good one to make measurements and observations.
     If we now consider the situation in the new reference frame, Jupiter is now motionless but O is moving toward Jupiter and arrives at ri at the time for which the strength of gravity is being measured. Since Jupiter is not moving, no time is needed for Jupiter to propagate its effect on the curvature of spacetime in the vicinity of O. Although ri appears to be the relevant distance and a contradiction seems to be emerging, this is not the case: Einstein's general theory of gravity includes velocity dependent effects that Newton's theory does not have. These effects behave like v/c for small v, where v is the speed of O but they cannot depend on cg. In Einstein's theory, the results from the two frames are able to be compatible because cg = c.
     One concludes that, if the speed of gravity is not equal to the speed of light, then measurements of the strength of gravity are reference frame dependent.
     Part of the problem is that it is not known how to modify Einstein's theory of gravity to accommodate a speed of gravity different from c. In the frame of reference for which Jupiter is moving, it is natural to substitute v/cg for v/c in a computation based on general relativity. Indeed, this produces the theoretical formula used by Dr. Kopeikin in the analysis of the experiment. See Eq.(29) of Dr. Kopeikin's paper qc/0212121.
     If cg were greater than c, it would violate the principle that physical effects propagate faster than the speed of light. So this case leads to violations of special relativity. Not a single such violation of special relativity has ever been observed.
     If the graviton, the hypothetical particle that is supposed to be exchanged to produce gravity, has a mass, then the effect of gravity propagates at a speed less than c. It is not easy to produce a consistent theory of massive gravitons, but it is known that Newton's inverse square law (the weakening of gravity as the distance squared) would be violated with gravity being eventually exponentially (highly) suppressed. The current bound on the graviton mass is that it must be exceedingly small – about 1034 times smaller than the mass of the electron! This produces a speed of gravity that differs so minutely from the speed of light as to be not measurable in any of the classic tests of general relativity including the Jupiter/quasar experiment.

Earlier Evidence that the Speed of Gravity Is the Speed of Light

     Dr. Clifford M. Will in the early 1970's argued that a certain phenomenological parameter of Einstein's theory would differ from its canonical value if cg did not equal c. Not surprisingly, the calculation of the parameter was frame dependent. Assuming a particularly natural frame, one finds cg equals c to less than one-tenth of a percent!
     The observation that the binary pulsar 1913+16 loses energy due to the generation of gravitational waves also validates Einstein's theory at the one percent level. Indirectly, this supports that gravitational waves propagate at c.
     The above theoretical and experimental evidence for cg = c undermines tests of this equality that are only accurate at the 20% level.

The Calculation for Jupiter/Quasar Measurement

     The Fomalont/Kopeikin measurement focused on the time delay and bending of quasar's light as it passed by Jupiter. However, as explained above, if cg does not equal c then results are frame dependent. In a frame moving with Jupiter, the strength of gravity does not depend on cg. In such a frame, the situation is static. It can also be selected to coincide with one on Earth at the time of the measurement. See the animation in Figure 2.

deflection of light
Figure 2. The Jupiter/Quasar Event as Seen by an Observer Moving with Jupiter.
In this frame, it is the Earth that appears to be moving.

The computation for an observer on Earth can be done by using the results for the static observer s if differences between the two systems are related.
     There are two effects. First, if ks is a unit vector pointing along the direction of propagation of the radio waves at any particular time as measured in the non-moving Jupiter frame (the one denoted by s in Figure 2), then the corresponding vector in the moving-Jupiter frame (that is, the one seen by someone on Earth (denoted E in Figure 2)) is merely shifted to kE = ks + v-/c to order (v/c)2, where v- is the component of v perpendicular to ks. This is just the familiar result that the velocity measured by one observer is shifted when measured by another by the relative velocity of the two observers when, in addition, the constancy of the speed of light is taken into account.
     Second, if v has a component in the direction of propagation of the radio waves, then the Earth system is moving toward Jupiter when viewed from the static system. This reduces any time delay by the time it takes the radio waves to travel the extra distance. If DTs is the time delay as measured by the static observer, then during this interval, Earth moves closer to Jupiter as measured in the static system by a distance vpDTs, where vp is the velocity component of Jupiter in the direction of the propagation of the radio waves. The time it takes the radio waves to travel this extra distance is vpDTs/c. Hence, the time delay DTE as measured on Earth is DTs - vpDTs/c, that is, it is reduced by a factor of 1 - vp/c when compared to DTs.
     Thus to compute the effect of a moving Jupiter, one can proceed as in the static case of a motionless Jupiter carefully noting the dependence of formulas on ks and then performing the above shift in kE and reducing time delays by 1 - vp/c: DTE(kE) = (1 - vp/c) DTs(ks). Indeed, this is precisely Eq.(22) of Dr. Kopeikin's paper qc/0212121 with cg replaced by c. This demonstrates that only v/c and not v/cg enters the calculation when computed in the static frame. This is probably why some astrophysicists believe that the measurement was actually that of the speed of light and not cg.
     In summary, if a speed of gravity parameter is introduced into general relativity that differs from the speed of light then measurements become observer dependent thereby rendering the theory inconsistent. The interpretation that the Jupiter/quasar experiment is a measure of the speed of gravity is valid only in a frame for which Jupiter is moving. In the frame in which Jupiter is not moving, the measurement is only sensitive to the speed of light. This issue undermines the thesis that the Jupiter/quasar measurement is a test of the speed of gravity.

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