Meaning of the "speed of gravity"

This is a complete set of exchanges between Tom Van Flandern (tvf) and S. Kopeikin (sk) facilitated by Stephen Speicher in connection with the 2002 September 8 Jupiter-quasar appulse and the "speed of gravity" issue. Further information about the views expressed here may be found in the just-published paper: “Experimental Repeal of the Speed Limit for Gravitational, Electrodynamic, and Quantum Field Interactions”, T. Van Flandern and J.P. Vigier, Found.Phys. 32(#7), 1031-1068 (2002).

[Note added 2002/01/08: See press release "Kopeikin and the speed of gravity".]

SUMMARY: Kopeikin uses the expression "the speed of gravity" for the speed of travel of changes in the gravitational potential field responsible for light-bending and radar/radio signal delay, also known as the speed of gravitational waves. No current dispute exists about this speed, which must be the speed of light (c). The Jupiter-quasar appulse may indeed be the first direct measurement of that speed. By contrast, the appulse can provide no information about the propagation speed of gravitational force, which is bounded by many experiments to be much faster than light, and by the most sensitive experiment to exceed 20 billion c. In general relativity, when the solutions to the Einstein equations (which govern the potential) are converted to equations of motion (which describe gravitational acceleration), the assumption of infinite speed of gravitational force is implicitly adopted by setting aberration in the gradient of the potential equal to zero.

In the following email exchanges, “the paper” refers to S. Kopeikin’s paper in ApJ Letters 556, L1-L5 (2002).

INITIAL LETTER:

[tvf]: The paper’s introduction makes some very confusing statements about the meaning of the concept “the speed of gravity”, because it speaks of the speed of gravitational waves interchangeably with the speed of gravity. Indeed, most of the paper is about the potential field and its consequences (light-bending and radio signal propagation delay), and not at all about gravitational force propagation. Of course, neither I nor anyone else that I know of has a credible notion that the speed of gravitational waves can be anything but c. That speed is pretty much undisputed. But at several later places in the text, references are made to consequences of a speed of gravity different from c (which must mean the speed of gravitational waves because standard GR doesn’t have any propagation delay for gravitational forces.) Then there is explicit mention that the propagation “speed of gravity” is infinite in Newtonian gravity (which must mean gravitational force speed because Newtonian gravity has no gravitational waves). I find that I am not able to separate out the confusing consequences of the lack of clarity on this point. Whenever the speed of gravity is mentioned in the paper, which places mean the propagation speed of waves in the potential (the space-time medium), and which places mean the propagation speed of gravitational force? (The latter, of course, does not even exist in the geometric interpretation of GR.)

[tvf]: Now let’s look specifically at SK’s equation (2). In essence, this says the time delay is the integral of 2Gm/(rc^2) dt, where only time spent close to significant masses is a major contributor to the delay. The fact that retarded times are used in the integral’s limits is irrelevant because (a) those are the retarded times for the electromagnetic signals, not for the active masses or their gravitational forces; and (2) the integral’s end points are of little concern because most of the action of interest occurs while the signal is passing Jupiter. I see nothing in distance that appears in the denominator of that equation to indicate that Jupiter acts from anything but its instantaneous position as the signal passes Jupiter. If there were any retardation to Jupiter’s gravitational force field, it would have to show up as a delay in the value of r proportional to Jupiter’s velocity (to the first power) and to the force field propagation speed. I see no such term.

[tvf]: Clearly, my first impression, based as it was on Kopeikin's own descriptions of what his results meant, was completely erroneous. Kopeikin repeatedly uses ambiguous terminology and is oblivious to the on-going discussion of the "speed of gravity" issue. For him, "gravitational field" means "gravitational potential field" and "the speed of gravity" means the propagation speed of changes in the "nearfield" potential. While it is true, as he says, that this latter speed has not been measured experimentally before now, I am unaware of any serious challenge to the idea that the propagation speed of these changes is the speed of light.

[tvf]: Kopeikin shows his unfamiliarity with the speed-of-gravity issue in many places, but perhaps none more telling than this candid remark: he says the Lienard-Wiechert potential "accounts for all possible effects in the description of the gravitational field". If that were true, there would be no controversy over the speed of gravity. But in fact, the L-W potential describes only the potential field, and by itself has nothing to say about the force field. To describe the force, one must at least take the gradient of the potential, which then (for the first time) raises the question of which gradient, instantaneous or retarded, that is so intimately tied up in the propagation-speed-of-gravitational-force question. Kopeikin never goes there.

FIRST ROUND OF RESPONSES:

{>> indicates quote from above; >[sk] indicates SK’s initial reply; unattributed paragraphs are tvf’s response.}

>> [tvf]: [The SK-ApJ paper] speaks of the speed of gravitational waves interchangeably with the speed of gravity.

> [sk]: There are two phenomena - (1) propagation of free gravitational waves in radiative zone of an isolated system emitting these waves, (2) retardation in propagation of gravitational fields inside the near zone of the isolated system. These two phenomena interrelated and the retardation time taken by gravity in the near zone to propagate can be calculated if one knows the speed of propagation of gravitational waves. In other words, near zone and wave zone propagation of gravity have the same speed.

I agree with everything in this statement. But neither of these phenomena, gravitational waves or Lienard-Wiechert-type retardation of gravitational fields inside the near zone, has anything to do with the propagation speed of gravitational force. This last phenomenon has no wave character, and therefore "nearfield" is an undefined concept for it. Nor does the L-W potential have anything directly to say about it.

Let me be as clear as I can about this central point. Different people mean one of two quite different things by the expression “gravitational field”. Relativists usually mean the gravitational potential field. However, there is no controversy about the speed of disturbances in the gravitational potential field. Those are the two phenomena described above, and they propagate at light-speed without dispute (although that is not experimentally verified). The speed of gravitational radiation is c, period; and any attempt to theoretically or experimentally parameterize any deviation from speed c does a great disservice to science if it does not carefully disassociate itself from the “speed of gravitational force” controversy. Gravitational potential is not gravitational force, nor is it even a contributor to ordinary gravitational force except through the usually miniscule vehicles of light-bending, perihelion advance, and other effects that are at least second order in (v/c). The relationship "force is the gradient of potential" does not imply a common propagation speed, any more than it would between any other physical parameter and its derivative.

By way of contrast with the preceding, dynamicists usually use “gravitational field” to speak of gravitational force or acceleration, which is a different concept entirely. It is common to hear that gravitational force must either not exist at all (as in the geometric interpretation of GR), or have infinite propagation speed (as in the force interpretation of GR). But it is undisputed that it cannot simply propagate at speed c, as any computer experiment will readily show. In fact, binary pulsars show that the acceleration of each component anticipates the future position, velocity, and acceleration of the other in much less than the light-time between them. The modern explanation for this, championed recently by Steve Carlip, is that gravitational force actually propagates at lightspeed also, but a counter-force arising spontaneously within the gravitational field itself (imposed by nature to conserve angular momentum) almost exactly cancels the effects of propagation delay from this gravitational force, making it only appear to propagate with infinite speed. The alternative possibility I have advocated is that gravitational force fields really do propagate strongly faster-than-light. This would mean replacing special relativity with Lorentzian relativity, but has no mathematical consequences for relativity theory, and changes little except that it lifts the universal speed limit. No existing experiment differentiates between these two variants of relativity (SR and LR).

Given Kopeikin's other remarks, I would hazard a guess that he is unfamiliar with Lorentzian relativity (LR), a key part of this issue. While there is no shame in that (LR is not being taught in universities these days), unfamiliarity with this issue does cripple SK's ability to contribute in a helpful way to the on-going dialog.

>> [tvf]: Indeed, most of the paper is about the potential field and its consequences (light-bending and radio signal propagation delay), and not at all about gravitational force propagation.

> [sk]: Gravitational force propagation does take part in the bending and time delay of light. This is because photons move through the time-dependent gravitational field which propagates with finite speed.

I disagree with the statement that gravitational force has anything to do with time delay of light, and little to do with light-bending. In Newtonian gravity, particles moving at speed c would have half the bending that light does in GR. But the light bending in GR is not caused directly by gravitational force, but rather by gravitational potential, as is obvious by examining the formula for the effect (proportional to 1/r).

As a specific example, if we send a light beam through the empty interior of a uniform spherical shell of matter, the gravitational force there is zero, but the potential is not, and still the light beam suffers propagation delay. Therefore, gravitational force cannot be causing that delay.

> [sk]: My paper suggest that the speed of gravity is the same as the speed of light.

Let us agree not to use the ambiguous expressions "gravitational field" and "the speed of gravity" without specifying whether we mean the potential field or the force field. That will prevent much misunderstanding.

> [sk]: Propagation of gravitational force is revealed as the effect of retardation after expansion of the post-Minkowskian expression for this force in power series with respect to v/c. Since we are talking about solutions of the Einstein equations "c" here means the speed of gravity. The situation is simple as I repeated many times in many different places (1) take retarded solution of the Einstein equations which describe propagation both gravity waves and gravity forces (depends on whether the wave zone or near zone are considered) (2) substitute this retarded solution in the equation of light propagation (propagating with the speed of light of course) (3) solve this propagation equation and watch carefully how the retardation effect (gravity propagation) from the Einstein equations enter solutions of the light propagation equations (4) conclusion is that the light must be deflected by gravitating bodies taken in their retarded positions and this retardation is due to the finite speed of propagation of gravity. That's it.

This restates the misunderstanding already described. Solutions of the Einstein equations contain the speed of light, not the speed of gravity. It is most unhelpful to use terms in a way that contributes to confusion instead of clarifying.

The field equations and their solutions do not address gravitational force. One must take the gradient of the potential, or set up a Hamiltonian and take partials, or some such process to develop equations of motion in 3-space with respect to coordinate time. These equations of motion describe gravitational "force" in the sense that expression is used in celestial mechanics.

In the process of deriving equations of motion, no terms in v/c ever arise. If c is the speed of gravity, then c = infinity produces the observed result, just as in Newtonian gravity. In fact, GR could not reduce to Newtonian gravity in the weak-field, low-velocity limit if forces in GR did not propagate at the same speed as in Newtonian gravity.

>> [tvf]: The fact that retarded times are used in the integral's limits [in equation (2)] is irrelevant because (a) those are the retarded times for the electromagnetic signals, not for the active masses or their gravitational forces;

> [sk]: Not true, because gravitational potentials enter the equation (2) and they are retarded solutions of the Einstein equations which describe propagation of gravity but not light. The same equation would be written for any massless particle. The retardation effect in Eq. (2) is a general phenomena associated with the finite speed of propagation of GRAVITY and valid for description of propagation of any kind of massless particles in the variable gravitational field.

You are speaking of gravitational potential, and I am speaking of gravitational force. So we are speaking at cross-purposes. Waves in the potential propagate at the speed of light, whether they be electromagnetic or "gravitational". But true gravitational waves would be utterly undetectable, which is why I referred only to electromagnetic signals here.

>> [tvf]: and (2) the integral's end points are of little concern because most of the action of interest occurs while the signal is passing Jupiter.

> [sk]: Eq. (2) is a rigorous solution of differential equations. In case of any concern mathematical mistake in my solution should be indicated. I did not invent the limits of integration by hand - they are coming up as a result of exact mathematical solution of propagation of light from quasar to observer. I now understand the sense in which you mean that, and do not disagree with you in a "potential field" context. I take it you do not disagree with my statement, or perhaps have no opinion, upon realizing that I was addressing gravitational force only.

It is easy to show with a "back-of-the-envelope" estimate that one need not waste time calculating the observable effects of any retardation in gravitational force caused by Jupiter being at its retarded instead of instantaneous position during the light-time from Jupiter to the quasar signal. Any such effect would be negligible in size compared with the effects you have computed rigorously.

>> [tvf]: I see nothing in distance that appears in the denominator of that equation to indicate that Jupiter acts from anything but its instantaneous position as the signal passes Jupiter.

> [sk]: Eq. (2) is the starting point. It must be integrated to see what is going on. Proceed to Eqs. (3),(4) and see that the retarded time in (3),(4) is due to the retarded Lienard-Wiechert gravitational potentials

This is another example of our semantic potential-field vs. force-field problem. My statement remains unrefuted. Your equations calculate the time delay based on essentially instantaneous positions of Jupiter, not positions that are retarded, from the perspective of the passing quasar signal (as contrasted with the perspectives of the observer or the source, which are ultimately of small importance to the effect we seek).

>> [tvf]: If there were any retardation to Jupiter's gravitational force field, it would have to show up as a delay in the value of r proportional to Jupiter's velocity (to the first power) and to the force field propagation speed. I see no such term.

> [sk]: You are talking about the Lienard-Wiechert retarded potential.

I am specifically *not* talking of Lienard-Wiechert-type retardation (which is retardation in the potential), but of retardation in the force field, which is roughly (v/c) times the distance from Jupiter to passing signal.

My conclusion is that we really have no disagreements except over terminology, the critical one being the meaning of "the speed of gravity". Kopeikin's paper is all about the speed of changes in nearfield gravitational potential, about which there is no controversy. Any quick estimate shows that the signal propagation delays will not change significantly if Jupiter's gravity acts from its retarded instead of instantaneous position with respect to the quasar signal. And doing so would be a violation of GR anyway, because GR requires gravitating bodies to act from their instantaneous positions, not retarded ones.

SECOND ROUND OF RESPONSES:

>> [tvf]: Let me be as clear as I can about this central point. Different people mean one of two quite different things by the expression "gravitational field".

> [sk]: Gravitational field is defined through derivatives of metric tensor - first and/or second.

That is one possible meaning. It is important to know it is not the only meaning of "gravitational field" in common usage.

>> [tvf]: The relationship "force is the gradient of potential" does not imply a common propagation speed, any more than it would between any other physical parameter and its derivative.

> [sk]: In electrodynamics differentiation of L-W electromagnetic potential gives an electromagnetic force which `propagates' with the speed of light, c.

The same arguments apply to electrodynamics as apply to gravitation. Electrodynamic (Coulomb) forces propagate almost instantly, which is why the angular momentum of charges is unchanged by their encounters. By contrast, all electromagnetic phenomena propagate at speed c, and forces applied by these phenomena (e.g., radiation pressure) alter angular momentum. The relationships between the two phenomena are very much like those between the force field and potential field for gravity. They are related as derivative (gradient) to function. Functions and their derivatives are *not* constrained to have the same properties, such as propagation speed. It is not even necessary that both propagate just because one of them does.

>> [tvf]: By way of contrast with the preceding, dynamicists usually use "gravitational field" to speak of gravitational force or acceleration, which is a different concept entirely. It is common to hear that gravitational force must either not exist at all (as in the geometric interpretation of GR),

> [sk]: There are tidal gravitational forces expressed through the Riemann tensor. Their existence does not depend on the point of view of anybody.

However, tidal forces are not relevant to the points under discussion here. If the word “force” carries too much baggage, substitute 3-space “acceleration”, which is (I think) unambiguous in its meaning.

>> [tvf]: or have infinite propagation speed (as in the force interpretation of GR).

> [sk]: I never heard such a statement from professional relativists whom I knew or know.

Nor have I. But then, most professional relativists today are educated in the geometric interpretation of GR, in which that statement is not true. Few of them today even know about the force interpretation, even though that is the one favored by Einstein, Dirac, and Feynman, among many others.

>> [tvf]: But it is undisputed that it cannot simply propagate at speed c, as any computer experiment will readily show.

> [sk]: Computer experiments are doubtful if they are not confirmed by analytic calculations.

I deal in a field (celestial mechanics) where analytical solutions are often impossible or non-convergent. Computer experiments and numerical integrations are normal tools of the trade, and their results are valued.

>> [tvf]: In fact, binary pulsars show that the acceleration of each component anticipates the future position, velocity, and acceleration of the other in much less than the light-time between them. The modern explanation for this, championed recently by Steve Carlip, is that gravitational force actually propagates at lightspeed also, but a counter-force arising spontaneously within the gravitational field itself (imposed by nature to conserve angular momentum) almost exactly cancels the effects of propagation delay from this gravitational force, making it only appear to propagate with infinite speed. The alternative possibility I have advocated is that gravitational force fields really do propagate strongly faster-than-light.

> [sk]: This does not work out. Lorentz-invariance is strongly violated under such assumption.

We both know that is true. What only one of us seems to appreciate is that gravitational force does not have the general property of Lorentz invariance, whatever its aesthetic appeal. That much should have been obvious from the fact that the Newtonian universal gravity law is not Lorentz invariant, and GR reduces to Newtonian gravity in the weak field, low-velocity limit.

Lorentz invariance was brought back to gravitation in GR by switching the subject from force fields to potential fields, and ignoring the fact that the gradients used to convert potentials back into forces were instantaneous gradients, not retarded ones. The use in GR of instantaneous gradients ends the reciprocal Lorentz invariant property for gravitational forces. Only solutions to the field equations, which involve gravitational potential, are fully Lorentz invariant. When converted to equations of motion, they lose that property.

> [sk]: I never saw equations of gravitational field which are covariant, assume Lorentz-invariance, and have solutions describing propagation of the field with infinite speed.

Nor have I. The equations of motion of GR, which have no force propagation delays in them for any distance between a source mass and a target body, are not Lorentz invariant. But you were no doubt speaking of solutions to the field equations, which are.

> [sk]: What is Lorentzian relativity?

Lorentzian relativity (LR) is a slight updating of the relativity theory of Lorentz, published in 1904 (one year before Einstein). It was the first theory to combine the relativity principle and the Lorentz transformations. It is mathematically identical to special relativity (SR). However, Lorentz developed his theory in the context of an aether. Einstein’s 1905 contribution was to hypothesize that the aether was unnecessary, in large part through his two unique postulates.

The essential difference between these two theories (SR and LR, which make identical predictions for observable phenomena in any one inertial frame viewed from any other) is the lack of reciprocity in LR because one frame (the local gravity field) is special. In SR, of course, all inertial frames are equivalent. The only consequence of this difference of importance here is that SR has a universal speed limit (c), whereas LR does not. This happens because in SR, time and space are changed by motion; whereas in LR, only clocks and rulers are changed, but time and space are unaffected. The physics of these two theories is quite different, even though the math is the same.

>> [tvf]: As a specific example, if we send a light beam through the empty interior of a uniform spherical shell of matter, the gravitational force there is zero, but the potential is not, and still the light beam suffers propagation delay. Therefore, gravitational force cannot be causing that delay.

> [sk]: I should look at this problem. I did not see mathematical solution of it. If indeed there is a delay in propagation it can be also caused by boundary conditions.

But the delay increases with the diameter of the spherical shell. Yet the boundary conditions for a simple light beam remain independent of the shell diameter. So it is difficult to see how the propagation delay can be affected by boundary conditions.

> [sk]: Force field is obtained as derivative of metric tensor. If metric tensor has a retarded argument so will the force have.

This second sentence is perhaps the most important sentence written by either of us so far in this discussion. If it were true, there would be no issue before us. Force is derived from a retarded potential by taking a gradient. In the gradient (which is just a set of partial derivatives), one can choose to use instantaneous or retarded coordinates in those partial derivatives. If one chooses retarded coordinates, the gradient points toward the retarded source position, and angular momentum conservation is lost. So in GR the choice is always to use instantaneous coordinates when taking this crucial derivative. This points the gradient toward the instantaneous source mass, and conserves angular momentum, allowing the theory to agree with observations at the expense of its Lorentz invariance.

The choice of instantaneous coordinates for taking a gradient (as is done in GR) is physically valid only if the target body has no detectable transverse motion during the force propagation time. Making that choice in the general case, i.e., pointing the gradient toward the instantaneous source mass even for a moving target body, is the logical equivalent of adopting infinite propagation speed for gravitational force (or whatever causes gravitational acceleration).

> [sk]: I do not see principal difficulties with my approach because i have solved equations, have written their mathematical solutions, etc. Any words can be used after that - they do not change mathematical presentation. For this reason, I prefer do not use words at all. Find mistakes in my mathematics.This is also an important point, one you touch on repeatedly. Study my previous answer closely. Notice that I have no objection to anything in the mathematics -- yours or that of GR. The only mistake is in the words used to describe the results. So I cannot honor your understandable wish to confine this discussion to equations, where you are no doubt correct. Rather, it is about words used to describe what the equations mean, which I maintain are certainly incorrect.

I hasten to add the fault is not yours alone. You are following the tradition that arose when the geometric interpretation of GR was first advanced to ignore such issues of physics, and to sweep them away with defective analogies and sound-byte rhetoric. By concentrating all attention on the potential field, one can keep the student from asking too many embarrassing questions about the gravitational force field -- questions for which there are no good answers.

> [sk]: Forget about light when you deal with gravity. Speed of gravity is numerically the same as the speed of light but gravity obviously is not light.

This reverts to the ambiguous language that is at the heart of the problem here. Only gravitational waves (changes in gravitational potential) propagate at light speed. (This is perhaps because those waves *are* very-long-wavelength electromagnetic waves. But whether gravitational waves are electromagnetic phenomena or not is a separate issue.) Gravitational force, by contrast, clearly exhibits no propagation delay in either theory or experiment. Various reasons are offered about why this is, but there is no dispute that it is so.

>> [tvf]: It is most unhelpful to use terms in a way that contributes to confusion instead of clarifying.

> [sk]: I do not use, I solve equations. Find mistake in my mathematical derivation.There is no mistake in your mathematical derivation. But I see a big mistake in your words. They have led Stephen Speicher and others to draw incorrect conclusions that the coming appulse of Jupiter and a quasar will experimentally test the "speed of gravity" (meaning gravitational force), when you meant only the speed of changes in gravitational potential, about which there is no dispute. How do you suggest that I address this?

My procedure is to allow experiments and reason to guide us in interpreting mathematical results. To ignore these constraints can lead to confusion, if not outright nonsense.

>> [tvf]: In the process of deriving equations of motion, no terms in v/c ever arise.

> [sk]: This is not true. What about EIH equations of motion, or equations of geodesics?

I am sorry for using yet more ambiguous language. I meant to say "no terms in (v/c) to the first power", such as would be required to cancel propagation delay effects. Much smaller effects proportional to (v/c)^2 and higher powers are of course the main content of those equations.

>> [tvf]: It is easy to show with a "back-of-the-envelope" estimate that one need not waste time calculating the observable effects of any retardation in gravitational force caused by Jupiter being at its retarded instead of instantaneous position during the light-time from Jupiter to the quasar signal. Any such effect would be negligible in size compared with the effects you have computed rigorously.

> [sk]: Well. Many people say this but nobody solved the problem before me. After solution is known many people say it is (almost) obvious. It is up to people how to behave, my task is to solve mathematics and to give its interpretation.

You did not address the problem I mentioned just above, dealing with delays in gravitational force. GR predicts no such delays, so naturally you are not motivated to parameterize them. You addressed only delays in changes in the potential field, which is also the only way in which "gravitation" delays electromagnetic signals. Force or acceleration has no effect whatever on these small GR features such as light bending and propagation delay of electromagnetic signals. For example, in accelerator (cyclotron) experiments, even at accelerations of 10^19 g, no new effects arose other than the usual ones attributable purely to speed and gravitational potential.

>> [tvf]: My conclusion is that we really have no disagreements except over terminology, the critical one being the meaning of "the speed of gravity". Kopeikin's paper is all about the speed of changes in nearfield gravitational potential, about which there is no controversy. Any quick estimate shows that the signal propagation delays will not change significantly if Jupiter's gravity acts from its retarded instead of instantaneous position with respect to the quasar signal. And doing so would be a violation of GR anyway, because GR requires gravitating bodies to act from their instantaneous positions, not retarded ones.

> [sk]: OK. Everything is written in my equations. Their solution was not trivial. Everybody can try to repeat my calculations step by step. This way brings about clarification of everything - terminology, limits of integration, position of Jupiter, quasar, etc., etc. Problem of propagation of electromagnetic signals in time-dependent gravitational fields was not solved before publications of my papers. What people were able to do was only estimates. At present time when we have super-precise clocks, interferometry, etc. these estimates had to be replaced with precise theory. It was the goal of my research and I have done it. Now people can judge my work to the best they can reach.

Unfortunately, your equations do not deal at all with what is now called the "speed of gravity" issue, yet your words say that they do. It is true that you may be the first to propose a way to measure the speed of gravitational waves in the solar system, and it would be of some interest to put that result on experimental footing. But your words say that you will test the "speed of gravity", which is the subject of a current debate throughout the field and in some published papers about the speed of gravitational force, which neither your equations nor this experiment have any chance to test. Because of the way you worded your claims, Stephen Speicher and others on the Internet are now of the opinion that your experiment will test the "speed of gravitational force" issue, and that is a false impression.

Please help to clarify this situation by choosing words that accurately tell what your experiment will test, and what it will not test. Leaving the full responsibility for understanding your equations and what they really mean up to the reader is unfair because few readers have the time, experience, motivation, and perhaps even skills needed to delve that deeply into what you have done. So the words you choose to say what you have done are perhaps even more important than the equations because their effect reaches farther.

THIRD ROUND OF RESPONSES:

> [sk]: As soon as one knows the metric tensor one can calculate gravitational forces etc.

Yes, of course, the relationship has been agreed upon by convention. To be precise, the force is the instantaneous gradient of the retarded potential. Note, however, that relationship is not part of, and does not follow from, the Einstein equations. It is an add-on assumption in GR.

> [sk]: Gravitational potentials and gravitational forces in GR are not different concepts - the latter are derived from the former.

Forces and potentials are different concepts in anybody’s physics. SK must mean they are not independent concepts. In GR, the exact relationship is assumed to be the one I specified just above. And as I explained in my last message, properties of a function and its derivative need not be similar. Moreover, no experiment has established whether this relationship exists because potential causes force or because force causes potential. The geometric interpretation of GR assumes the former, and the field interpretation can be made to work either way. However, reasoning based on both causality and momentum conservation principles require that the arrow of causality be the latter -- force must cause potential. This point is unimportant to the math of GR, but vital to the physical interpretation of the GR equations.

> [sk]: If gravitational potential has speed of propagation c_g the same speed will have the gravitational force.

While I agree this is generally assumed (which is to say that people who have not thought the matter through tend to assume that the speed of gravitational force is probably the same as the speed of gravitational waves), nothing compels such a conclusion, and all experimental evidence goes against it. Obviously, the propagation speed of a force in particular or any cause in general need not be the same as the propagation speed of the potential in particular or any effect in general. (E.g., if an asteroid from space passes through our atmosphere and hits the ground, it sets off sonic waves. There is no particular relation between the speed of the asteroid and the speed of the sonic waves, even though the one caused the other. The force of impact and the sonic waves are good analogs of gravitational force and gravitational waves, respectively.)

> [sk]: It is possible to invent other theory where gravitational potentials and forces will be independent. This is not the case of GR - the only theory I use in my calculations. If other theory is implicitly discussed I need to see its mathematical structure – field equations, etc. Then that theory should be used to repeat my calculations.

No such “other theory” has been suggested by any party to this discussion. And there are certainly no alternate equations. (One could put in a light-speed propagation delay for gravitational force in GR, but then the equations would be wrong, so why do that?) We are discussing one single theory, GR, with one single set of equations, which has two quite different interpretations. This dual interpretation has been much discussed since Eddington’s 1920 book, most recently in Feynman Lectures on Gravitation, R.P. Feynman, Addison-Wesley, New York (1995). Section 8.4, p. 113: “It is one of the peculiar aspects of the theory of gravitation, that it has both a field interpretation and a geometrical interpretation. ... the fact is that a spin-two field has this geometrical interpretation: this is not something readily explainable -- it is just marvelous. The geometrical interpretation is not really necessary or essential to physics. It might be that the whole coincidence might be understood as representing some kind of invariance. It might be that the relationships between these two points of view about gravity might be transparent after we discuss a third point of view, which has to do with the general properties of field theories under transformations. This point of view will be developed more fully later -- we discuss it here so as to get a feeling for some directions which we might take in attempting to understand how gravity can be both geometry and a field.”

> [sk]: If this is not done there is no sense to discuss anything - GR is self-consistent theory and its interpretation is unique and unambiguous as well as the result of my calculation of the light deflection in the field of moving self-gravitating bodies and my interpretation of that effect.

Only the math of GR is unique and unambiguous. The interpretation (or equivalently, the physics) has no such clarity, which is why there is an issue about the speed of gravity. This question arises in GR because of its assumption that the gradient of the potential should be an instantaneous gradient, not a more natural retarded gradient. This choice is seldom questioned because observations demand that the instantaneous gradient is the only correct choice. But the physical meaning of this choice is very much open for discussion and testing.

I take the shortness of SK’s reply to mean that he is unfamiliar with this issue, and not especially interested in it. However, the matter Stephen Speicher and I wished to resolve is now fully resolved in my mind, and I hope in his too at this point. The coming quasar appulse may allow the first experimental measure of the speed of changes in the gravitational potential field, but will not provide an opportunity to measure the propagation delay of gravitational force. In fact, no aspect of the proposed experiment deals with anything affected by gravitational force. (Light-bending and signal propagation delay are related to potential strength only, not to force strength, as my spherical shell and cyclotron examples illustrated.) By my back-of-the-envelope calculation, the largest force propagation delay effect is a few orders of magnitude smaller, well below the threshold of detectability for this type of experiment. And as Jim Graber argued, GR already assumes zero propagation delay (instantaneous gradients) for gravitational force anyway, so all we could hope to show is that the assumption of retarded gradients is wrong, which is already known.

FOURTH ROUND OF RESPONSES:

[sk]: (1) The experiment is designed to check whether the fundamental constant c_g in the Einstein equations is equal to the fundamental constant c_l entering Maxwell equations. Any difference would give non-zero value of the parameter delta introduced in my paper.

[sk]: (2) My calculations have been done in the framework of General Relativity (GR). This theory deals with tensor potential - metric tensor. As soon as one knows the metric tensor one can calculate gravitational forces etc. Gravitational potentials and gravitational forces in GR are not different concepts - the latter are derived from the former. If gravitational potential has speed of propagation c_g the same speed will have the gravitational force. It is possible to invent other theory where gravitational potentials and forces will be independent. This is not the case of GR - the only theory I use in my calculations. If other theory is implicitly discussed I need to see its mathematical structure – field equations, etc. Then that theory should be used to repeat my calculations. If this is not done there is no sense to discuss anything - GR is self-consistent theory and its interpretation is unique and unambiguous as well as the result of my calculation of the light deflection in the field of moving self-gravitating bodies and my interpretation of that effect.

[tvf]: I just came across "The light cone effect on the Shapiro time delay" by H. Asada, Astroph.J.Letters for 7/20, v. 574, pp. L69-L70 (2002). He concludes "S. Kopeikin argued that the excess time delay was due to the propagation speed of gravity. The present Letter shows that the excess comes from nothing but the propagation delay of light, namely, the light cone effect."

It's always nice to see that others have independently arrived at a similar conclusion. Best wishes. -|Tom|-