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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|-
|
