On Apr 22, 8:10*pm, wrote:
Tom Van Flandern wrote:
[...]

and Steve Carlip writes:
[Tom VF]: no one is disputing that changes in gravitational potential
(the subject of the field equations) propagate at the speed of light, c.
I am always careful to state that "the speed of gravity" measured by the
six available experiments always means the 3-space propagation speed of
gravitational force, and has nothing to do with changes in gravitational
potential.
[Carlip, summarizing Low's paper]: change the source of gravity in R any
way you want. According to GR, nothing at all happens at a point p outside
R until the time for a light signal to reach p from R has passed. By any
sensible definition of "speed" I can imagine, that means that gravity
propagates at the speed of light.
I agree completely with Low's mathematical reasoning that you summarize
here, and have never claimed anything different from that.

If you really agree, then we are arguing over semantics. *But lets see....

Let R contain a single mass M moving at a constant velocity. *Let's suppose
it has been moving at this velocity for a long time -- much longer than the
light travel time to p. *Then both GR and Newtonian gravity agree that a
test mass at p will experience an acceleration toward the "instantaneous"
position of M. *In particular, the direction of that acceleration will track
the motion of M.

Now, at time t=0, make the following change in R: stop the motion of M.
You apparently agree that this change will have no affect at p until the
time for a light signal to reach p from R. *In particular, if you really believe
this, you now accept that the acceleration of a test mass at p will continue
to "track" the previous motion of M, even though M is no longer moving,
until a time t=d/c, where d is the distance to p.


I don't think he agrees, but anyway, I am coinfused with what he
agrees and what he doesn't.

I do not agree. The change will have an instantaneous effect at p.

What does light have to do with gravitational effects anyway? Can you
show the connection first?

Write down the equations.


If you really "agree completely with Low's mathematical reasoning," then
you accept this direct consequence of that reasoning. *But we can go
further. *Write down the exact solution of the Einstein field equations for
a mass M that initially moves at a constant velocity and then abruptly
stops. *(This is not too hard -- you can take the Kinnersley solution, which
I wrote down in my Phys. Lett. A paper, *gr-qc/9909087, for a mass with an
arbitrary motion, and put in this special case.) *Now just compute the
acceleration at p. *(Again, not too hard -- you can use equation (2.2) of
my paper for the Christoffel connection.) *You will find exactly the behavior
that I described above, as implied by Low's theorem.


Your oversimplifications of field equations for the purpose of
obtaining the solutions you have in mind a priori do not concern
anyone. If you want to solve a REAL WORLD problem forget about test
mass and point mass and solve the two-body problem. But you can't.

Are you now ready to accept this? *If so, I'll believe that you accept the
mathematics of GR. *If not, then you don't, period. *This is not a question
of an "interpretation" -- it is a direct, unambiguous mathematical
prediction. *If you don't accept it, fine. *But then stop hiding your beliefs
and pretending that you really accept GR, and are just arguing about an
interpretation of the math.

[...]

Now, consider the 2-body problem with one source mass and one (nearly)
massless target body. By construction, the source mass represents Low's R, *
a collection of smaller masses that are the source of a gravitational
potential field. We all agree that changes in the potential field propagate
or update at speed c. So there is no issue there. Now let's look at the
gravitational force generated outside R, assuming R is a single, fixed
mass - the simplest case. Even with nothing changing at the field source, we
still have a problem about the force applied to the target body.
* * The one and only mathematical question of importance here to the speed
of gravity issue is this: For a target body with a transverse motion
relative to the source mass, should we use the retarded gradient or the
instantaneous gradient to get the force?

There is no such thing as a "retarded gradient." *The gradient of a function
is the vector of its spatial derivatives. *Time doesn't come into it.


Yes, I agree, there are only retarded people.


It is also an elementary mathematical fact, of course, that if a function at
x at time t is determined by the behavior of some source at an earlier
time t', then the gradient of the function of x at time t is also determined
by the behavior of some source at time t'.

* * If this force, or "gravitational influences" (your term), propagates
from source mass to target body at speed c, then we must use the retarded
gradient, which leads to wrong answers (outward spiraling orbits).

You are managing to thoroughly confuse yourself about some fairly
elementary mathematics. *Apparently you find the use of potentials
-- which are just auxiliary functions used to simplify computations --
confusing. *You can, in fact, do all of the calculations without ever using
a potential. *For electromagnetism, for instance, you can directly solve
Maxwell's equtions for the electric and magnetic fields, without ever
using potentials; you again find that the fields are completely determined
by the retarded behavior of the sources (in this case, the current and
its first derivative). *For GR, things are harded, since the equations are
nonlinear, but you can again derive wave equations for the full curvature
tensor.

Steve Carlip

Mike