[I ignore the embroidery and only discuss Van Flandern's primary confusion.]

Tom Van Flandern wrote:
We then go on to explain why
physics has an issue that math does not, with these words:
“Note that the gradient of a scalar field is a vector, not another
scalar. But if the field source begins to move, does the field gradient
point toward the instantaneous or retarded position of the source? That
depends on whether the field updates or regenerates instantly or with
delay. So when we say that the gravitational acceleration of a test body
follows the field gradient, we must ask which gradient it will follow
-- instantaneous or retarded. Physics has an issue that math does not.
Retarded potentials in math allow for delays only in the mass
distribution and in changes of distance between masses in a scalar
field. Retarded potentials in physics must allow also for delays in the
vector direction of the field – normally the dominant effect of
retardation.”

Why is it that whenever we switch from talking math (equations) to
talking physics (concepts, properties, principles), we seem to have so
much difficulty communicating? -|Tom|-

[For simplicity, I'll discuss the Lenard-Wiechert
potential of classical electrodynamics (ignoring
magnetism). I'll relate this to GR below.]

"We" don't have "so much difficulty", only YOU do, probably because your
description above is just plain wrong. Your question "does the field
gradient point toward the instantaneous or retarded position of the
source?" is SUPERFLUOUS AND MAKES AN INVALID ASSUMPTION -- the math is
unambiguous, and the answer is: NEITHER, the field gradient points to
the retarded source position EXTRAPOLATED LINEARLY to the time at which
the gradient is evaluated.

The correct way to use the L-W potential for a single point source, and
to compute the E field from its gradient, is as follows:

To determine the E field at spatial point P and time T (both specified
in a given inertial frame which is used throughout) one needs to know
the potential at every point within a spatial neighborhood of P at time
T. One does that using the L-W formula, noting that the retardation is
different for each point in the neighborhood of P, so that the source
position is evaluated at slightly different times for each point in the
neighborhood (one can relate this difference to the velocity of the
source at the retarded time). Now one takes the gradient of the
potential at P. Note there is no ambiguity whatsoever in taking that
gradient, because the potential was evaluated at time T for every point
within the spatial neighborhood of P.

Let me repeat that: there is no ambiguity whatsoever in taking the
gradient at point P and time T, because the potential was evaluated at
time T for every point within the spatial neighborhood of P. In
particular, properties of the source are evaluated only in an
infinitesimal neighborhood of the retarded time T-R(T)/c.

[The spatial neighborhood of P is 3-dimensional; the
neighborhood of the retarded time is 1-d; for each
point in the latter there is a 2-d locus in the former.]

When one does this, one finds that for a moving source (i.e. moving
relative to the inertial frame used above), E does not point at the
location of the source at the retarded time. When one linearly
extrapolates the position and velocity of the source at the retarded
time in the obvious way to time T, one finds that E points at this
latter point.

I'm saying nothing new, and this is explained in every
graduate-level textbook on classical E&M.


In the weak-field linear approximation to GR the situation is more
complicated, but basically the same. The gravitational force vector at
point P and time T points to the QUADRATICALLY EXTRAPOLATED position of
the source at time T. That is, the position, velocity, and acceleration
of the source are involved in an infinitesimal neighborhood of the
retarded time.

In GR itself, without approximation, there is no general
Green's-function method to solve the field equation, because it is
nonlinear. There can be no scalar "gravitational potential" because in
GR gravitation has more degrees of freedom and a scalar field is
inadequate to represent it. In some sense the metric can be considered
to be a "generalized gravitational potential", and the analogy to the
gradient is the Christoffel symbols -- this has the necessary property
that the "generalized gravitational force" is zero when evaluated in
locally-inertial coordinates.



This has been explained to you numerous times. I don't know why it is so
hard for you to understand this. Apparently you don't actually have the
requisite background: this E&M is very basic physics taught to every
graduate student of physics (ditto for the GR, but not every graduate
student takes that course).

[Your problem seems to be that you cannot imagine that the
gradient does not point at the source. It doesn't -- this
is NOT a central force, and your notion that retardation
screws up planetary orbits is just plain wrong. Indeed,
the quadratic extrapolation is PRECISELY what is needed
to make such orbits almost but not quite stable (e.g. the
precession of perihelia).]


Tom Roberts