### Another View of Relative Motion

*
Rejecting Action at a Distance Resolves Precession of the
Perihelion of Mercury without Requiring General Relativity
*

by Fred Vaughan

*"…What would happen if the
Earth were suddenly dropped into place, at its proper distance
from the Sun? How would the Sun
'know' that the earth was
there? How would the Earth respond to the presence of the
Sun?…But the Sun would not
'know' that the Earth had
arrived until there had been time (Faraday had no way of guessing
how much time) for the Earth's gravitational
influence to travel across space… and reach the
Sun."*^{1}

*the author encounters the
gravity of the physical world*

Newton's action at a distance involving instantaneous transmittal of forces was problematic from the first. If one is to eliminate action at a distance from the classical equations acknowledging that it takes time for forces of interaction to be transmitted between objects but retaining Newton's other concepts of a centrally directed gravitational force, one is faced with a dilemma — the elliptical orbits will precess. But once this has been taken into account by the gravitational force, general relativity is not required because the precession implied by removing action at a distance accounts for observations without further emendation.

Much of the simplicity that Newton was able to incorporate into
his laws of nature might seem necessarily to have become
obfuscated if action-at-a-distance does not apply, however. We
encounter, for example, the situation of the
"central"force being directed, not
along the line of centers of two massive objects, but offset at
an angle in a direction to which there would seem to be no source
for a force. If the transmittal speed of the potential energy
that drives the force is equal to that of light in a vacuum, then
the angle of relativistic aberration determined by the relative
velocity of the objects would determine the
"line of sight"to where each
object would *appear*. This then would serve also as the
realized direction any associated force. This has the merit, of
course, of both objects experiencing the force to and from the
direction at which the other object *appears*, even if not
where it *is*. But the framework in which this is true is
*not* an inertial coordinate frame; it is an accelerating
frame of reference - at least with respect to
the location of the center of mass of the two objects as it is
usually accounted. Newton's laws of motion
had, of course, been accepted as applying exclusively with
respect to such inertial frames of reference.

Determining the center of mass of the two relatively moving
massive objects is problematic from the start. Without
action-at-a-distance, wherever there is relative motion there
will be alternative perspectives caused by the direction from
which the light (or force in this case) arrives, which will be in
the direction to which the object *had been* located when
the light (force) was emitted — *not* to
where it is located when it arrives. This "aberration" will
result in assessments of the center of mass being at variance.
Look at the system of two equally massive objects depicted in
figure 1. Since the minimum separation of their two straight line
trajectories as drawn from a third perspective of an observer C
half way between them (defined so as to provide symmetry) is
non-zero, aberration distorts their assignment of a center of
mass to the system (i. e., CM_{A}, CM_{B}, and
CM_{C}) as shown. About which of these points will the
objects orbit?

In figure 2 we have drawn the circular orbit of radius R of what
could be a planet p about a star S. Of course, the planet would
be moving at the astronomically high speed of 0.449 times the
speed of light, c to produce this much aberration. The darkened
circle S is drawn where we conceptually envision the star to
"be" and S' is
drawn where it would appear from the vantage of the planet in its
path about the star (assumed to be much more massive than the
planet so the center of mass of the system is approximately that
of the center of the star). The planet's
orbital speed was merely chosen in this case to accommodate the
planet completing one orbit in exactly 14 times the length of
time it takes the force field to travel from the star to the
planet so it could be easily visualized. (However, if the
realities of actual planets being unable to achieve viable orbits
at distances compatible with this speed is a drawback to the
reader in understanding what is at issue here, then assume the
integer 14,000, or 140,000, or an irrational number for that
matter; it really makes no difference to the point of this
article.) So if we evaluate the status of the situation 14 (or
140,000) consecutive times during the course of an orbit as
shown, we see that in a reference frame stationary with respect
to the center of mass, the orbit of the planet whether about the
*actual* star or about the *apparent* star is the same!
See panels a and b of figure 3. However, in the latter case
(panel b) the orbit is out of phase by one fourteenth (or one
hundred and forty thousandths) of the orbital period such that
p_{1} is to S as p_{2} is to
S_{2}', etc.. (Also S as well as p
orbits S' in this case.) This relationship of p_{i} being
to S as p_{i+1} is to S', etc. will be
the same no matter what the relationship of the speeds v and c as
long as the separation of p_{i} and p_{i+1} is R
v/c, but if c is not an integral multiple of v, there will be a
change in the phase shift from one orbit to the next that is
proportional to the residue.

Whether the *actual* star S (which is not seen from the
planet) or the *apparent* star S' (that
*is* seen) is envisioned as orbiting *also* (at the
radius R v/c) about the center of the planet's
orbit may seem of little import since the
planet's path will be the same in either case.
Of course, there are major epistemological differences in these
perspectives. One tends to care little whether mere ephemeral
conceptual constructs gyrate in strange ways to accommodate
mathematical models, but for an *actual* star to orbit a
void point other than the center of mass of the system from the
perspective of comoving surrounding systems might seem to involve
some travesty of thought. It is, after all, only in the
accelerated frame of the ephemera that the unseen but
"actual"star orbits. However, it
is only in *that* frame of reference that our familiar
concept of an inverse square law force
"actually"applies. So, after all
these mental gyrations, where are we? Maybe this just seems to
work because we are dealing with a circular orbit here rather
than the more general conic elliptical orbits. How would this
alternative to action-at-a-distance play out with an elliptical
orbit?

The answer is that the elliptical orbit (as seen from the
comoving surrounds of the star) would involve the precession of
an otherwise stationary closed elliptical orbit, whereas the
elliptical orbit solution about the appearance of the aberrant
star for which the central force equations apply would not
precess. And this brings us to the famous test of
Einstein's general theory of relativity with
regard to the precession of the perihelion of Mercury by 41
seconds of arc in a century, which is unaccounted for in
Newtonian mechanics. We are repeatedly reminded that the
phenomenon is finally resolved by general relativity with the
determination having been made by Schwartzchild of the
appropriate gravitational metric tensor from which to determine
the Ricci tensor, Ricci scalar, and stress-energy tensor from
which the result can supposedly be computed. Needless to say,
"That ain't necessarily easy
for neophytes!"It is much easier, in fact, to merely
acknowledge that it takes time for forces of nature to
travelthrough space and determine the relative locations of the
interacting bodies at appropriate times as shown above.
*That* makes sense!

In looking up the data on the precession of
Mercury's perihelion to check out the
viability of all this, I ran across a very learned article
published in arXiv.org:physics/0510086, January 20, 1906 by Jaume
Giné on the internet^{1}, which
exhaustively describes the history of such efforts as mine and
extols the efforts of a German school teacher named Paul Gerber
who in 1898 proposed just such an approach. It resulted in
accounting for only 14 seconds of arc and not the entire 41.
However in analyzing those results Giné was able to show
that using the round trip time instead of the one way force
interaction time as suggested by the collaboration of Wheeler and
Feynman on absorber theory in 1945, that the entire phenomenon is
thereby completely accounted. See figure 4 taken from the
reference.

In the diagram of figure 4 the retardation parameter
*τ* is equal to what we would have referred to
as R v/c^{2} above. And the fact that the two pie-shaped
segments differ in the second panel in figuring the round trip
delay is that the orbit is not assumed to be circular as we did
for didactic purposes in figure 1. In
Giné's article he does not associate
retarded potentials directly with the special relativistic
aberration phenomena, but this association is inevitable.

To update my own analyses as Jaume Giné did for Paul Gerber,
the relationship would have to be made between p_{i}
being to S as p_{i+2} is to S', etc..
What this says in interaction terms is that the force
"signal"sent out by S is in
response to a complementary force signal received from p, so that
the force received by p from S will be in response to where p was
located two intervals back. It should be obvious that this makes
sense once action-at-a-distance is done away with.

So, although I feel somewhat "scooped" (albeit narrowly by only one hundred and eight years), it is refreshing to know that, however rare, perspicacity has always lurked around every corner. And there is some, however, vestigial memory of the right answers to real problems.

## Footnotes

1 John Gribbon, ** The
Scientist**, Random House, New York, 2002, p. 423. In
reference to a presentation made by Michael Faraday to the
Royal Institution on 19 January 1844.

1 http://www.citebase.org/cgi-bin/fulltext?format=application/pdf&identifier=oai:arXiv.org:physics/0510086

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