Wednesday, December 27, 2006

Another View of Relative Motion

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

Fred Vaughan headshot 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

Fred Vaughan
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., CMA, CMB, and CMC) as shown. About which of these points will the objects orbit?

Figure 1

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 p1 is to S as p2 is to S2', etc.. (Also S as well as p orbits S' in this case.) This relationship of pi being to S as pi+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 pi and pi+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?

Figures 2,3

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, January 20, 1906 by Jaume Giné on the internet1, 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/c2 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.

Figure 4

To update my own analyses as Jaume Giné did for Paul Gerber, the relationship would have to be made between pi being to S as pi+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.


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.


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