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 p₁ is to S as p₂ is to S₂', 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 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.

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.

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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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Skepticism with Regard to an Astrophysical Trend

by Fred Vaughan

"If our friendship depends on space and time, then when we finally overcome space and time, we've destroyed our own brotherhood." - Jonathan Livingston Seagull¹

Once again a friend has driven me to abstraction - a recurring situation for which I am repeatedly in his debt. (I guess that is a major criterion for entry into my inner circle of friends.) His articles place philosophical issues in a context that I find not only titillating, but damned unnerving at times as well! My search for irreversibility in microscopic interactions that have always been considered completely reversible by virtually everyone was spurred on as an objection to his counter claims. I am very grateful for the opportunity to have sought (and I flatter myself in believing that I actually found!) the source of irreversibility in the usually negligible Doppler shifted energy losses of the photon exchanges by which collisions between molecular constituents of substances are effected. The many hundreds of enjoyable hours of investigation into irreversibility were in direct response to the stimulating discussion of "the astrophysical trend."² I must say that I truly believed I had handled the "hardest" problem he had posed in his article. Thus, I considered the total scope of my efforts to have been an adequate disputation (if not total refutation) of his notion of the inevitability of dire long term "trends" in our universe's behavior as a whole. These apparent trends included, of course, that the universe must necessarily be winding down, and that photons generated in the heat of interaction while our universe is still interesting are being more or less "sucked out" into a vast chill in which he evidently conceives the universe to be immersed. If he and most modern cosmologists are correct, the universe will, as Robert Frost opined as alternative, "end in ice" - only colder with individual atoms continuing to separate themselves endlessly. Although there is still a heated contingent that "favors fire!" I argue instead for a "cosmocentric" equilibrium that avoids both these drastic extremes.³

It is fashionable in physics nowadays to consider the Big Bang as the origin of everything otherwise unaccountable in physics and in this regard the Astrophysical Trend was perhaps a trendsetter. Preoccupation with deducing from a presumed origin of the universe what formerly would have been determined by more inductive means only after extensive experimentation and observation is a bit presumptuous if you ask me. This backwards perspective (what I consider to be looking down the wrong end of telescopes) has given rise to extreme extravagance in physics: Searches for heavy particles whose failures only precipitate proponents insisting that the particles do in fact exist but they must involve higher energies and if this superstring, brane, or mini black hole cannot be detected by current instruments then it must be because it is even larger than we suspected - excuses ad infinitum. This disrespect for accurate predictions and the results of experimentation is rampant in physics today. But my friend went even further to suggest an accomplice to the Big Bang: "The Big Bang would thus provide the 'push' while the cold nonreflecting space would serve as the 'pull,' for expansion-contraction processes..."⁴"

There are other places where this allusion to a "pull" of cold outer space is employed. This fiction results from a theoretical model refuted by facts of a hot plasma known to facilitate our view of the universe out to 10 or 15 billion light years...so far. Intergalactic space is not cold or the low levels of dispersed hydrogen and helium would have absorbed the light by which we witness the cosmos. Conservative estimates of its temperature are between 10⁴ and 10⁶ K, but to effect such complete stripping of electrons it may well have to be 10⁸ K. In fact it is as hot as the interiors of stars we observe - just much less dense. Virtually all electrons must indeed be "stripped" to accommodate the transparency of our view notwithstanding islands of Lyman α forests where protogalaxies form! Certainly this data was not available in the sixties when this "astrophysical trend" was introduced.⁵ To be fair, the current view is that there was a time after ambient temperatures from the Big Bang cooled to about 10⁴ K (very similar to Luger's indication of the temperature for electron capture), and after stars formed, the intergalactic medium was reheated by the resulting radiation. I would argue this but it is unnecessary in this context. The lesser argument stands - where is this cold sink for radiation? But it is not clear in what sense Luger sees cold outer space as contributing to any local physical process.

There is also, as he points out, the need for links to explain why entropic phenomena apply at the local level on a time-scale for which any evolution of the universe is irrelevant. This is the problem, not just an irrelevant corollary of the problem! The other way around is physics on its head.

Recently I found that my friend had not been convinced in the slightest by my arguments nor apparently had my efforts to establish a sound basis for irreversibility given him pause to even reconsider his position or attempt refutation of my hypothesis. In a recent e-mail he opined that:⁶ "The evidence for arrows is so overwhelming that I don't know where to begin, and there's little point in boring you with elaborate lists. A simple intuitive example that springs to mind is the surface temperatures of the Sun, which range from an inner one over a million degrees, whereas the outermost 'layer' is maybe a few thousand degrees."

The facts associated with there being vast variations in temperature and density throughout the multifarious domains of our universe were of course not news to me. It was only when he then succinctly asked, "How could such a steep gradient be possible without cold outer space?" that I realized what was at issue between us - the scope of the philosophical dilemma with which we wrestled. (The age-old problems of philosophy will never go away; conjectures that attempt to solve them will only cause these truly meaningful problems to be reformulated with successively more relevance accruing as time goes by, but forever nagging at our heels nonetheless.) With a renewed understanding of the nature of the gulf between us, i. e., the consequences of irreversibility originating at the bottom or the top, I decided to begin again with renewed vigor to attack the horns of the Parmenidean dilemma - Heraclitus's river that is always the same and always different. So I now proceed with my current understanding of how irreversible changes in variations of characteristic aspects may persist even in a continuously stable universe that never collapses, does not expand indefinitely, nor run down as a grandfather clock in need of some grandfatherly figure to rewind it.

Certainly to accurately assess whether a trend exists one must sample suspected behavior over time and space with samples that can be justified as representative of phenomena for which the trend is presumed to apply. To this end one must have a valid model of the behavior of the system being sampled. This is exemplified by the shortcomings in the perspectives of two blind men who argue the nature of elephants from their own happenstance-tactile-limited experiences with a hind leg, trunk, or tusk. So in a real sense one should have a working global knowledge of what is being sampled before averring too sanctimoniously to have comprehended its inherent nature, let alone, its "trend" into the undefinable future. Of course when the system under test is the entire universe one can run into unique modeling problems. Hawking, in defining what I have referred to elsewhere as a "Hawking sphere" and then claiming that it would appropriately represent the gravitational characteristics of an entire "infinite" Newtonian universe, erred by misrepresenting such a universe as having an inside and an outside which is patently absurd. The terms "universe" and "infinite" by definition preclude the void "exterior" from which collapse derives in Hawking's derivation. This error precipitates many erroneous notions including universes beyond the realm of our universe as though it were a mere galaxy, and other conceptions suggesting the hoarse shallow depth in the rattle of a dying man's last breath. Einstein had, of course, as Hawking knew, proceeded from just such assumptions:⁷

"As I have shown in the previous paper, the general theory of relativity requires that the universe be spatially finite. But this view of the universe necessitated an extension of equations with the introduction of a new universal constant λ, standing in a fixed relation to the total mass of the universe (or, respectively, to the equilibrium density of matter). This is gravely detrimental to the formal beauty of the theory."

I personally think it ludicrous to presume that one's methodologies and theoretical model might appropriately dictate requirements on the actual universe that one is attempting to model as suggested in Einstein's remark. This is a much more major error than what Einstein considered to have been his "greatest error" in the above quotation. We must limit our models to valid descriptions of actual phenomena from which to extract explanations and accept them only to the extent that they are valid descriptions, if we would have the entire universe acquiesce to such pronouncements. One easily falls prey to gibberish otherwise.

In Heraclitus's analogy of the river one must model much more than the solid banks of a river and the fluid that flows between if one is to resolve the paradox of identity in flux. It takes more than the addition of mountains, foothills, and valleys through which tributaries flow into the river, and more than models of the occurrence of seasonal rain and snow if one is not to eventually have it run dry or fill the seas to overflowing. One must complete the loop in any valid model if one is ever to have a chance to understand an equilibrium situation. Without completing such logical loops, equilibrium will always be seen as an impossibility. I see it as no different with the astrophysical trend to which Dr. Luger defers. Certainly gradients and change are essential to our nontrivial world, but that does not preclude cosmological stability. Certainly there are gradients of temperature in the universe just as there are gradients associated with the flow of rivers, but that does not in itself suggest either that rivers will all one day run dry nor that the oceans will overflow in a material manifestation of Olbers paradox. There is more subtlety in heaven and earth than that! Olbers had not the slightest conception of magnitudes involved in either the separations in space nor the lifetimes of stars or he would not have conjectured as he did, and others would not have wasted so much time on this supposed paradox.⁸

It is no surprise that open loop models run dry. In resolving irreversibility at the microscopic level it is necessary to extend Einstein's blackbody radiation model to close that loop. This model then had to be extended to incorporate the complimentary mechanical aspects of the system as well to show that although any and every individual process dissipates energy, that energy goes somewhere and energy from elsewhere can keep the system going - yes, even indefinitely.⁹So it is definitely conceivable for there to be gradients of change in perpetuated systems, but it can not, as Dr. Luger would quickly counter, be assumed as a foregone conclusion! This is particularly so when identified processes such as a "big bang" and "black holes" are claimed with some credibility to be, respectively, an exhausted one-time resource and irreversible sinks of the energetics that drive the whole system. But the presumed characteristics of these two processes as currently modeled are too obviously contradictory to allow such presumptions to limit discussion. According to virtually any version of the standard model of cosmology, we are either just within, or have just recently escaped from, the Schwartzchild radius of the "black hole" of our own universe. So either black holes are not singularities into which matter is sucked endlessly to a mathematical point never to escape as established theory predicts or the "big bang" never happened. Take your pick. A healthy "conceptual skepticism in irreversible energetics" would not allow one to rashly embrace both such conflicting models after a hardy breakfast.

One must incorporate a valid model of every known process consistent with assumptions of all others within the system before one has a valid model of the system itself from which to declare "alpha and omega!" Glibly imposing requirements on a universe that happens quite defiantly to exist without regard for our conjectures is totally absurd! This is particularly the case if we are to avoid naïve presumptions of limited open loop models of the entire universe! We can reach no valid conclusions without completeness, and this does not bode well for increasingly popular "theories of everything" (TOEs) that are being hawked by Hawking and his lessors recently. With outstanding questions of such magnitude concerning the nature of the primary processes of our universe, we must emphasize observation. The overwhelming scope of our ignorance should certainly humble theorists. It seems to me, however, rather to have emboldened those who should know better to greater and greater levels of pugnacity. So I will not attempt to stick out my own big TOE to be stepped on here. I will rather content myself with labeling as presumption suggestions that currently observed "trends" imply that the metaphorical river is ineluctably running dry.

1 Richard Bach, Jonathan Livingston Seagull, Avon, New York (1970), p. 87.

2 Frank Luger, "Conceptual Skepticism in Irreversible Energetics," Gift of Fire, #119, October 2000, pp. 10-24.

3 This comment is, of course, an ironic reference to Dr. Luger's subsequent article that addressed the same trend, i. e., "Anthropocentrism vs. Cosmocentrism - groping toward a paradigm shift," Gift of Fire, #120, November 2000. Pp. 24-33.

4 Op. cit. Frank Luger, p. 22.

5 [See for example, J. V. Narlikar, Proc. R. Soc. London A270, 553 (1962).]

6 Private e-mail communication from Frank Luger to Russell F. Vaughan dated Sun, 17 Nov 2002.

7 A. Einstein, "Do gravitational fields play an essential part in the structure of the elementary particles of matter," repuplished in The Principle of Relativity - a collection of original papers on the special and general theory of relativity, Dover, New York (1952) p. 193.

8 We have that 10¹⁰ years is a reasonably long lifetime for a star, and 10²³ light years is the average distance of a line of sight to encounter a stellar object with the densities of stars encountered in our universe. Thus if we represent "sky cover ratio" X as defined by average night time intensity along a line of sight divided by what it would be if directed directly at the sun, we obtain the likelihood of a line of sight encountering a bright star as 10¹⁰/10²³ = 10^-13. This is only an estimate good to within a factor of a few thousand. So that, 10^-16 < X < 10^-10. In contrast, at mid day the sun provides a total sky cover ratio of about 3x10^-4 so that it is predicted to be about a billion times darker at night even if the universe were infinite. And one would not require sun glasses to enjoy the splendors of the night sky! We are just situated in a "hot spot" in the universe.

9 If we reverse all velocities involved in an interaction and try it again, the situation does not reverse. We've lost energy in the form of escaping radiation and in doing it again [in reverse], we'll lose some more. This escaping radiation may be absorbed within the boundaries of the defined 'system,' but unlike atomic matter that can be confined, it may escape into or beyond boundaries of any jar or laboratory. Unless the amount of radiation from outside the boundary makes up the deficit there will be a deficit.

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