Non-Laplacian Interactions

 

According to Newton’s law of gravitation, two material particles exert a mutual force of attraction on each other such that, at any given instant, each particle is subject to a force pointing directly toward the position of the other object at that same instant. Subsequently several physicists, notably Laplace, examined the possibility that the force of gravity propagates with a finite speed U, such that the force at the time t acting on a particle at a distance D from the source is directed toward the place where the source was located at the time t – D/U. This is illustrated in the figure below, which shows two identical massive objects revolving around each other with orbital speed v in a circular path of diameter D.

 

 

The aberration angle between the direction of the force F and the true instantaneous line between the two particles is approximately to v/U, such that the force tends to accelerate the particle slightly forward rather than purely toward the center of the circle. Laplace showed that any such “non central” acceleration would lead to observable instabilities of the planetary orbits unless U was at least millions of times the speed of light. For many years this analysis of Laplacian aberration was thought to imply that the force of gravity must propagate either instantaneously (as in Newton’s formulation) or else far faster than light.

 

However, on closer examination, it is clear that Laplace’s analysis is applicable only to one particular kind of interaction, one that is rather implausible (if not incoherent) for other reasons as well. (Historically, Laplace was actually prompted to conduct his analysis in response to the “shadow theory” advocated by Lesage.) We should first note that the situation described above, viewed as elementary, entails an explicit violation of Newton’s principle of action and re-action, and hence violates the conservation of momentum. The forces on the two particles are not equal and opposite, so there is a net torque on the system, which of course accounts for the dynamic instability.

 

Laplace has actually posited two separate interactions, one between particle A at time t and particle B at time t – D/U, and another between particle B at time t and particle A at time t – D/U. Each of these two interactions is totally asymmetrical, like one hand clapping, in the sense that (for example) particle A at time t is acted upon by particle B at time t – D/U, but particle B at time t – D/U is not acted up at all by particle A at time t. Thus the strict reciprocity of the forces between particles envisaged by Newton is not upheld (assuming we regard these interactions as elementary). Another way of expressing this is to say that Newton’s laws are temporally symmetrical, i.e., the time-reversed version of events satisfies Newton’s laws just as well as does the forward-time version. This accounts for the conservation laws implicit in Newton’s theory. In contrast, the interaction evaluated by Laplace is obviously not temporally symmetrical.

 

It’s easy to conceive of a temporally symmetrical generalization of Newton’s law of gravity such that particles separated by a distance D interact with each other only if separated by a time D/U, as illustrated in the figure below.

 

 

We have depicted just the force interactions involving particle A at time t. As in the Laplacian model, particle A at time t is subject to a force directed toward the past location of B, but now we posit an equal and opposite force on particle B. Likewise, particle B at the time t + D/U is subject to a force directed toward the current location of A, but we posit an equal and opposite force on particle A. This generalization is arguably more in accord with Newton’s original conception, since each interaction between particles is mutual, preserving perfect action and re-action, and the law is also temporally symmetrical. Not surprisingly, we also find that the net force on particle A, consisting of forces directed toward the “advanced” and “retarded” positions of B, is directed straight towards the current position of B.

 

Lest anyone think this kind of interaction is impossible to achieve by any realistic physical process, notice that precisely this kind of “advanced and retarded” interaction is easily achievable is quite ordinary circumstances. To facilitate the description in terms of ordinary impulse exchanges, consider the force of repulsion between two people sitting at opposite points on the perimeter of a rotating circular disk, bouncing a ball back and forth between themselves. The ball arrives at person A from the direction of the past location of person B, so it imparts an impulse directed along that line. Person A then reflects the ball toward the future location of person B, so there is a recoil directed along that line. The net impulse on A from this reflection event is an impulse directed toward the position of B at that same instant. Thus the absence of aberration in the force between two bodies does not imply that the force is propagated instantaneously. Of course, this kind of interaction can exist only if the interacting bodies are moving in a sufficiently predictable way, so that the reflection can be performed without the need of information (such as visual cues) propagating faster than the impulse.

 

It’s interesting to examine more closely the two kinds of mechanisms discussed above, one leading to aberration of v/U, and the other leading to essentially no aberration. In the first case, consider two spherical bodies, labeled A and B, each of which is constantly emitting streams of tiny particles uniformly in all directions. This emission process is purely isotropic, so there is no net recoil on the emitting bodies. However, B will be struck by particles coming from the (past) direction of A, so this absorption is a highly directed process, and results in a net impulse on B along the line between the present position of B and the past position of A, as depicted below.

 

 

According to this model, each body such as A is constantly emanating impulse-carrying entities in all directions, but only a very small fraction of those (at least in the immediate vicinity) are intercepted by another body such as B, which thereby is subjected to a directed force. Of course, the reciprocal process is also taking place. But even if we assume the impulse-carrying entities striking B from A are absorbed by B, and vice versa, the vast majority of the emitted entities are simply dissipated into space. Since these entities are required to possess momentum and energy in order to convey impulse, each body must be expending its presumably finite store of these entities – unless we hypothesize continual creation of them from nothing. Recall that if a body is actually emitting some kind of conventional radiation, it does indeed become depleted, i.e., it’s mass-energy is diminished by the amount of the radiation. In contrast, no such depletion is apparent for bodies exerting a force on other bodies. Therefore, this type of model is inherently implausible as a mechanism for the production of stationary forces.

 

Another, more subtle, objection to this kind of Laplacian mechanism is that, assuming the impulse-carrying entities are discrete and emitted with a finite angular and temporal density, the force would become progressively less uniform at great distances, because the number of entities per unit area would be diminished as the radial distance increases.

 

A variation on the Laplacian type of interaction – and in fact the model that prompted Laplace to investigate such interactions – is the “shadow” model, according to which the universe is filled with a flux of tiny corpuscles moving at high speed in all directions. This is essentially just the “negative image” of the model described above, yielding a representation of a force of attraction. However, the hypothesis of an omni-directional flux of superluminal ultra-mundane corpuscles (as Lesage called them) raises more difficulties than it resolves, and even on its own terms it doesn’t eliminate (barring an even more problematical infinite regress) the necessity of postulating elementary forces of attraction to maintain the extended structures that serve as the elements of the Lesage mechanism.

 

The second type of model we described involves a directional process both for the emissions and the receptions. In this type of model, an impulse-carrying entity is emitted from A to B, where it is reflected back to A, and so on, as depicted in the figure below.

 

 

The bodies are not continually depleted in this model because the mechanism doesn’t involve discarding vast numbers of impulse-carrying entities. Hence the exertion of a stationary force between two distant bodies is a perfectly conservative process, as long as the motions of those bodies are sufficiently predictable. If and when one or both of the bodies departs from their predictable paths, we would expect the emission of radiation, which is qualitatively consistent with the actual observed phenomenon of radiation and aberration for accelerating bodies.

 

Of course this mechanism requires the impulse-carrying entities to be reflected in precisely the right directions so that they continue to bounce back and forth between the two moving bodies (assuming those bodies are moving with sufficient uniformity). A typical case of two uniformly moving bodies with an on-going force of repulsion between them is illustrated below.

 

 

At the instant depicted in this figure, there is a body at B and a body at C. The body at B is being struck by a particle from the position of the other body when it was at location A, and this particle is reflected at B and sent in the direction of D, where it will strike the other particle and be reflected towards E, and so on. Notice that the net impulse on the body at B, comprised of the incoming impulse of the particle arriving from A and the recoil due to the re-emission of that particle in the direction of D, points directly toward the other body at C, i.e., the position of the other body at the same instant, despite the fact that the impulse is being conveyed by particles with finite speed.

 

One can certainly question whether such a process could form the basis of a realistic mechanism for fundamental forces such as electromagnetism and gravity (see below for a discussion of some of the difficulties), but at the very least this simple process proves that Laplace’s method for inferring propagation speed from aberration angle does not apply to general force mechanisms, even if we limit ourselves to simple mechanical models based on impulse-carrying entities. We have here a process by which two bodies exert a mutual force on each other by means of tiny particles traversing the distance between them at an arbitrarily slow speed, and yet there is essentially no aberration in the direction of the force.

 

As noted above, this mechanism relies on suitably directed reflections so that the impulse-carrying particles are repeatedly intercepted by the interacting bodies. This doesn’t violate the conservation of energy or momentum, but it might seem to require some active “intelligence” to suitably guide the particles, somewhat similar to the functioning of Maxwell’s daemon, a hypothetical gatekeeper at an aperture in the wall separating two containers of gas. The daemon examines the speeds of the individual gas particles as they approach the aperture, and either reflects them or allows them to pass through, depending on their speeds, thereby enabling violations of the second law of thermodynamics. By allowing only the high-speed particles to pass in one direction, the temperature on one side of the partition can spontaneously increase while the temperature on the other side decreases. Such gate-keeping is not actually feasible at the fundamental quantum level because, on that level, the physical process by which the daemon “observes” the approaching particles interferes significantly with the outcome, such that no gross violations of the second law can occur.

 

It might seem as if the same sort of reasoning would rule out the possibility of a purely spontaneous and fundamental process of particle exchanges between moving bodies. However, the coordination required for these reflections is not based on the speed of the particles, but only on the direction, and it leads to no apparent violations of the second law. Furthermore, in the relativistic context, the lightlike exchanges take place along null intervals, so there is really no need for a choice of direction, as can be gathered from the spacetime diagram below, where the cone represents the lightcone of event C.

 

 

According to the advanced and retarded action model, the two exchanges involving the particle P1 at the event C are the reception of a lightlike impulse emitted from particle P2 at the past event A, and the recoil from the transmission of an impulse absorbed by P2 at the future event B. Notice that particle P1 at event C is null-separated from particle P2 at two (and only two) distinct events, namely A and B. Hence there is no ambiguity as to endpoints of these exchanges. Now, if particle P2 moves uniformly from A to B, the net effect of the two interactions involving C will be a force pointing directly toward the current location of P2 (in between A and B). On the other hand, if P2 accelerates significantly during the time between A and B, the force on C will not (in general) point directly toward the current position of P2, although even in this case the aberration will be much less than Laplace’s analysis would predict for the propagation speed of c.

 

The above model is similar to the “retarded potential” approach to electrodynamics arising from the “distance action” tradition of Ampere, and developed by Weber, Leinard, Weichert, and others. In contrast, a “field” approach was developed by Faraday, Maxwell, and others. The field theory gives yet another model of force propagation that does not exhibit Laplacian aberration. Note that models based on retarded (and advanced) distant action can be regarded as “local” (i.e., every effect is propagated contiguously between null-separated events) only when viewed in the context of Minkowski’s non-positive-definite metric of spacetime, in which lightlike intervals have zero magnitude. Prior to the advent of special relativity the possibility of this causal topology was never imagined, and so those who insisted on local contiguous propagation of action considered it necessary to hypothesize a mediating field capable of supporting stresses and strains. According to this conception, the field itself embodies both energy and momentum. Each charged particle simply responds to the field in its immediate vicinity – and it also contributes to the field to which other particles respond. (For slowly accelerating particles the self-field effects can be neglected, although in the general case the self-field effects lead to difficulties that have never been fully resolved in classical electrodynamics, as discussed in the note on accelerating charges. These difficulties have occasionally prompted re-consideration of the retarded-advanced action model.)

 

It’s important to note that the energy and momentum conveyed by the field is not limited to radiation. Whenever charges are in motion (even uniform motion), the electric and magnetic fields embody energy and momentum. With electric and magnetic fields e and b, the energy density is (|e|2 + |b|2)/(8p) and the momentum density is (e x b)/(4pc). Integrating these expressions over any finite volume of “empty space” gives the energy content and the momentum of the electromagnetic field contained in that volume. Needless to say, this is a very non-Laplacian model of force propagation between charged particles, because the particles are not the only entities possessing momentum. The overall flux of momentum is spread out over the entire region of space surrounding the charges, and momentum flows in many different directions at different points. This is true even for uniformly moving charges, which do not emit any “radiation”. Merely by virtue of the motions of those charges, the electric and magnetic fields re-configure, and this re-configuration entails a re-arrangement of the energy of the field. This movement of energy from one place to another corresponds to a flow of momentum.

 

Now, any changes in the fields are explicitly propagated at the speed c, in accord with the time-dependent field laws, regardless of whether these changes are “radiation” or simply secular re-configurations of the fields. Thus there is only one meaning that can be assigned to the term “speed of propagation” of a force. For example, given two initially stationary charged particles separated by a distance D, if one of the particles is suddenly displaced, the force to which the other particle is subjected will not change until a time D/c later, where c is the speed of propagation of the force. This is certainly the case for the Maxwell-Lorentz force of electromagnetism, and yet, this force (for uniformly moving charges) does not exhibit any aberration – nor would we expect such aberration, considering that the flow of momentum in this model is utterly dissimilar to the kind of mechanism that Laplace had in mind.

 

In fact, as both Lorentz and Poincare showed around 1904, any fundamental force that is consistent with observed relativistic phenomena, including the Michelson-Morley experiment and the Trouton-Noble experiment, must propagate at the speed of light and must exhibit no aberration for uniform motion. Lorentz’s molecular force hypothesis, which accounted for the Fitzgerald contraction of all objects in the direction of motion (thereby allowing Lorentz to achieve consistency with the Michelson-Morley experiment), is nothing but a consequence of the so-called Heaviside ellipsoids of the Leinard-Weichert retarded potential for the electromagnetic force. As Lorentz explained in his “Theory of Electrons” in 1909

 

It is especially important to observe that the values of [the charge density] existing at a certain point Q at the time t – r/c do not make themselves felt at the point P [located at a distance r from Q] at the same moment t – r/c, but at the later time t. We may therefore really speak of a propagation taking place with the velocity c.

 

The lightspeed delay of the force is responsible for contraction of the spherical potential into an ellipsoid. Furthermore, Lorentz showed that all forces, including the nuclear forces holding the constituents of matter together, must also propagate at the speed c, because otherwise the theorem of corresponding states (by which he accounted for the experimental results) would not hold good.

 

One occasionally sees claims (in “fringe” publications) that Lorentz’s interpretation of relativity is consistent with superluminal propagation of gravity, electromagnetism, and the nuclear forces. Such claims are based on a fundamental misconception. As explained above, Lorentz’s theory was firmly based on the lightspeed propagation of all forces having any appreciable effect on the structure of matter, including the shape of the Earth, because if any of those forces (e.g., gravity) propagated at a significantly different speed, experiments such as the one by Michelson and Morley would yield markedly different results. It is true that Lorentz argued in favor of an interpretation in which the concepts of absolute rest and absolute time were retained, but he acknowledged that this was merely a philosophical preference, having no empirical significance.

 

The only sense in which Lorentz’s interpretation is more congenial than Einstein’s to superluminal action is that, in Lorentz’s interpretation, the relativity of the known forces of nature (i.e., electromagnetism, gravity, and the strong and weak nuclear forces) is just due to a conspiracy of coincidences, and if a new, previously unknown, physical effect were discovered tomorrow, it might conceivably violate local Lorentz covariance, making it non-relativistic. If such a effect were discovered, it could be introduced into the Lorentzian conceptual framework, making it a non-relativistic theory. In contrast, with the Einsteinian interpretation, the relativity of all fundamental forces is an unavoidable consequence of the structure of space and time, so the discovery of a non-relativistic effect would falsify this interpretation. This is why Einstein’s interpretation is stronger, because it isn’t just a catalog of existing observations, it places those observations within the most stringent and economical framework possible, and thereby is able to make predictions about the characteristics of previously unknown physical effects – all of which have (so far) turned out to be correct. Lorentz’s catalog of conspiracies is incapable of making such predictions, since it could be modified to accommodate just about any empirical result.

 

A good analogy can be made between the principle of relativity and the principle of energy conservation. It is possible to adopt the principle of energy conservation only provisionally, just as Lorentz adopted the principle of relativity. We could say that every phenomenon we’ve ever observed so far has conserved energy (or mass-energy in the relativistic context), but we might expect to find a violation of energy conservation tomorrow. This is analogous to Lorentz’s approach to relativity. Individual who believe they have invented perpetual motion machines often prefer this attitude toward scientific principles, since it permits them to argue that their ideas may be confirmed tomorrow, even though they are refuted by all past experience.

 

But regardless of what might transpire upon the discovery of some hitherto undetected physical effect (an effect which would have to be so inconspicuous as to have no relevance to the outcome of any prior observation) was discovered, it’s worth bearing in mind that the known forces of electromagnetism, the nuclear force, and gravity, must all be locally Lorentz covariant, because otherwise the incongruity between how these forces transform from one frame of reference to another would immediately reveal the effects of absolute motion. As discussed above, the fundamental forces of nature are all non-Laplacian in the sense that Laplace’s analysis of the relationship between propagation speed and aberration angle is not applicable to them.

 

Return to MathPages Main Menu