Central Forces From Retarded Action 

According to the mechanical world view of the eighteenth and nineteenth centuries, all physical interactions should be reducible to central inversesquare forces acting instantaneously at a distance. The archetypal example of such a force was Newton’s gravity. Newton himself was equivocal about the intelligibility of this model, and there was never a shortage of skepticism about the concept of “action at a distance”, but this skepticism was undermined in the early 1800s by a calculation of Laplace, showing that if the “speed of gravity” is not infinite, it is at least several million times faster than light. This was based on the fact that the Sun orbits the center of mass of the solar system, just as the planets do, and if the Sun’s “pull” on a planet took several minutes to propagate from the Sun to the planet, then by the time this force reached the planet the Sun would have moved on to a slightly different location, so the force would not point directly toward the Sun’s present position. Instead, it would point toward the place where the Sun had been located when it “emitted” the force. The cumulative effect of this misdirected force over thousands of years would be to make the planetary orbits unstable, contrary to observation. Even the severest critics of actionatadistance theories had to accept that there is virtually no aberration in the force of gravity, and this was taken to support the idea that the force of gravity must act instantaneously at a distance. 

Subsequently the mechanical world view was replaced by more sophisticated theories, to account for the effects of electrodynamics, quantum phenomena, general relativity, and so on. Still, it’s interesting to note that the absence of aberration in the force between two separated bodies need not imply instantaneous action at a distance, even in the purely mechanical context of eighteenth century physics. In the following discussion we examine one particular scenario in detail, to show how force can be exerted by a classical exchange of momentum carrying particles of finite speed without aberration. 

Consider two particles A and B, each in uniform (inertial) motion. With respect to inertial coordinates x,y,z,t in which A is at rest at the origin, the position of B is given by 




The impulses imparted to B at the times t_{1} and
t_{3} due to the photon reflections point directly away from the
stationary particle A, so there is no aberration of the "force" of
this interaction at B. This is because in each case the net change in the
momentum of the reflected photon is purely in the direction of A. Naturally
this would not be the case if a photon from A was absorbed by particle
B, because in that case the net change in the photon’s momentum would have a
component in the direction of B’s motion, so it would exert a drag on B,
producing aberration. However, for pure reflections, with the incident and
reflected rays both pointing directly along the line to A, the net impulse is
also along this same line. Now, we can imagine the photon being repeatedly
reflected back and forth between these two particles, and we can view this
process with respect to the rest frame of B, in terms of which it is
selfevident that the impulses on A are always directed away from the
position of B at the same instant. By symmetry, it follows that the impulses
are central on both particles. 
_{} 

Adding the first two together gives 

_{} 




_{} 

_{} 

_{} 


_{} 

Thus, letting s_{i} and c_{i} denote sines and cosines for brevity, we have 

_{} 


_{} 

Noting that the righthand factor in this last expression is unity, we arrive at the result 

_{} 

which confirms that the equations for x_{i} and x_{0} given by equations (1) and (2) are in fact identical, meaning that the net impulse on A at the reflection event at t_{2} = 0 is directed toward the actual position of B at that instant. This is a very agreeable result, because it implies that, for a force mediated by the exchange of photons reflected in this specific way, the force of each impulse is always directed toward the instantaneous position of the other particle, with no aberration at all, with respect to the rest frame of A, and of course the same result applies to the rest frame of B, or of any other inertial coordinate system. 

This result was to be expected, based on a very simple conservation argument. We are given a system consisting of two particles in inertial motion, and a photon being thrown back and forth between them, which causes them to receive impulses tending to drive them apart. To maintain their inertial motions, equal and opposite impulses are applied from outside the system. Now, since the particles are in inertial motion, the angular momentum about their center of mass does not change, so the external impulses must be directed through the center of mass, which implies they are directed through both particles at their true positions. (In a relativistic context this argument no longer holds, and the external forces on a nonrotating system my be skewed, but this corresponds to the flow of energymomentum, as explained in the note on the right angle lever paradox.) 

It’s interesting to consider why this kind of mutual force mediated by photons does not exhibit aberration, whereas ordinary radiation of photons does exhibit aberration. The difference is obviously the reflection angles (assuming the photons are not absorbed). For ordinary radiation, the photons strike the atoms in a particle with random reflection angles, and the mean reflection angle is just directly back along the incoming direction, so we get a “radiation pressure” in that direction, as shown in the lefthand below. This pressure exhibits aberration as would be expected based on the speed of the photons and of the transmitting particle. In a sense this can be regarded as an incoherent interaction, and most of the photons emitted from the moving particle never return to that particle. 


The righthand figure shows a fundamentally different kind of interaction. In this case we still have photons departing from the moving particle, moving uniformly at their characteristic speed, and being reflected (or absorbed and reemitted) by the stationary particle. However, every photon emitted by one of the particles is reflected (or absorbed and reemitted) by the other. This is a highly coherent process. Each interaction takes place between nullseparated particles. 

In spacetime the coherent twoway exchange of a photon can be depicted as in the figure below. 


Needless to say, the events 1 and 2 are nullseparated, as are the events 2 and 3. Also, events 1 and 3 are the only two events on both the worldline of B and the nullcone of event 2. This coherent twoway exchange corresponds to a force field, whereas a oneway exchange (or an incoherent set of oneway exchanges) represent radiation. 

The condition of particle A at event 2 is conducive to coherent reflection if the intersection events 1 and 3 of particle B with the null cone of A are consistent with uniform motion. This means that the elapsed proper time of B  and therefore the advance of the phase of B’s wave function  between events 1 and 3 is maximal. This phase relation facilitates coherent reflection of a photon from event 1 to event 3 on the null cone of event 2. However, if particle B undergoes acceleration between events 1 and 3, the elapsed proper time (and phase change) will be strictly less, disturbing the coherence of the twoway force interaction. Instead of a perfectly coherent reflection, the interaction becomes scrambled and diffused, resulting in the emission of radiation, i.e., photons that are absorbed and “thermalized”, making them available for exchange with any other particle, rather then being phaselocked in a coherent forcelike interaction between two specific particles. 

Attractive forces could be modeled in essentially the same way, using timereversed exchanges of photons as discussed in Attractive Forces from Quantum Exchanges. 
