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 inverse-square 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 mis-directed force over thousands of years would be to make the planetary orbits unstable, contrary to observation. Even the severest critics of action-at-a-distance 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

At the time t₁ suppose a photon from the direction of particle A strikes particle B at position x₁ and is reflected directly back toward A. The photon reaches A at time t₂ = 0, and is reflected in such a way that it collides again with B at location x₃ and time t₃ and is reflected back in the direction of A as shown below.

The impulses imparted to B at the times t₁ and t₃ 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 self-evident 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.

To verify this, it’s worthwhile to explicitly determine the direction of the force on A with respect to the rest frame of A (in terms of which B is moving with speed v). The photon departs from B at time t₁ and arrives at A at time t₂, and during the intervening time the particle B has advanced a distance of v(t₂−t₁), so it might seem that this would result in aberration of the force applied to A. However, our supposition is that the photon isn't reflected directly back from A toward the position of B at t₁, but rather it is reflected toward the position that B will have (if it maintains constant velocity) at the future time t₃. Therefore, the net impulse on A is in the direction of an intermediate point x_i between the past and the future positions (x₁ and x₃) of B.

We know the angle of the impulse on A is mid-way between the angles to particle B at times t₁ and t₃. Letting θ₁ and θ₃ denote these two angles (from the direction of closest approach) we have

Adding the first two together gives

Substituting into the equation for the intermediate position, we have

This is the position toward which the impulse on A is directed at time t₂ = 0. This confirms that, as one would expect, if θ₁ is very close to θ₃, then x_i approaches the average of x₁ and x₃.

Now we wish to know the actual position of particle B at the time t₂ = 0 in terms of x₁ and x₃. If we solve the equations

for t₁ and t₃, and substitute into the light-like equations (with units chosen so that the speed of light is 1)

and then subtract one of these equations from the other, we get

Thus at the time t₂ = 0 the instantaneous position of B and the point toward which the force on A is directed are found by taking the average of x₁ and x₃ and then scaling the result, in one case by (1−v²) and in the other case by the ratio of cosine products in equation (1). The speed v equals the ratio of the distance traveled by B to the distance traveled by the photon during the time interval from t₁ to t₃, so we have

Thus, letting s_i and c_i denote sines and cosines for brevity, we have

We can simplify the trigonometric expression to give

Noting that the right-hand factor in this last expression is unity, we arrive at the result

which confirms that the equations for x_i and x₀ given by equations (1) and (2) are in fact identical, meaning that the net impulse on A at the reflection event at t₂ = 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 non-rotating system my be skewed, but this corresponds to the flow of energy-momentum, 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 left-hand 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 right-hand 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 re-emitted) by the stationary particle. However, every photon emitted by one of the particles is reflected (or absorbed and re-emitted) by the other. This is a highly coherent process. Each interaction takes place between null-separated particles.

In space-time the coherent two-way exchange of a photon can be depicted as in the figure below.

Needless to say, the events 1 and 2 are null-separated, 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 two-way exchange corresponds to a force field, whereas a one-way exchange (or an incoherent set of one-way 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 two-way 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 phase-locked in a coherent force-like interaction between two specific particles.

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

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