Advanced Potentials, Quantum Theory and Special Relativity

The advanced potential solution of Maxwell's equations has produced major intellectual difficulties for electrical engineers and physicists for over a century. Until very recently all authorities had claimed that individual advanced potential solutions were invalid. This rejection relied upon a simple causal argument because no theoretical analysis, based on Maxwell’s equations, was available. It has only recently been accepted that individual advanced potential solutions are valid for the particular case of a one-dimensional signalling process, where energy is beamed directly from a source to a detector with only an insignificant energy loss in between. Advanced potential solutions are then essential to understanding the relationship between special relativity, quantum theory and the limitations imposed on Maxwell’s equations and special relativity by the ultraviolet catastrophe effect.

Both retarded and advanced potential solutions appear whenever Maxwell's equations are applied to predicting the electric and magnetic fields associated with the radiation of energy produced by a transmitting antenna (or aerial). The simplest case to analyse is when an alternating current flows in an infinitesimal, thin-wire, antenna (a Hertz dipole). One may then integrate to obtain the solution for a given current distribution in any larger antenna.

During the analysis for a transmitting antenna the advanced potential solution is ignored. The retarded potential solution is selected to predict the fields produced at a distance r  from an antenna when the antenna is transmitting energy. The fields are "retarded" because of the time taken for any signal to get from the antenna to the point in question. This time delay, t, is equal to r/c, where c is the velocity of light.

The retarded potential solution involves only +t, whereas the advanced potential solution involves only -t. The advanced potential solution is mathematically identical to the time-reversed retarded potential solution. Until recently all authorities had ruled out the use of any advanced potential solution on its own, because they considered that the solution applied only when time was running backwards (i.e. in the opposite direction to the usual direction of the arrow of time, when an event occurs before the action that caused it, and a glass would lie broken on the floor before a person had dropped it). But this conclusion was not wholly correct. Individual advanced potential solutions are sometimes valid in the real world, when time is flowing in its usual direction.

The advanced potential solution predicts that any change in the fields at a distance r  from the antenna occurs before the related current change in the antenna. The fields are then advanced in time by r/c relative to the current in the antenna. For a transmitting antenna this would mean that time was running backwards. However, suppose we are dealing with a receiving antenna and not a transmitting antenna. We then expect any change in the signal fields at a distance r  to be advanced in time relative to the current change which is later going to be induced in the antenna by these fields. Analytically, in the time-delay equation, t=r/c, the sign of r is now negative because the direction of propagation of the signal has been reversed relative to the antenna. But time continues to flow in its usual direction in the real world. At first sight we then appear to have an obvious application for the advanced potential solution. We use the retarded potential solution for a transmitting antenna and the advanced potential solution for a receiving antenna.

But things are not so simple, and this is why problems have arisen. The analysis required when dealing with a receiving antenna is inherently more complicated than the analysis required for the simplest case of just a single transmitting antenna situated in an otherwise empty universe. When receiving energy there must be both a transmitting antenna and a receiving antenna in an otherwise empty universe! Most of the energy radiated by the transmitting antenna disappears off into free space and does not reach the receiving antenna. One is then dealing with a complex problem involving both the radiation and reception of energy.

If we try to use the advanced potential solution for the reception of energy from an infinite, three-dimensional, region of free space, it is easy to show that it is an invalid solution. Suppose one has a transmitting antenna and a receiving antenna, separated by a finite distance in an infinite free space region, with both antennas being situated inside a very large imaginary sphere. The energy everywhere on the imaginary spherical surface must be travelling outwards. Hence, any fields arising from an individual advanced potential solution must be ruled out as they would result from energy travelling inwards.

However, there is a unique, and important, three-dimensional problem where an individual advanced potential solution must be applied. But, as might be expected from the above argument, one has to restrict the energy that may escape from the system through the imaginary spherical surface.

Let a Hertz dipole antenna radiate a single pulse of electromagnetic energy into a free space region which is bounded by a very large, spherical, conducting shell centred on the dipole. The relationship between the current in the dipole and the fields of the outgoing pulse is given by the retarded potential solution. On reaching the spherical, conducting shell the pulse is reflected and the direction of travel of the pulse is reversed. For the returning reflected pulse the relationship between the fields of the pulse and the current that is later going to be induced in the dipole is given by the advanced potential solution. But time continues to flow in the usual direction in the real world. The advanced potential solution is needed to account for the direction of travel of the pulse being reversed relative to the dipole. It is a valid solution because no energy escapes from the closed system. This specific three-dimensional configuration is the only case where an individual advanced potential solution may be used in three dimensions. It is essential that Maxwell’s equations should predict an individual advanced potential solution, applying in real time, to cover this case. The negative sign associated with the time delay of a signal must clearly be interpreted as applying to the reversal of the direction of propagation of the signal in real time. Although the negative sign for the time delay could also be interpreted as a reversal in the direction of the arrow of time for a signal travelling in the positive direction of r, this solution must be discarded when one is dealing with any analysis applying in real time in the real world.

Now comes the key point. If we use the advanced potential solution for the reception of energy in only one dimension there are no difficulties. By stating that we are using one dimension we are stipulating that a signal is being beamed directly from a transmitting antenna to a receiving antenna with only an insignificant loss of energy in between. The simple causal argument given above, that forbids advanced potential solutions when energy may be radiated off into free space, is no longer relevant. There is then no problem in using the advanced potential solution to relate the fields in the beam to the current that is later going to be induced in the receiving antenna.

Why is this very limited version of the advanced potential solution important? The answer is clear. A one-dimensional signalling system is directly related to the assumptions made in establishing special relativity theory. Special relativity assumes that the velocity of light is always equal to c "in empty space". Einstein realised that such a statement was meaningless on its own, as one cannot measure the velocity of light "in empty space". He then set up some additional time-assigning assumptions which involved signalling between inertial frames of reference. In particular, Einstein assumed that the out-and-return transit time of a signal travelling between two inertial frames A and B is equal to 2AB/c. It has not been appreciated that this signalling system is only applicable to steady-state, or near-steady-state, electromagnetic signals, and will not be valid for extreme, precursor, transient signals where time delays may occur at the reception and the re-radiation of any signal.

A precursor transient occurs with the arrival of a nearly perfect rectangular pulse signal at any material boundary. A rectangular pulse may be analysed in terms of the sum of a large number of sinusoidal (Fourier) frequency components. For a perfectly shaped rectangular pulse the frequency components would have to extend to infinity and an infinite frequency photon, having infinite energy, would be required! In practice, some of the Fourier frequency components that make up a near-perfect pulse signal will be in the ultraviolet region or higher. This is where the well known "ultraviolet catastrophe" effect enters, and this effect limits the application of both Maxwell’s equations and special relativity when predicting the electromagnetic energy radiated and received at very high frequencies.

Classical radiation theory originally demonstrated the ultraviolet catastrophe problem when analysing the radiation of energy from a black body. This classical theory predicted that the radiated power per unit frequency increased as the square of the frequency, and the total power radiated tended to infinity as the frequency is increased! It was clear that classical radiation theory could not be applied to frequencies in the ultraviolet region and beyond. This problem has to be solved using quantum theory. But there is another, unrecognised, ultraviolet catastrophe problem associated with the reception of precursor transient signals, as these signals also contain frequency components in the ultraviolet region and above. Quantum theory is essential for analysing both the reception and the radiation of frequencies in the ultraviolet region and beyond. This brings us back to Einstein’s signalling system that is needed to set up special relativity.

In order to observe the velocity of a pulsed electromagnetic signal, a material detector has to be placed in an otherwise empty region of space. As mentioned earlier, all detectors of electromagnetic waves may be analyzed in terms of an infinitesimal, or Hertz, dipole antenna. But such an antenna will only have a finite effective area, and hence can only start to detect a pulsed signal, once the scattered (or perturbation) field has had time to expand after the arrival of the pulse. After the scattered field has been generated by the arrival of the first few high energy, high frequency photons, the solution for the received signal is then given by the advanced potential solution of Maxwell's equations. This solution predicts that the steady-state arrival velocity of the signal is equal to c relative to the detector.

For a nearly perfect rectangular pulse signal, both Maxwell's equations and special relativity are unable to predict the initial, precursor, transient solution associated with the initial scattering of the field by the dipole. The precursor transient solution has to come from quantum theory, and quantum theory predicts that there will always be an uncertainty about the time of arrival of the first few photons associated with the signal.

It may be noted here that the transients observed with the pulses used for typical digital communication systems and radars are very far from being precursor transients, and special relativity will correctly apply to the near-steady-state analysis applicable to these systems. However, the latest systems are reaching the level where precursor transients are significant.

In essence, the observed velocity of a signal approaching a material detector is equal to c because the act of observation makes it so in the electromagnetic steady state. The advanced potential solution requires that any steady-state measurement of the velocity of light must yield a value of c. There is no need to make any assumption about the velocity of light in empty space. As well as demonstrating that special relativity and quantum theory are separate parts of a common solution, this analysis also puts special relativity on the same footing as quantum theory, in that the act of observation changes what is observed.

There is a common misconception that the appearance of the velocity c in the uniform plane wave solution of Maxwell's equations requires that the velocity of light must be equal to c in empty space. But this deduction is incorrect. Any hypothetical uniform plane wave experiment would require an unrealizable infinite plane detector. Such an infinite plane detector hides the nature of the interaction of a signal with any finite detector. When attempting to analyse special relativity, and its relationship with both Maxwell’s equations and quantum theory, it is essential to consider how realistic observations of both the constant velocity of light, and precursor transients, may be made. To analyse how the steady-state velocity of light is observed one must always start with a Hertz dipole detector, and then integrate up for a larger receiving antenna. It is the material of the dipole that dictates that the steady-state velocity of any approaching wave is equal to c relative to the dipole.

The observation of any electromagnetic wave travelling in free space must be based, in the limit, on analysing the output of an infinitesimal, thin-wire (or Hertz) dipole antenna. The individual advanced potential solution of Maxwell's equations requires that the observed velocity of the electromagnetic wave must be equal to c relative to the Hertz dipole, and  independent of the source velocity. No assumption is needed regarding the velocity of light in empty space.

However, for the case of a near-perfect, rectangular-pulse, signal a precursor transient precedes the steady-state condition associated with the arrival of the electromagnetic wave. The initial precursor transient consists of the arrival of a few individual, high-energy, photons. These photons will not produce any output from the Hertz dipole, but will be absorbed in the form of energy stored in the electric and magnetic near-fields around the dipole. Any analysis of these first few photons requires the application of quantum theory, and there will be an uncertainty about the time of arrival of the photons. It may be noted that a Hertz dipole is transparent to any signal until its effective area becomes finite and this  only occurs after absorption of energy from the precursor transient to form the perturbation (or near) fields.

It is then clear why, in developing special relativity, Einstein had to assume that the velocity of light was equal to c in empty space, and also  assume a time-assigning function that restricted signalling between inertial frames to the use of steady-state electromagnetic waves. In 1905 quantum theory and precursor transients were unknown and, even in 2008, individual advanced potential solutions are misunderstood. Both of Einstein's assumptions are now unnecessary, but the mathematics of special relativity is perfectly valid provided it is restricted to the steady-state, electromagnetic wave, solution.

We may now summarise how electromagnetic waves propagate. The retarded potential solution of Maxwell's equations predicts that the departure velocity of an electromagnetic wave from a source must be equal to c. The advanced potential solution predicts that the arrival velocity of an electromagnetic wave at a detector must be equal to c. When there is relative motion between the source and the detector, the arrival velocity of the  electromagnetic wave at the detector will still be equal to c, and independent of the source velocity. But how can this be so? Simply because the insertion of a material detector into a region of empty space is a major change. The act of detecting the signal will bring the wave to an abrupt halt. A radiation pressure force is exerted on the detector and an equal reaction force is exerted on the wave. These forces will change if the source is in motion relative to the detector, and will ensure that the steady-state arrival velocity of the signal (i.e. the arrival of the wave) is equal to c relative to the detector. In the true steady state (after infinite time) the perturbation fields produced by the presence of the detector will extend past the source and to infinity. It is then not surprising that the steady-state approach velocity of a wave is equal to c relative to both the detector and the perturbation fields, and independent of the source velocity.

The preceding analysis, using advanced potentials, was based on an initial assumption that any signal is beamed directly from a source to a detector, with only an insignificant loss of energy in between. However, the analysis may also be applied to the general case of a wide-angle transmitting antenna that radiates energy into a free-space region. Provided there is linearity, one may consider the component of the energy that is intercepted by the detector on its own, as the relevant observed part of the signal to which the advanced potential solution may be applied. The remaining component of the energy, that escapes to infinity, then requires a separate analysis.

The details of what has been discussed in this section are given in the following papers:

Paper 2:
The Relevance of Advanced Potential Solutions of Maxwell's Equations for Special and General Relativity

Paper 3:
The Significance of Precursor Electromagnetic Waves in Special and General Relativity.