Key words: special relativity, general relativity

1. INTRODUCTION

Maxwell's equations predict symmetrical retarded and advanced potential solutions⁽¹⁾ which respectively relate to the radiation or absorption of energy by a Hertz dipole antenna. The retarded potential solution is associated with the radiation of energy from a single source into an infinite region of free space, and is generally accepted as being valid in classical electromagnetic theory. But the advanced potential solution produces difficulties if attempts are made to apply it to the generalised absorption of energy by a Hertz dipole antenna. These difficulties will be discussed later. As a result, all authorities rule out any valid application of individual advanced potential solutions^(2,3), but they are only able to deduce this result by appealing to causality arguments. Hoyle and Narlikar⁽²⁾ even go so far as to suggest that the existence of individual advanced potential solutions, which are never specifically excluded by the theory at a later stage in the analysis, indicate that there is a fundamental weakness in Maxwell's equations. However, it will be shown that this conclusion is not wholly correct.

It is important to appreciate that the advanced potential solution is a physically significant solution which applies, in certain circumstances, to positive energy, travelling in positive real time (i.e. with the usual direction of the arrow of time) towards a Hertz dipole antenna. First, it may be noted that for any receiving antenna the fields associated with the energy approaching the antenna will be advanced in time relative to the corresponding current which is later induced in the antenna by these fields. All that is signified by the time-reversal of the advanced potential solution, relative to the retarded potential solution, is that one is now treating the fields in free space as the cause and the current in the antenna as the effect. Thus, the advanced potential solution would be expected to apply to a sink of electromagnetic energy⁽⁴⁾ in limited circumstances. It is true to state that the passing of an alternating current in an antenna cannot produce advanced waves converging on the antenna from infinity⁽³⁾. Nevertheless, suitably positioned advanced waves, converging on the antenna from a distance, can produce an induced current in the antenna.

One trivial example of a valid advanced potential solution exists, where all of the energy is contained in a closed system. 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 need for both the retarded and advanced potential solutions is immediately clear. The relationship between the current in the dipole and the fields of the outgoing pulse is given by a retarded potential solution. For the returning reflected pulse the relationship between the fields of the pulse and the current which is later induced in the dipole is given by an advanced potential solution. The advanced potential solution is mathematically identical to the time-reversed retarded potential solution. But such a time reversal does not necessarily imply that real time is running backwards. For this example, the advanced potential solution applies to the dipole when acting as a sink of energy and to the reflected pulse as it travels with time, in the normally accepted direction of the arrow of time, towards the dipole. There is no question of negative energy being involved, or real-time reversal⁽³⁾, if the advanced potential solution is applied to the current induced in the antenna by the positive energy received by the dipole from the proposed reflected pulse. It is essential that Maxwell's equations should predict an advanced potential solution to cover this case. However, such a valid advanced potential solution is based on an impractical source of energy and the advanced potential solution has no practical applications in three dimensions. All other three-dimensional solutions have to be eliminated by a direct appeal to the logic of cause and effect, as Maxwell's equations, on their own, are unable to eliminate these solutions. The argument used is simple. Suppose one has a transmitting antenna and a receiving antenna, separated by a finite distance in an infinite free-space region, both being situated inside a very large imaginary sphere. The energy everywhere on the spherical surface must be travelling outwards. Hence, any fields associated with an individual advanced potential solution are ruled out as they would result from energy travelling inwards. However, it is clear that such a deduction, based on causality arguments, only applies to problems where energy may escape to infinity.

It is usually considered that there are no problems associated with individual retarded potential solutions, because it is assumed that a physical meaning may be attached to the concept of an isolated source radiating energy into an infinite region of free space. But there are deeper, Machian-like, arguments that question the significance of having just a source of energy in an otherwise empty universe, where no absorber is present. It seems likely that both retarded and advanced potential solutions require a further physical interpretation.

At this stage it is interesting to note that Wheeler and Feynman⁽⁵⁾ have demonstrated that valid solutions exist at the microscopic photon level for a propagator which produces a combination of half-retarded and half-advanced potentials, provided that all electromagnetic disturbances are ultimately absorbed. Their approach throws no light on the macroscopic problem of interpreting the validity of individual advanced potential solutions associated with the absorption of energy by a receiving antenna. However, there is an indication from their approach that valid advanced potential solutions will only exist in a system where no energy is permitted to escape to infinity.

The point being made in this paper is that, if one limits any analysis to one-dimensional solutions, there are valid individual advanced potential solutions when both a transmitting dipole and a receiving dipole are present in a free-space region. There are then cases where no energy can escape to infinity. One solution is for the case of an open resonator⁽⁶⁾. Another solution applies to a detector where signal energy is beamed directly from a source to the detector⁽⁴⁾, and no energy is lost in the transfer. It is permissible to use the advanced potential solution if all of the signal energy is transferred from the source to a chosen sink of the energy. Furthermore, it will be shown that the advanced potential solution is critically significant when analysing the reception of an electromagnetic signal if there is relativistic motion of the source, as it then explains the limitations of the assumption made in special relativity concerning the constancy of the velocity of light relative to empty space.

To observe a signal a material detector has to be introduced into a free-space region. But the presence of the detector will change the steady-state field pattern out to infinity. Hence, it is essential to consider how the act of observation may change the observed velocity of the signal. Consider the proposition: "no physical meaning may be attached to the statement that an electromagnetic signal travels at a given velocity relative to its source until the signal is observed". This proposition is contrary to the usual method of classical physics. Nevertheless, this proposition bears a very close relationship to the assumption made in special relativity, namely: "that light is always propagated in empty space at a definite velocity c which is independent of the source velocity". Special relativity's indeterminate assumption, that light travels at c relative to empty space, is unnecessary. In place of this assumption one may introduce a valid advanced potential solution by directly applying Maxwell's equations to the boundary condition imposed by the material detector that must be present when a signal is observed.

Let a source of electromagnetic waves produce a signal consisting of an ideal cylindrical beam of energy in free space, all of which is intercepted by a Hertz dipole receiving antenna. Such an arrangement is possible if a parabolic reflector is placed behind the receiving antenna. Provided no energy is lost in the transfer of the signal, one may apply the advanced potential solution to the receiving antenna and deduce that the steady-state velocity of the approaching wave will be equal to c. The advanced potential solution requires that this velocity must be relative to the receiving antenna. The limitation of the advanced potential solution to the steady state arises from the inherent finite response time of a receiving antenna. If the signal to be observed consists of a pulse of energy then, on a classical field theory approach, one assumes an instantaneous response of a Hertz dipole to the electric field stimulus. But the dipole will have a negligible effective area until the scattered field has had time to expand⁽⁷⁾. New "near" fields will become established around the dipole which correspond to the near fields of a transmitting dipole. Hence, Maxwell's equations are restricted to predicting that the observed steady-state, or near-steady-state, velocity of a signal will be c in free space. Quantum theory must be used to predict the velocity of the initial transient associated with the arrival of the first few photons. In passing, it may be noted that quantum theory indicates that an indeterminate time delay will occur before the first photon may be detected. To conclude, Maxwell's equations dictate that a Hertz dipole is initially transparent to a signal pulse. The act of observation with a material dipole generates a changed steady-state field pattern, and it is these new fields which determine the observed steady-state approach velocity of the signal. The advanced potential solution requires that this velocity must be equal to c relative to the detector.

If one extends this argument to the case where there is uniform translatory motion of the receiving antenna, then the advanced potential solution of Maxwell's equations still dictates that the creation of a sink of energy ensures that the steady-state approach velocity of the signal must be equal to c relative to the current induced in the material of the receiving antenna and independent of the source velocity. No assumption concerning the velocity of light in empty space is necessary when considering steady-state observations.

1.   THE ONE-DIMENSIONAL LIMITATION PLACED ON LORENTZ TRANSFORMATIONS

A direct application of Maxwell's equations to the analysis of uniform translatory motion in special relativity only yields a one-dimensional Lorentz transformation. All physical derivations of the Lorentz transformation deduce it in one dimension. To maintain invariance of transformation in three spatial dimensions an additional rotation of the axes has to be assumed to form the Lorentz group, whose structure differs from the Galilean group. The assumption of this rotation of the axes has generally been considered to imply that a physical rotation of an inertial frame occurs when it is travelling at a relativistic velocity. For a material body located in the inertial frame, such a physical rotation would require a torque and an energy input, but there is no apparent source for this energy. These deductiorequire a torque and an energy input, but there is no apparent source for this energy. These deductions arise solely because it has been assumed that any analytical deductions from Maxwell's equations do not require to be further interpreted in terms of their physical validity.

However, Terrell⁽⁸⁾ has produced an analysis which eliminates all difficulties associated with the rotation of axes in the Lorentz group. Terrell's analysis has not received sufficient credit because it has not been appreciated that it provides the physical interpretation of the mathematics of the Lorentz group. It is well known that Terrell has shown that there is no visible Lorentz contraction in the direction of motion. But his analysis also requires that there is no physical rotation of the axes of a body in uniform translatory motion. If one considers observing a spherical, incandescent, material, moving body then the leading edge of the body will appear to trap some of the visible radiation. From an observer's point of view, the photons that define the leading and trailing edges of the body, on the image he or she observes on a flat screen, come from points on the body corresponding to a rotated diameter. Hence, the rotation of the axes of an incandescent moving body is a mathematical requirement which allows for the apparent distortion of the paths of the photons reaching the observer. Nevertheless, no actual interception of photons occurs at the leading edge of the body in the frame of reference of the body.

2.   CONCLUSIONS

No observation of radiated electromagnetic energy is possible until a material sink is inserted into a free space region. Fields and photons do not exist, as such, in free space. They are simply two alternative mathematical concepts which enable one to solve for different parts of any problem relating to the observation of electromagnetic energy in free space using a material detector. Quantum theory provides the extreme transient solution and electromagnetic wave theory provides for the near-steady-state and steady-state solution.

One may extend these concepts to cover the case where a source and a detector are in relative motion. Maxwell's equations cannot be used to analyse the problem associated with the detection of an electromagnetic wave unless a material sink is first introduced into a free space region. To obtain the solution for what is observed it is then necessary to produce an analysis applicable to a material detector. The advanced potential solution dictates that the steady-state approach velocity of a signal beamed onto a Hertz dipole must be equal to c, relative to the detector, and independent of the source velocity. There is no need to make any assumption concerning the velocity of light "in empty space". It is the material that must be present to form a sink of energy which ensures that the observed steady-state velocity of light is always c. It is, therefore, the act of observation with a material detector which governs the observed steady-state velocity of light. The advanced potential solution is vital because it provides the essential solution for the observed steady-state velocity of the energy received by a material detector. However, for an extreme transient, where only the quantum nature of the energy is relevant⁽⁷⁾, the observed velocities are not limited to c. Such superluminal velocities have been observed, and tunnelling velocities of individual photons as large as 4.7c have been noted^(9,10). There is no reason why Maxwell's equations (and hence special relativity) should restrict the velocity of such a transient to c.

Terrell's analysis demonstrates that the complication of the rotation of axes, in the formation of the Lorentz group, arises because an observer viewing a moving, incandescent, material body considers that the energy emitted at the leading edge is intercepted by the body. A mathematical rotation of the axes of the body has to be made to allow for this apparent interception of the energy. But there is no actual interception of the energy. Hence, there is no physical rotation or contraction of a moving body. This deduction has clear consequences. If there is no physical rotation of the body then there can be no anomalous energy required to produce the rotation.

The derivation of special relativity from the advanced potential solution restricts the theory to material inertial frames of reference. All anomalies associated with the theory are then eliminated. Thus, if special relativity is restricted to material inertial frames, both special and general relativity may be anchored in the real universe. Any material frame must have a finite mass. But a uniformly moving frame of finite mass could not have achieved its motion unless there was, at some time in the past, an exchange of momentum between the material of the frame and the rest of the universe. A clock will then only show a time difference, as a result of its motion, following an exchange of momentum between the clock's material inertial frame and the rest of the universe, which accords with Mach's principle. Special relativity is thereby anchored in a unique preferred inertial frame provided by the distant stars. As all aspects of time transformation arising from motion in general relativity have their origins in special relativity, general relativity will also be anchored in the real universe.

If special and general relativity are anchored in the real universe then it is likely that a classical theory must exist to explain both the origins of the value of the gravitational constant, G, and also the observed identical nature of the gravitational mass and inertial mass of any given body. The gravitational theory of Hoyle and Narlikar⁽¹¹⁾ proposes that the value of G may be inversely proportional to the mean density of all matter in the universe. Their theory is viable because they show that local matter would have an insignificant effect on the value of G. However, to explain both the origin of G, and the equality of inertial mass and gravitational mass, it is necessary to extend their static theory so as to produce a dynamical theory relating G to the density and the motion of distant matter. Only two large-scale, inertial motions of distant matter exist. One is the expansion velocity of a given region, and the other is the rotational inertial motion of a given galaxy. We will consider the consequences if the terrestrial value of G were to be generated, in a classical cause-and-effect manner, by the rotational inertial motion of distant matter in our galaxy. If the value of G had its origin in the rotational inertial motion of distant matter in our galaxy, then the equality of gravitational mass and inertial mass would follow as a necessary consequence. A specific dynamical theory for the value of G is possible^(12,13). This theory relates the free-space value of G within any gravitationally-stable, rotating galaxy to the mean angular velocity (w) and density (r) of the galaxy, such that G = w²/2pr. The theory will also apply to gravitationally-stable, rotating galactic clusters. G will then have different values in other galaxies and galactic clusters. It is interesting to note that an idealised, gravitationally-stable, rotating fluid body has two unique characteristics. First, all particles within the body are in continuous free fall. Secondly, the equivalence principle applies throughout the volume of the body as a whole and not just at individual points.

General relativity assumes that G is a universal constant, but this assumption may be unnecessarily restrictive as it requires the assumption of dark matter to explain the stability of many galactic clusters. There is no inherent requirement in general relativity that insists that G should be a universal constant, and this deduction may be easily demonstrated. Suppose that, on another planet in some distant galaxy, G had a value that was, say, 10 percent larger than our terrestrial value. Provided that the law predicting the value of G is the same in all inertial frames then the principle of relativity is satisfied. On this distant planet scientists would gradually develop the laws of physics, up to and including general relativity, and all of the tests of general relativity would be satisfied with the larger value of G. There is, therefore, no inherent physical reason for continuing with the assumption, originally made by Newton, that G is a universal constant. The larger value of G, predicted by the theory for all galactic clusters which have a low density, would then explain their stability without the need to assume the presence of dark matter.

The proposed dynamical gravitational theory is fragile to experimental testing, as it predicts that a perturbation of G will arise from the rotation of the near-fluid Earth. A terrestrial measurement of G will indicate a value that is about0.4 percent higher when it is measured below the geoid compared to when it is measured in free space above the geoid. The Bouguer anomalies present in observations of the acceleration due to gravity (about a 0.4 percent increase below the geoid) might then arise partly from actual variations in G. Such a specific check on G has never been made, but early measurements are consistent with this prediction⁽¹⁴⁾. Two very recent measurements^(15,16), made in Colorado, USA, and Wuppertal, Germany, show unexpected differences in the measured values of G well in excess of the expected experimental error of about 0.05 percent. While these experiments are still being undertaken it would be valuable to conduct two additional experiments, using the same apparatus, one located at a height of about 2km above the geoid and the other located at about 2km below the geoid.

Acknowledgement

I am very grateful to Professor C. W. Kilmister and Professor A. L. Cullen for many invaluable comments and suggestions. I also wish to thank a referee for proposing a number of clarifications.

Received 1 February 2000