Physics Essays, volume 11, number 4, 1998
The Significance of Precursor Electromagnetic Waves in Special and General Relativity
Lawrence M. Stephenson
Although the mathematics of special relativity is not open to question, the interpretation of the mathematics, especially regarding its relationship with quantum theory, is far from complete. A study of precursor electromagnetic waves, which are closely associated with quantum effects, offers a new insight into special and general relativity.
Key Words: special relativity, general relativity.
Special relativity is based on two primary assumptions. The first assumption is the principle of relativity, which states that the laws of physics should be the same in all inertial frames of reference. The second assumption is a postulate concerning the velocity of light in empty space, namely: light is always propagated in empty space at a definite velocity c relative to any mathematical, matter-free, inertial frame of reference. But it is impossible to test this postulate experimentally because any observation of the velocity of light requires the presence of a material detector. Einstein was clearly aware of the imprecision involved in defining the velocity of light relative to empty space. He introduced additional assumptions by establishing a viable system of signalling and time assigning, so as to achieve a method of synchronising a number of clocks located in inertial frames which are in relative motion. Einstein recognised that these additional time-assigning assumptions might create limitations for the theory because he concluded: "We assume that this definition of synchronism is free from contradictions...".
Einstein's additional time-assigning assumptions are essential to qualify the primary assumption concerning the velocity of light in empty space. It is incorrect to assume that special relativity is inherently a mathematical theory of matter-free, four-dimensional, space-time, and then to introduce matter back into the theory at a later stage. Such an approach can offer no explanation of the one-dimensional nature of the Lorentz transformation, and it requires a further assumption of a mathematical rotation of axes, to form the Lorentz group, in order to maintain invariance of transformation in three spatial dimensions. If special relativity is a theory of physics, then it is reasonable to require that only material inertial frames should be considered.
It is possible to avoid all assumptions in special relativity, other than the principle of relativity, if one returns to Einstein's approach and reassesses Maxwell's equations in terms of the effects that precursor transients have on any realistic experimental measurement of the velocity of light.
2. PRECURSOR TRANSIENTS
Precursor transients arise whenever a step-function of electromagnetic energy, initially travelling in free space, interacts with any material that is able to store energy. The best known example1 of a precursor transient occurs when a step function of electromagnetic radiation enters a slab of dielectric medium. The front edge of the first precursor will continue to travel in the medium at c, the velocity of light in free space, because a dielectric cannot exhibit a value of relative permittivity greater than unity until dipoles have been formed in the medium. The formation of these dipoles takes a significant time and depends on energy being extracted from the precursor. Although classical electromagnetic theory may be used to find the velocity in the dielectric of both the front edge of the first precursor and the observed signal, there are difficulties in analysing how the precursor transients develop before the arrival of the true signal. A definitive analysis is limited by the usual restrictions placed on Maxwell's equations at the atomic level and approximations are necessary concerning the formation of the dipoles.
A precursor transient will also arise whenever a step-function of radiated electromagnetic energy, travelling in free space, is observed using a practical, material detector. In the limit, the analysis of any electromagnetic wave detector may be reduced to considering a Hertz dipole antenna. Consider the effect of such a simple, thin-wire, receiving antenna. A thin-wire antenna distorts the free-space field pattern that would have existed in its absence and, in the steady state, the antenna will be surrounded by new "near" fields which contain stored energy. On the arrival of a step-function of energy at the antenna the precursor transient introduces a time-delay before the signal can be detected; during this time-delay energy from the precursor transient is building up the near-field stored energy around the antenna. Looked at another way, a thin-wire antenna presents a negligible physical area. Even on a classical field theory approach, which assumes an instantaneous response of the antenna to the electric field stimulus, it is still predicted that the antenna will have a negligible effective area until the scattered field has had time to expand and the near fields have begun to be established. Until the effective area of the antenna is finite no signal can be detected.
The time-delay introduced by the precursor transient, before a signal can be detected, may also be readily examined on a quantum basis. A single photon, appropriate to an infinite-frequency Fourier component, associated with a perfect step-function of energy will have infinite energy. Even for a near-perfect step-function of energy a single photon will have sufficient energy to establish the stored energy in the near fields produced by the steady-state current induced in the antenna. But the precise time of arrival of the first photon is an indeterminate quantity. Hence, both quantum theory and classical electromagnetic theory predict a finite time-delay before a step-function signal can be detected or reradiated by any material object situated in free space.
The prediction of such a finite time-delay, before a step-function of energy can be detected or reradiated, has an impact on special and general relativity. In order establish special relativity, and a method of signalling between two observers A and B, Einstein made two assumptions. The time-assigning assumption defines the one-way transit time of a signal travelling between A and B as being equal to one half of the out-and-return transit time. In addition, special relativity assumes that the out-and-return transit time is equal to 2AB/c. But for a practical, material reflector at B the observed out-and-return transit time of a true square-edge signal pulse will be greater than 2AB/c because of the extra precursor transient time-delay introduced during the process of detection and reradiation. It is not until the new near fields have begun to be established at the material surface of the reflector at B that a return signal can be generated. Hence, if special relativity is considered to be based on the signalling system initially envisaged by Einstein to establish clock times in inertial frames, then special relativity is invalid for any case where a step function of radiant energy is used to signal between material inertial frames.
The conclusion just reached may be viewed in two distinct ways. At a trivial level it indicates that, for non-ideal circumstances, a measurement problem may exist when applying special relativity. It is necessary to analyse any measurements appropriately. If an inertial frame containing a material detector exists in a free-space region, the steady-state, or near-steady-state, solution for a signal received by the detector will be given by Maxwell's equations and special relativity. But the extreme precursor transient solution associated with a near-perfect step-function signal, and the arrival of the first few photons, can only be achieved using quantum theory.
Hence, when analysing an inertial frame containing a material detector of energy, it is clear that, from a measurement point of view, special relativity must be restricted to steady-state, or near-steady-state, solutions.
But such a trivial explanation of the significance of precursor transients is at variance with the essential part played by the presence of a material sink in any observation. Special relativity should be based on studying observable quantities. However, no observation is possible in a non-material inertial frame. To make an observation a material detector has to be introduced into a free-space region. The resulting radically changed steady-state field pattern then has a further specific consequence. The steady-state approach velocity of an electromagnetic wave to an infinitesimal (Hertz) dipole receiving antenna must be equal to c, and independent of the source velocity, as a result of Maxwell's equations. The solution is derived from the retarded potential solution2 for a transmitting Hertz dipole followed by the application of reciprocity for a linear, isotropic, homogeneous, free-space region. This result may also be deduced from the advanced potential solution3 , provided one avoids causal restrictions by limiting the analysis to signal energy that is beamed directly from the transmitter to the detector. Hence, Maxwell's equations dictate that the observed steady-state velocity of light must be equal to c, relative to an observer's material detector, because the act of observation makes it so. In quantum theory the effect of the observer on the outcome of any observation is inherently significant. But in special relativity the effect of the presence of a material observer on the outcome of an observation has not been previously considered. It is now clear that it is the material of the observer's detector which ensures that the approach velocity of any electromagnetic wave is equal to c, relative to the observer, in the steady state.
It is often implied that the uniform plane wave solution of the wave equations requires that the velocity of light must be equal to c “in empty space”. Such a deduction is misleading as no observation can be made in empty space. The uniform plane wave solution is a useful idealisation, but it is only valid for the unrealisable case of an infinite plane source and an infinite plane detector. The solution for an infinite plane detector fails to illustrate how a practical detector interacts with the wave in any finite region of space. However, the solution for a realisable detector, outlined above, shows that the steady-state velocity of any observed electromagnetic wave travelling in free space must be equal to c relative to an infinitesimal dipole.
As this last point is so important it is worth elaborating further. If a finite-sized transmitting antenna is used, instead of a Hertz dipole, then the steady-state, free-space signal velocity of any radiated electromagnetic wave will be marginally less than c close to the antenna and will only asymptote to c at infinity. At first sight this deduction might be thought to imply a source theory of propagation, but exactly the same result applies to the observer's viewpoint of the steady-state approach velocity of an electromagnetic wave to a finite-sized receiving antenna. To deduce what will be observed it is the solution relating to the sink that is required, not the solution relating to either empty space or the source. Special relativity is able to ignore the importance of considering the observer's material sink of the energy, and the effect of precursor transients, by making two assumptions. But the consequence of these assumptions is to restrict special relativity to near-steady-state solutions. Furthermore, there is then no need for Maxwell's equations to be Lorentz invariant under all conditions. Lorentz transformations are simply the transformations required to allow for the steady-state boundary conditions imposed by a material detector when in uniform translatory motion.
Care is needed if special relativity is applied to transient problems. In addition to steady-state analysis, special relativity may be applied to most established transient signal analysis because, in nearly all cases, the significant part of the signal is observed after the arrival of the first few photons and these photons will have established the near fields of the detector. However, special relativity and Maxwell's equations will not apply to the precursor transient. As the main relativistic aspects of time appearing in general relativity are wholly dependent on the relativistic aspects of time developed in special relativity, care is also needed when applying general relativity to some transient problems. General relativity may have limitations when applied to a theoretical examination of the origins of a "Big-Bang" type of Universe, as the initial stage of this problem may involve precursor transients. But general relativity will apply to the three main tests of the theory because the perihelion precession of the planet Mercury, the deflection of a light beam by the Sun and the gravitational red-shift of the frequency of emitted radiation are all steady-state phenomena.
The proposed alternative way of interpreting special relativity, based on considering only material detectors, goes further and explains why the relationship between Lorentz transformations and the rotation group is different from that holding for Galilean transformations. In the limit, Maxwell's equations require that any observation of an electromagnetic wave may be based on analysing a Hertz dipole receiving antenna, followed by integration for any larger antenna. A one-dimensional current element is thus always associated with the practical radiation or detection of energy. Three-dimensional, spherically-symmetrical, phase-coherent radiation of electromagnetic energy is geometrically unobtainable because it is impossible to arrange a current distribution within a spherical shell of material which appears identical when viewed from all directions and which radiates.
The Lorentz transformation is a convenient way of abbreviating the steady-state solution for a moving, one-dimensional, current element. To maintain invariance of transformation in three spatial dimensions an additional mathematical rotation of the axes has to be introduced to form the Lorentz group, whose structure differs from the Galilean group. Physical derivations of the Lorentz transformation (as distinct from abstract approaches based on the assumption of the group structure) always deduce it in one dimension. The full group then arises from more or less clumsy arguments involving rotations. This is not a sign of inadequacy in the mathematics; it is because there is no physical rotation of the axes of a material body. Terrell's analysis4 deserves much more credit. He showed that the mathematical rotation is precisely allowing for the fact that a moving, incandescent, material body intercepts some of the radiation produced at the leading edge. The perceived diameter of such a body, if spherical, is not Lorentz contracted in the direction of motion. The diameter in the direction of motion has to be mathematically rotated so that it coincides with the point sources of radiation whose outputs, after different transit times, define the leading and trailing edges of the circular image that appears on an observer's flat screen. Hence, all mathematical aspects of the Lorentz transformation summarise the full steady-state analysis arising from the application of Maxwell's equations to a particular class of realisable experiments employing material bodies in uniform translation.
By restricting special relativity to material frames it is possible to anchor both special and general relativity in the real Universe. Any material frame will have a finite mass. A uniformly moving frame of finite mass cannot 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. It is then clear that a clock will 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. If special and general relativity are anchored in the real Universe, a direct relationship between general relativity and other gravitational models is available5 ,6 . One model indicates that the value of the gravitational constant may be related to the density and angular velocity of the galaxy.
Conventional special relativity assumes that light travels in empty space at a velocity c relative to any hypothetical, matter-free, inertial frame. Such a proposition can never be checked experimentally because any observation of the velocity of light requires the presence of a material detector. Einstein partially recognised this fact by additionally assuming the need for a viable system of signalling and time assigning. But he did not consider the limitations imposed on special relativity by the presence of material detectors. It is incorrect to assume initially that special relativity is inherently a mathematical theory of matter-free, 4-dimensional, space-time, and then to reintroduce matter back into the theory at a later stage.
Two alternative approaches to special relativity are possible, both based on considering the use of practical, material detectors to make observations of the velocity of light. First, one may return to developing special relativity so as to be consistent with Einstein's suggested methods of electromagnetic signalling and time assigning. If special relativity is based on physically realisable observations, made between material inertial frames, then any application of the theory must be restricted to steady-state or near-steady-state electromagnetic analysis. This restriction arises because the initial assumptions of special relativity forbid its application to the precursor transient which arises when a true step function of electromagnetic radiation is incident on any material medium. Special relativity then becomes the steady-state part of the solution to any given material boundary value problem. Quantum theory provides the extreme transient solution associated with the arrival of the precursor.
A second alternative approach to special relativity is also possible which is independent of any argument concerning precursor transients. One may examine the implications of Maxwell's equations for a material receiving antenna. For a material inertial frame the assumption of a constant value of c in empty space becomes redundant because Maxwell's equations dictate that the free-space, steady-state, approach velocity of an electromagnetic wave to an observer's receiving Hertz dipole must be equal to c and independent of the source velocity. In this context, Maxwell's equations only need to be Lorentz invariant to meet the requirement of the steady-state solution applicable to the boundary conditions imposed by an observer's material detector. Not only does this approach eliminate the need for any assumption concerning the velocity of light in empty space, but it also explains the one-dimensional characteristic of Lorentz transformations in terms of the one-dimensional nature of the current element always associated with the radiation or reception of electromagnetic waves.
Special relativity may be considered to be a theory of physics arising directly from the application of Maxwell's equations to the material boundary condition imposed by an observer's receiving antenna. There is no need for special relativity to be based on an experimentally unverifiable initial proposition concerning the constancy of the velocity of light in empty space. Special relativity then becomes directly related to quantum theory in the sense that it provides the final steady-state solution of any given material boundary problem for which quantum theory has provided the transient solution.
It is essential to appreciate that physics has moved on from basic Maxwell's equations to quantum electrodynamics, and from Einstein's original form of general relativity to the beginnings of quantum gravity. Nevertheless, the limitations and assumptions introduced at the formulation of any theory must always be considered when interpreting the final predictions of the theory.
I am very grateful to Professor C.W.Kilmister and Professor A.L.Cullen for many invaluable suggestions and comments. I am also grateful to a superb referee whose detailed comments have resulted in many improvements to the text.
References1. J.A.Stratton, Electromagnetic Theory, (McGraw Hill, New York, 1941).