Waves and causality in higher dimensions

We give a new, wave-like solution of the field equations of five-dimensional relativity. In ordinary three-dimensional space, the waves resemble de Broglie or matter waves, whose puzzling behaviour can be better understood in terms of one or more extra dimensions. Causality is appropriately defined by a null higher-dimensional interval. It may be possible to test the properties of these waves in the laboratory.


Introduction
Despite data from the double-slit and related experiments, the theory behind de Broglie or matter waves is not well understood [1][2][3][4]. For example, a fundamental feature is that the product of the particle velocity and the wave velocity equals the square of the speed of light, so if the former is subluminal then the latter must be superluminal. This and other puzzles can be better understood if the de Broglie waves observed in ordinary 3D space originate in five or more dimensions [5,6]. The extension of general relativity to five dimensions, as in Membrane theory and Space-Time-Matter theory, is now well established [7]. It is in agreement with extant observations and is widely regarded as a viable step towards a grand-unified theory of all the interactions of physics. In the present work we will present a new, exact solution of the field equations of five-dimensional relativity, and compare this with the approach in four-dimensional spacetime. Our conclusion will be that phenomena involving de Broglie waves may be better understood in terms of the physics of one or more extra dimensions, where causality is defined by setting the extended interval to zero. It might be possible to study such higher-dimensional waves in the laboratory.

An Exact 5D Wave Solution
The field equations of five-dimensional relativity are usually defined by the 5D Ricci ing a constant extra potential [5; 8-10]. It would be of special interest to find a solution which has wave-like properties and involves the extra dimension in a meaningful manner. This because the extra potential represents a scalar field, modulated by spin-0 quanta, which is believed to be of potential importance for both particle physics and cosmology [7]. In this section, we will present such a solution.
Consider the following 5D line element:  (1) above. A discussion of the physics of 5D wave solutions may be found in connection with a previous study [5]. The criterion for the acceptability of any 5D complex solution is that the 4D properties of matter calculated from it should be real. We will find below that the solution (1) satisfies this criterion.  (1), the wave number is / qL , and the metric has signature ()      so the extra dimension is spacelike. When q is taken out of the complex phase via q iq  in (1), the motion in l is not wavelike but monotonic, and the metric has signature ()      so the extra dimension is timelike. Both options are allowed in 5D relativity [7]. We will concentrate on the former case, since we will find that (1) shares several properties with de Broglie waves.
One characteristic property of de Broglie waves is that the product of their phase velocity p v and group velocity g v is equal to the square of the speed of light, where the group velocity is identified with the speed of the associated particle [1][2][3][4]. The same relation is implied by (1), as may be seen by considering the spacetime part of the wave travelling along the x-axis (say).

This is described by exp[ ( )]
x i ft k x  , where f is the frequency and x k is the wave number. Here, 5 as noted above, the frequency in conventional units is / f c L  . To fix L, we take Planck's law and apply it to the energies of the wave and its associated particle: , which is the Compton wavelength of the particle whose mass is m. To fix the wave number x k , we take de Broglie's relation between the wavelength and the momentum of its associated particle, / xg h mv   , and invert it to write the wave number as Combining the frequency and the wave number now gives a relation between the phase velocity of the wave and the velocity of the particle: This is the aforementioned relation for a matter wave and its associated particle. Ordinary particles observed in the laboratory have velocities , where  is arbitrary and can be less than unity.
Causality in 5D is most logically defined by the 5D null paths given by This includes the conventional 4D paths for both photons and massive particles, given in terms of the 4D interval or proper time by 2 0 ds  . It has been known for a while that certain 5D metrics admit superluminal velocities, the simplest example being 5D Minkowski space with a timelike extra coordinate. However, such velocities are covered by the condition 2 0 dS  , which ensures that all events in the manifold are in causal contact. 6 The metric (1) shows that motion along (say) the x axis is simple harmonic in nature. If this motion were present in a mechanical system, it would be governed by a 'spring constant' 2 1/ L . The question arises of whether the waves in (1) exist in empty space, or whether they are supported by some kind of fluid. Campbell's theorem, mentioned above, helps to answer this.
For it implies that any solution of the apparently empty 5D field equations 0 The equation of state is that of radiation or ultra-relativistic particles. The properties of matter depend on the wave number in the extra dimension (q) and the magnitude of the scalar field (  ). They do not depend on the wave-numbers of the motion in ordinary 3D space. The oscillations described by the 5D metric (1) are not therefore ordinary electromagnetic waves or conventional gravitational waves (the latter have different properties and propagate through truly empty space). But they exist in 3D space, and share characteristics with a previously-studied case whose background metric describes the Einstein vacuum [5,6]. In that case, the waves 7 were identified as de Broglie or matter waves. These can exist in any kind of medium, so we believe that the new solution (1) also describes de Broglie waves.
The new solution (1) may be usefully compared with the 5D de Sitter solution, which has been much studied. In 4D, this solution is the basic one with vacuum energy as measured

Matter Waves and Causality in 4D and 5D
De Broglie waves as they are understood in 4D raise questions to do with causality [2,3,6]. In this section, we wish to give a brief discussion of how matter waves are viewed in 4D spacetime, and then indicate how solutions like (1) above relate to causality in a 5D manifold.
It should be recalled that de Broglie was led to infer that particles have associated waves by essentially comparing the time and space components of the 4-vectors associated with the particle (mass, momentum) and the wave (frequency, wavelength). This procedure leads basically to Planck's law and the definition of what is now called the de Broglie wavelength of a particle. The combination of these relations results in the equation 2 pg v v c  between the phase velocity of the wave and the group velocity, which latter is identified with the ordinary velocity 8 of the particle (see above). Some workers have been suspicious of the inference that p vc  , since it appears to violate the tenets of special relativity. However, Rindler has argued that the noted relation is indeed valid, using an unusual interpretation of the Lorentz transformations [2].
In the remainder of this section, we will use an alternative and concise method to derive the rela- Here  is a typical property of the medium, such as the potential if the wave travels through a field, and the motion is taken to be along the x-axis at (5) In relativistic particle physics, the underlying metric is commonly taken to be the 4D Minkowski one, and the 4-velocities are defined in terms of this and normalized to unity: This relation has been extensively tested in accelerator experiments. Therefore, the behaviour of the group velocity of the wave (or the velocity of the particle as included in p) is well established. However, the phase velocity of the wave is not directly measurable and must be inferred. Using previous relations in (5), the latter reads 1 0 00 This is again de Broglie's relation between the phase velocity of the wave and the group velocity of its associated particle. The same result may be obtained from (5) by expanding (6) in the low-velocity case in terms of /1 g vc  , or in the high-velocity case by using Einstein's energy/mass relation. There is no plausible way to avoid the conclusion that particles which can be seen moving at speeds less than c should be accompanied by waves which cannot be seen and are moving at speeds greater than c.
This result becomes more plausible if we take into account the effects of dispersion in the medium through which the wave propagates. Most media show dispersion at some level, because their microscopic structure causes the phase and group velocities to differ. This is manifested as the spreading of a wave packet, causing the associated particle to become delocalized [1,3]. Even the vacuum shows behaviour which can be attributed to dispersion [6]. To better understand the implications of the new wave solution (1), it is instructive to consider as a com-parison the propagation of light through a dispersive medium [3]. In that case, the strength of the dispersion, or the difference between g v and p v , is measured by the variation of the refractive index n as a function of the frequency  or equivalently the wave-number k. Considered as a phenomenological parameter, we would like to evaluate () nk for de Broglie waves.
The standard relations between the relevant parameters for electromagnetic waves are: To these three relations should be added the de Broglie formula (7) which is independent. To obtain () n n k  from the preceding four relations, it is best to proceed as follows. Multiply (9) and (10) together and use (7) to eliminate c, and find Obviously / dn d changes sign at 1 n  , so there are two types of behaviour for () n  . Integrating (11) gives 11 The sign inside the square brackets is reversed for 1 n  . The constant 0  is mathematically arbitrary, but physically sets a cutoff frequency in devices such as waveguides, where waves with lengths greater than the size of the device cannot propagate. For practical purposes, it is useful to change the variable in (12) to the wave number, using (8). The result is where as before the sign inside the square brackets is reversed for 1 n  .
In the above account, the refractive index is used as a convenient phenomenological parameter to describe the behaviour of de Broglie waves, and these should not be expected to behave in the same way as electromagnetic waves. In particular, the results (12) and (13)  show what is defined to be normal dispersion ( /0 dn d  ) for 1 n  , while de Broglie waves in general have this behaviour by (11) for 1 n  . The reason for the differences between light waves and de Broglie waves can be traced to the fact that the latter are related to particles with finite rest mass.
When a particle with finite rest mass travels through any kind of dispersive medium, its associated wave has phase and group velocities which obey the relation  Both of these objections are suspect, as can be appreciated by the following short thought experiment. Suppose a source S of vacuum waves emits in the direction of an observer O, and that near O there is a particle which is free to move. According to de Broglie's theory, the phase velocity of the waves emitted by S affects the ordinary velocity of the particle at O. In particular, when S emits waves with phase velocity p v , the particle at O responds and is seen to move with speed 2 / p cv , which can be measured. This process represents at least a crude transfer of influence from S to O, and an extension of the method to a series of on/off switches can be used in principle to transfer information in binary code, with an effective speed exceeding that of light. It is not difficult to imagine a more sophisticated process, in which the phase speed of the waves is modulated at S, representing information which must be recorded with fidelity by the response of the particle at O. It should be noted that the spatial separation of S and O is not 13 really pertinent, even though it can formally approach infinity. This and other aspects of the situation are compatible with the condition 2 0 dS  mentioned above for causality as defined in 5D.

Discussion and Conclusion
In Section 2 we presented a new solution of the 5D field equation whose properties lead to an interpretation in terms of de Broglie or matter waves. In Section 3 we re-examined the status of these waves in 4D, focussing on dispersion and confirming that their properties are indeed unusual compared to other phenomena in spacetime. From these studies, it appears that de Broglie waves are better understood in 5D than 4D.
This conclusion was implied by earlier work [5,6]. But that work employed a 5D metric quite dissimilar in form to the one used here. This suggests that de Broglie waves may be a common feature of 5D metrics. In this regard, it should be mentioned that material velocities greater than that of light are implied by both modern versions of the Kaluza-Klein approach, namely Space-Time-Matter theory and Membrane theory [7; both approaches can be extended to higher dimensions]. These current theories are in agreement with observations and explain certain aspects of cosmology not covered by general relativity [11][12][13][14][15][16]. However, the implications of 5D relativity for laboratory-accessible physics have not so far been much investigated.
It appears, after consideration of the physics involved, that a feasible experiment is to measure the phase shift of the eigenstate of a quantum system induced by the extra dimension.
This would be technically part of the Berry or geometric phase, and might be detected using approaches developed to investigate the scalar Aharonov-Bohm effect [17][18][19][20]. It is hoped to re-14 port on this and other possible tests in future work. We hope that others will take up the practical side of this subject, since it appears to us that new laboratory experiments may be possible that could detect the influence of an extra dimension.