Mechanical Model of Maxwell’s Equations and of Lorentz Transformations

Abstract

We present a mechanical model of a quasi-elastic body (aether) which reproduces Maxwell’s equations with charges and currents. Major criticism (in: Sommerfeld, Mechanics of deformable bodies, lectures on theoretical physics, Academic Press, Inc., London, 1964) against mechanical models of electrodynamics is that any presence of charges in the known models appears to violate the continuity equation of the aether and it remains a mystery as to where the aether goes and whence it comes. We propose a solution to the mystery—in the present model the aether is always conserved. Interestingly it turns out that the charge velocity coincides with the aether velocity. In other words, the charges appear to be part of the aether itself. We interpret the electric field as the flux of the aether and the magnetic field as the torque per unit volume. In addition we show that the model is consistent with the theory of relativity, provided that we use Lorentz–Poincare interpretation (LPI) of relativity theory. We make a statistical-mechanical interpretation of the Lorentz transformations. It turns out that the length of a body is contracted by the electromagnetic field which the molecules of this same body produce. This self-interaction causes also delay of all the processes and clock-dilation results. We prove this by investigating the probability distribution for a gas of self-interacting particles. We can easily extend this analysis even to elementary particles.

Appendices

Appendix A: A and B-Theory of Time

In order to understand the differences between the three interpretations of relativity theory, one needs first to take into account that there are two models of time [2], called tensed theory of time (also A-theory of time) and tenseless theory of time (also B-theory of time).

According to A-theory of time only the present is real (i.e. only the present exists), the future does not exist (it will exist) and the past does not exist (it no longer exists). This is the common sense notion of time. Let us imagine a staircase and let each stair represents a moment of time. According to A-theory of time one particular stair (present) exists, the stairs below this stair (past) no longer exist and the stairs above this stair (future) do not exist yet. When the next moment of time comes (and it becomes present), it comes into being and the previous stair (which becomes past) ceases to exist. Such is the classical notion of the flow of time.

According to B-theory of time, the whole staircase exists and is real, i.e. not only the present exists but also the past and the future. The flow of time is a subjective illusion in B-theory. Such a model of time allows the hypothetical possibility of going back in time (getting down to lower stairs), while the A-theory of time does not allow this possibility (since the past does not exist). The Minkowskian interpretation as we shall see below rests on the assumption of B-theory of time.

Appendix B: Lorentz–Poincare Interpretation

Lorentz–Poincare interpretation starts with the notions of space and time according to Newton. This means that there exists an absolute time and absolute space. Absolute time flows uniformly of its own nature and without reference to anything external. It is different than the physical time, which is the measure of absolute time by physical clocks and material bodies. The physical clocks are delayed when in motion but not the absolute time. The same with space. According to Newton there are two kinds of spaces—absolute and physical. Absolute space is homogeneous and immovable, it exists without reference to anything external. However, physical space measured by physical processes (light signals or physical rods) is merely a measure of the absolute space. This distinction shows immediately that there exists a special reference frame which should give physical time and physical space in coincidence with absolute time and absolute space.

According to LPI Lorentz transformations merely describe how physical rods and clocks are contracted and delayed when in motion. They connect reference frames made by physical rods and clocks amenable to alteration when in motion.

LPI has been further developed (neo-Lorentzian interpretation) to as few as possible assumptions. In fact it has become as simple as the relativistic and Minkowskian interpretations.

This interpretation gives us physical causes for the clock dilation and rod contraction, namely physical forces. Electromagnetic force literally acts on the arrow of the clock and slows it down. The interpretation assumes A-theory of time and standard notion of the flow of time.

Appendix C: Relativistic Interpretation

The second interpretation (the relativistic interpretation) is the original Einstein’s interpretation of his 1905 paper [20]. In this interpretation space–time is merely an instrument, a helpful tool, and is not interpreted realistically (as in Minkowskian interpretation). It assumes A-theory of time. Einstein dropped this interpretation later in favor of the Minkowskian interpretation.

It is customary to present this interpretation in terms of the postulate of the relativity of all inertial reference frames and the postulate of the constancy of the velocity of light c. However, in order to define properly the meaning of the words ’reference frame’, one needs more preparatory work and we shall see that instead of two axioms, we need in fact eleven, eight of which are mere conventions (at least in this interpretation) and three are empirical. We shall follow Reichenbach [7].

1.1 The Relativity of the Simultaneity of Distant Events

Fig. 2

Clock synchronization according the relativistic interpretation. Light signal is sent from clock A (the vertical axis is the time axis) to clock B and reflected back to A. The moment $t_{2}$ (as measured by A) when the signal reached B is chosen by convention, i.e. $t_{2}=t_{1}+\epsilon \left( t_{3}-t_{1}\right)$ for any $\epsilon$ ($0<\epsilon <1$). Einstein has chosen the convention $\epsilon =\frac{1}{2}$

We can easily establish whether two events at different locations are simultaneous if there were infinitely fast signals. We simply send such a signal from point A at moment of time $t_{1}$ to point B and reflect it back to A. Clearly this signal, being infinitely fast, returns back to A again at the moment $t_{1}$. However we do not have such signals, and thus absolute simultaneity of such a type cannot be established. Therefore we can use the fastest possible signal—light signal and send it from point A at a moment of time $t_{1}$ as shown in Fig. 2. Then the light signal reaches point B and is reflected back to A. It returns to A at moment $t_{3}$. What moment of time $t_{2}$ measured by a clock in A is simultaneous with the event when the light signal reached B? Obviously $t_{1}<t_{2}<t_{3}$. But since there are no infinitely fast signals, nor there are signals faster than the light signal, then it is impossible even in principle (according to the relativists) to establish $t_{2}$. Thus, the relativist claims that the moment $t_{2}$ is chosen by convention! In other words $t_{2}=t_{1}+\epsilon \left( t_{3}-t_{1}\right)$ and we can choose by convention any $\epsilon$ such that $0<\epsilon <1$. The choice $\epsilon =\frac{1}{2}$ is one such possibility. If we choose $\epsilon =\frac{1}{2}$ it appears that we have assumed that the light signal travels in both directions with the same speed. However this is not true. We have in fact defined it to travel in both directions with the same speed. The choice of any $\epsilon$ is a convention and it defines simultaneity of distant events. Therefore the constancy of the speed of light in both directions (being the fastest signal) is a convention, not an empirical fact.

1.2 Definition of Reference Frames: Lorentz Transformations

Let us imagine a continuum of points in the whole of space, each endowed with an observer. Let us consider a particular point A. The observer at A defines his unit of time by some periodic process and let this unit of time be the second.

Axiom 1: (convention) Time flows uniformly at all points in space.

Next, the observer sends a light signal to some point B and reflects it back to A. Let us denote with $\overline{ABA}$ the time interval for the whole trip of the light signal $A-B-A$ as measured by the clock at A.

Axiom 2 (convention) If the point B has the property that the time interval $\overline{ABA}$ is always the same as measured by a clock at A, no matter when the light signal is sent from A, we define such a point of being at rest relative to A.

Please note that this is a mere convention and in fact a definition of rest. Now, the observer at A finds other points C, D, etc. being at rest relative to A. We call such a system of points at rest relative to A. However, just because $\overline{ABA}=\text {const.}$, $\overline{ACA}=\text {const.}$, etc. it does not follow that $\overline{BAB}=\text {const.}$ or $\overline{CAC}=\text {const}$. In other words, the points B, C, etc. are at rest relative to A but it does not follow that A is at rest relative to B or to C or to any other point. That such systems of points exist with the special property that all points are at rest relative to each other is an empirical fact (we do not consider general relativity here).

Axiom 3 (empirical fact) There exist special systems of points A, B, C,..., such that all points are at rest relative to each other.

Note, there is not just a single system of points but infinite such systems.

Axiom 4: (convention) We select such a system of points which are at rest relative to each other.

Next, the observer at A sends his time unit (second) to the other observers at B, C, etc. He may do so by merely sending light signals every second. Please note that the unit of time is thus transferred to the other observers, but clocks are not yet synchronized, i.e. the notion of simultaneity of distant events is not established yet.

Let us choose three points, A, B and C of our selected system of points. Therefore these points are at rest relative to each other. And let us send two signals simultaneously from A. One of the signal travels the trip $A-B-C-A$ and the other $A-C-B-A$. Now, generally the two signals will not return to the point A simultaneously (measured by the clock at A) even though the points A, B and C may be at rest relative to each other. That there exist such systems of points that the round trip journey takes the same amount of time is an empirical fact (again, we exclude general relativity here).

Axiom 5: (empirical fact) There exist special systems of points, at rest relative to each other such that the round-trip journeys $\overline{ABCA}=\overline{ACBA}$ are always the same.

We are finally ready to define the simultaneity of distant events by light signal synchronization.

Axiom 6: (convention) Distant clocks are synchronized using light signals. In other words if we choose two arbitrary points A and B of our selected system of points which are at rest relative to each other, we send a light signal at a moment of time $t_{1}$ measured by the clock at A. It travels the distance $A-B-A$ and returns at A at a moment of time $t_{3}$ by the clock at A. The moment $t_{2}$ at A simultaneous with the moment at B when the signal reached B is defined to be $t_{2}=t_{1}+\epsilon \left( t_{1}-t_{3}\right)$ for $\epsilon =\frac{1}{2}$. In this manner clock B is synchronized by the clock in A. The clocks in all other points can be synchronized by the clock at A in the same way.

The above definition may seem to have chosen a special point A. But it can be easily proved that the above synchronization procedure is symmetric. This means that the point A is not special in any way and in fact if we were to choose any other point to synchronize all clocks, both synchronizations will agree, provided we choose the same $\epsilon$ (in our case by convention $\epsilon =\frac{1}{2}$) . In addition this synchronization is transitive, i.e. if two clocks at different points B and C are synchronized by A they are synchronized by each other.

Thus far we have dealt with the concept of time in our selected system of points. Now we continue with space. The first notion is the topological notion of between.

Axiom 7:(convention) If we choose three points A, B and C in our selected system of points we define point B to be between A and C if $\overline{ABC}=\overline{AC}$.

Axiom 8 (empirical fact) If points $B_{1}$ and $B_{2}$ are between A and C, then either $B_{2}$ is between A and $B_{1}$ or $B_{2}$ is between $B_{1}$ and C.

The above two axioms help us to define the notion of straight line.

Axiom 9 (convention) The straight line through A and B is the set of all points which among themselves satisfy the relation between and which include the points A and B.

With this preparation in hand, we can define the equality of distances in our selected system of points.

Axiom 10 (convention) If the time interval $\overline{ABA}=\overline{ACA}$ for three different points A, B and C in our selected system of points, then we define $|AB|=|AC|$.

This concludes the geometry of space. The above axioms are quite sufficient to prove that space becomes Euclidean.

Axiom 11 (convention) Let us choose two inertial systems K and $K^{\prime }$ as defined by the above axioms in different states of motion. Let l be a rest-length in a system K and $l^{\prime }$ be a rest-length in $K^{\prime }$. If l is measured by observes at rest in $K^{\prime }$, they will not in general measure the same length l as observers at rest in K. There will be some expansion or contraction factor. The same principle is true if $l^{\prime }$ is measured by observers at rest in K. We require by convention the identity of these expansion (or contraction) factors obtained by the observes at rest in K and $K^{\prime }$.

With these eleven axioms at our disposal we finally have a correct meaning of the notion of reference frame. Obviously the above axioms define the light signal to have the same velocity in each reference frame. Not only that but the geometry is Euclidean (we are still in special relativity) and the distance traveled by a light signal from point (x, y, z) to point $(x+dx,y+dy,z+dz)$ is $c^{2}dt^{2}=dx^{2}+dy^{2}+dz^{2}$, where the right hand-side is the distance between two infinitesimally close points and dt is the time required for the light signal to traverse that distance. In another reference frame we have the same speed, thus $c^{2}dt^{\prime 2}=dx^{\prime 2}+dy^{\prime 2}+dz^{\prime 2}$. Given our axioms, the only transformations between x, y, z, t and $x^{\prime },y^{\prime },z^{\prime },t^{\prime }$ that obey the above two equations simultaneously are the familiar Lorentz transformations. All familiar results follow from here.

Imagine a rod placed in x direction in a reference frame K and let it move with a velocity V along x direction relative to K. How is the length of the rod measured? One simply places two observers at some moment of time t (in K) placed at both ends of the rod and measures the distance between the observers. However, if one performs the same experiment in a reference frame $K^{\prime }$ which moves with the rod (i.e., the rod is at rest relative to $K^{\prime }$) the very notion of the same moment of time $t^{\prime }$ in $K^{\prime }$ is quite different than that in K and thus different length is measured. Therefore the difference of the length of an object in different reference frames is connected with the relativity of simultaneity in different reference frames (according to the relativists).

The interpretation uses A-theory of time. This concludes the relativistic interpretation.

Appendix D: Minkowskian Interpretation

Minkowskian interpretation unites time and space into a four-dimensional manifold, called space–time. The space–time is not merely a helpful instrument but is interpreted realistically. The physical objects are four-dimensional. This interpretation assumes B-theory of time. The four dimensional distance between two points (x, y, z, t) and $(x+dx,y+dy,z+dz,t+dt)$ in space–time is $ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$. The geometry in space–time is thus defined, as being pseudo-Euclidean geometry. Going from one inertial reference frame to another is again given by Lorentz transformations, but they are here interpreted as a change of coordinates in the space–time manifold.

Appendix E: Assessment of the Three Interpretations

We shall examine carefully the various interpretations of relativity theory.

Fig. 3

The reality of FitzGerald–Lorentz contraction. As the rod, with rest-length l moves with velocity $\mathbf {V}$ in x direction it is shortened to $l\sqrt{1-V^{2}/c^{2}}$. On the other hand, the diameter l of the metal ring which moves with velocity $\mathbf {V}_{1}$ in z direction is not. Therefore the rod can pass through the ring!

We start with the relativistic interpretation. Let us imagine two objects [2]—a rod and a metal ring in a reference frame K in the configuration shown in Fig. 3. If there were no Lorentz contraction, the rod would not have been able to pass through the ring because its length is equal to that of the diameter of the ring. However, due to FitzGerald–Lorentz contraction the rod is shortened to $l\sqrt{1-V^{2}/c^{2}}$ while the diameter of the metal ring is not changed, since the velocity $\mathbf {V}_{1}$ is perpendicular to the plane of the ring. Therefore, the rod will be able to pass through the metal ring! Of course, if one examines what happens from the reference frame of the rod, it is trivial to show that the ring will be inclined due to Lorentz contraction and the rod will still pass the ring. However in K we see that the rod passes through the ring and so the FitzGerald–Lorentz contraction is a real physical phenomenon, not simply a result of the relativity of simultaneity as claimed by the relativists.

Let us examine another famous example—Bell’s spaceship paradox [2]. Two spaceships moving with the same velocity in an inertial reference frame K. Therefore the distance L between them remains constant as they move. If these spaceships accelerate simultaneously (in K) with the same acceleration, then the distance between the spaceships obviously will remain the same L even after they accelerate. Now, let us consider this scenario again but this time let us imagine a delicate string or thread that hangs between the spaceships, i.e. the string has a length L. Now, if the ships accelerate again with the same acceleration in K the string will be subjected to FitzGerald–Lorentz contraction, i.e. its length will tend to be less than L, while the distance between the ships remains L and the string will break! That it will break can be seen from the momentary inertial frame of the spaceships $K^{\prime }$, where due to the relativity of simultaneity the ships will not begin their acceleration simultaneously even though they accelerate simultaneously in K. Therefore FitzGerald–Lorentz contraction can break delicate strings.

Both of these scenarios can be multiplied [2] and people who are trained to think in terms of the relativistic interpretation will be quite startled at first. The reason for their surprise is that the FitzGerald–Lorentz contraction is quite real—as real as the contraction of metal rods when their temperature is decreased. Lorentz contraction is a true physical contraction. Within LPI these two examples are not difficult to explain because bodies that move with a velocity relative to the aether are indeed contracted by physical forces. There is a true physical force that causes the contraction and it may well break delicate strings and threads. In Minkowksian interpretation the bodies are not three dimensional but four-dimensional objects. And when the objects move it is like seeing them in the four-dimensional space–time from different ’angles’. Thus effects like the above are explained also in Minkowskian interpretation better than the relativistic interpretation. Examples like that show that Minkowskian interpretation has more explanatory power than the relativistic interpretation. And for that reason the practitioners of relativity theory favor the Minkowskian interpretation rather than the relativistic interpretation.

Therefore these examples show that the relativistic interpretation is explanatorily impoverished as compared with the LPI and the Minkowskian interpretation. However there are more problems. Indeed, since the relativistic interpretation assumes A-theory of time only the present exists. But the very notion of the present (and thus of what exists) is frame dependent. In one reference frame, a person may be shot dead, while in another he may still be alive (not yet shot). If the two reference frames are to have an equal status, then each reference frame is like a new world in which different things are real! Going from one reference frame to another is the same as going from one world to another. Such a pluralistic ontology is fantastic. Even worse, the relativistic interpretation is based upon arbitrary conventions. The relativist believes that he is compelled to choose $\epsilon$ by convention because one cannot establish empirically distant simultaneity. However the philosophy behind that is the old defunct philosophy of positivism (according to which things that one cannot measure are meaningless). However, this philosophy has been abandoned [2] by the majority of the philosophers of science since it is too restrictive and is contrary to the scientific endeavor. A scientist quite often postulates the existence of many things which are not yet empirically established in order to give explanations of a phenomenon—the molecular hypothesis in statistical mechanics has easily explained thermodynamics and chemical reactions well before these molecules were detected directly. Many other examples could be multiplied—the Higgs boson, great many elementary particles, chemical elements, the prediction of the existence of the planet Neptune, etc. In addition, positivism confuses epistemology (what we can know) with ontology (what exists).

Neither does Minkowskian interpretation solves the above problems satisfactorily because it is beset with other difficulties. Indeed, the first difficulty is the union of space with time. Just because one can write space and time coordinates on the same coordinate system, one cannot consider the space–time as real. One can unite pressure and volume on a single coordinate system. This does not mean that there is such a thing as a pressure–volume space. Neither does it help to claim that space–time is different than volume–pressure space by the presence of four-dimensional metric. But how has one detected this metric in the first place? One had to apply the clock synchronization procedure first, which is quite arbitrary and rests on arbitrary conventions (the choice of $\epsilon$) and on defunct positivistic principle. Different conventions of $\epsilon$ will lead to different metrics (Reichenbach [7] gives such examples). In addition, if one is to accept the realism of the space–time one has to accept the possibility that $ds^{2}<0$, i.e. space-like four-dimensional intervals exist and are complex numbers, which is quite incredible. But even worse than that is the acceptance of B-theory of time which flies in the face of our experience of time. B-theory assumes that past and future exist, that there is a hypothetical possibility of time-travel in the past. But there is no evidence of such things. In fact, one can argue that the A-theory of time is a properly basic belief [2] and the burden of proof lies upon the shoulders of the B-theorist. What is the evidence for B-theory? There is none. B-theory is simply postulated without any evidence. Thus, it is quite save to say that space–time is merely a good instrument, already used in Newtonian physics and is not to be accepted as the true reality.

Things are aggravated greatly if quantum mechanical considerations are taken into account. Bell’s inequalities seem to point that only non-local hidden variable theories are a reasonable alternative to Copenhagen interpretation, while these theories seem to be in great deal of tension with relativity theory. This is not so however in LPI, which can easily accommodate superluminal velocities with Lorentz transformations. Quoting Bell [3]: "I think it’s a deep dilemma, and the resolution of it will not be trivial; it will require a substantial change in the way we look at things. But I would say that the cheapest resolution is something like going back to relativity as it was before Einstein, when people like Lorentz and Poincare thought that there was an aether—a preferred frame of reference—but that our measuring instruments were distorted by motion in such a way that we could not detect motion through the aether... The reason I want to go back to the idea of an aether here is because these EPR experiments there is the suggestion that behind the scenes something is going faster than light. Now, if all Lorentz frames are equivalent, this also means that things can go backward in time... this introduces great problems, paradoxes of causality, and so on. And so it is precisely to avoid these that I want to say there is a real causal sequence which is defined in the aether”.

The introduction of general relativity as a proof that the space–time is necessary not just as an instrument but as a reality is also implausible since there is a perfectly reasonable field theoretical explanation of gravity, the so called bimetric theory of gravity [21, 22]. Such a bimetric approach to gravity makes possible to consider gravity as a field and energy–momentum tensor can be written. Not only that but the field approach unites all forces of nature under a single unified framework. Even more, according to Logunov [23, 24] one is compelled to consider gravity as a field in flat space–time such that the gauge is organically built into the theory. Otherwise Einstein’s gravity equations will not give unique predictions.