Having spoken of the rays of the sun, which are the focus of all the heat and light that we enjoy, you will undoubtedly ask, ‘What are these rays?’ This is, beyond question, one of the most important inquiries in physics.

Leonard Euler

At the turn of the twentieth century, a revolution occurred. Thousands of years of slow and steady progress in understanding the nature of physical laws had led many physicists to conclude that their work in the theoretical realm was nearly finished. Albert Michelson declared at the University of Chicago in 1894 that "It seems probable that most of the grand underlying principles [of physics] have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles." Yet it was Michelson himself who had planted one of the seeds of the revolution six years earlier in an experiment he performed with Edward Morley. Almost simultaneously new discoveries were made concerning the nature of atomic structure, electricity, magnetism, and the energy and velocity of light. Attempts to correlate these discoveries led to the special and general theories of relativity and laid the foundation for quantum theory.

Relativity theory proved very difficult for most people to understand. It was thought in the early days of the theory that there were perhaps only a handful of individuals alive who could comprehend it fully. A New York Times headline in November 1919 proclaimed, "A book for 12 wise men" in an article on relativity. As difficult as the special and general theories were, they quickly gained dominance as a description of the way the universe is. The success in predicting the deflection of starlight by the sun as well as an anomalous advance in the perihelion of Mercury’s orbit, were initial triumphs of the theory. In fact, it was the experimental verification of the deflection of starlight by the sun during a solar eclipse in 1919 that made Einstein an instant celebrity to the general public. The graphic demonstration of the famous equation E = mc² over Hiroshima and Nagasaki made the implications of the theory dramatically accessible to the lay person. The explosions in Japan not only closed the Second World War but also seared the concept of relativity as a fundamental axiom in the minds of a generation of scientists in the fifty years to follow.

Despite the almost universal acceptance of the special and general theories of relativity, there is a problem—Einstein was wrong. Beginning in the 1920’s, the field of Quantum Mechanics began to dominate physicist’s attempts to understand the basic workings and nature of the physical world of which we are a part. Einstein was very uncomfortable with the precepts of this new theory, stating at one point that "God does not play dice," referring to the probabilistic nature of the rules governing the physics of the quantum. He collaborated with Podolsky and Rosen on a thought experiment that demonstrated the foolishness (or incompleteness) of the theory. This experiment is referred to as the EPR paradox. Einstein’s conclusion from this hypothetical situation was that the theory of quantum mechanics, though not necessarily completely wrong, is at best incomplete.

Recent advances in experimental tools have allowed tests of the EPR paradox to be performed, most notably by Alain Aspect at CERN in 1982. The results of the experiment are quite striking. Either the notion of what we call reality is false, and the ideas of physical objects, sequenced events, history, dogs and planets are meaningless; or special relativity is incorrect. Specifically, that portion of special relativity that deals with the velocity of light being an absolute limit to the speed of objects or information transfer must be false. In short, the model of light proposed by Maxwell, Lorentz and Einstein, though not necessarily completely wrong, is at best incomplete.

Einstein developed the special and general theories of relativity to reconcile the amazing mathematical derivations of Maxwell’s electromagnetic theory with the experimentally observed properties of light and gravity. The Michelson and Morley interferometer experiments demonstrated that light has an apparent constant velocity independent of any particular frame of reference. Lorentz and Einstein took this one observable characteristic of light, and, treating it as an absolute characteristic developed a theory by which clocks in motion slow down, lengths contract in the direction of motion, and velocities of objects do not add in a common sense way. Combining this new model with Newton’s laws of conservation of energy and momentum then required also that mass increases with velocity. This set an upper limit on attainable velocities at c, the "speed of light," since reaching this speed would require infinite energy. Generalization of the special theory of relativity to the case of free-fall in a gravitational field resulted in the theory that gravity curves space and time. The end result of all these adjustments is a universe that is not only counterintuitive, but is practically inconceivable to the layperson.

The weakness in the foundation of Einstein’s theories lies in the assumption that the observed or measured invariant velocity of light represents an actual behavior of the light itself. This observed characteristic forms the basis for Einstein’s second postulate: "The velocity of light is constant from all inertial frames of reference and is independent of the motion of the source." We begin by modifying the second postulate to more precisely state: "The observed velocity of light is constant from all inertial frames of reference and is independent of the motion of the source." In order to understand the distinction, we must develop a model that obeys the modified second postulate (with the word observed), but violates the original. Our initial approach is to consider the case of an idealized rubber band.

If you place a cup on a table, the cup will remain there, at rest, until some outside force, say a cat, moves it. Even if the table moves, the cup may remain at rest in its place on the table. The cup will appear stationary to you whether you are seated at the table, or running past the table in any direction. The reason is that you are using the room you are in as a point of reference for you and for the cup and table. When you move, you are aware of your motion with respect to the room, and your mind takes this into account in determining that the cup is not moving. Such accommodating reference frames cannot always be found. We have all had the experience of pulling into a parking space and coming to a complete stop, only to slam on our brakes as the movement of the car next to us caused us to think we were rolling forward. In this case our mind used the adjacent car as a stationary reference frame and judged our motion relative to it. When the stationary reference moved, which it was not supposed to do, we reacted.

Imagine sitting in a train, looking out a window at another train adjacent to you on a parallel track. Suddenly your train begins pulling away. If the motion is smooth enough, it is impossible for you to tell whether it is the other train moving or your own. All you know is that in your reference frame, the other train is moving. The speed you assign to the other train depends on the relative velocity between you and that train. Imagine another passenger on a third train on the other side of the one adjacent to you. That person will assign a different velocity to the middle train if its own velocity does not match yours. With no external reference frame we can only judge motion relative to ourselves. If the velocity of the third train is not equal to yours, it is practically impossible, except in error, for that passenger in the third train reference frame to assign the same velocity to the middle train as you assign in yours. This said, we now propose an experiment in which this assignment is possible. The experiment involves several passengers traveling at different speeds who will each assign a velocity of zero to an object outside their windows.

Suppose we take a piece of clear elastic, very resilient and pliable, and one foot in length. We fasten one end of this elastic to a pole, and stretch the other end to a distance of one thousand miles. While it is stretched to this length, we place a faint white line every foot from the pole to the thousand-mile point. The elastic then looks like that in figure 1-1. Once we have completed marking the elastic, we allow it to return to its original one-foot length, still anchored at point O on the pole.

Figure 1-1 Each automobile will remain adjacent to a specific, mark on a piece of elastic stretching alongside them as long as they maintain a constant velocity

An important point about the way that an elastic material stretches is that any two points on the elastic always maintain the same relative separation. For example, if we place marks dividing the elastic into thirds, then as it is stretched these marks will continue to delineate three equal sections, as in figure 1-2. An implication of this is that each point on the elastic has a unique, unchanging speed as the elastic is being stretched. Thus if we pull the end of the elastic at three feet per second, the other marked sections will be traveling at one foot per second and two feet per second, respectively. These ratios of velocity and spatial separation hold for any combination of points on the elastic. In addition, for whatever speed the end of the elastic is moving forward, a unique point can be found somewhere on the elastic that is traveling at any speed we choose between zero and the speed of that end. In the example of figure 1-2, suppose one end is anchored while the free end is moving at three feet per second. If we wish to find a point traveling at two feet per second, that point will always be located at two-thirds of the distance from the anchored end to the moving end.

Referring again to figure 1-1, suppose we take the loose end of the marked elastic and begin pulling it forward at a velocity of one thousand miles per hour. At the same instant, two automobiles driven by Alice and Bob pass the starting pole, traveling in the same direction as the stretching elastic. Alice, in the first auto, is traveling at twenty miles per hour, while Bob, in the second, is traveling at fifty miles per hour. Further, each automobile is carrying a camera and pointing it directly at the elastic stretching alongside. We assume a very low light level, such that a long time exposure is required to obtain any detail in a photograph taken by either camera. Any object not exposing the same surface of the photographic plate for at least twenty minutes will not appear in the photograph. Thus any object that is in motion at even a very slow speed would, with sufficient light, appear only as a faint blur on the photographic plate. But, with the low light level we a using, any image moving with respect to the camera will not expose the plate long enough to be detected, and will not appear on the photographic plate at all. Each automobile begins a time lapsed photo thirty minutes after passing the starting pole, and allows the exposure to continue for thirty minutes.

After the experiment is complete and the photos are developed, Alice and Bob each have a photo containing one distinct white line and nothing else. The reason for this is as follows: Given an elastic with one end stationary and one end moving forward at one-thousand miles per hour, a unique point can be found on the elastic whose velocity corresponds to any given value between zero and one-thousand miles per hour. Further, an automobile traveling at twenty miles per hour and passing the pole at the same instant the elastic commences being stretched will remain adjacent to the very point on the elastic that is also traveling at twenty miles per hour for the duration of the trip. Since there is a white line on the elastic at this point, this line will appear to be stationary with respect to the camera in the car, and will therefore appear as a distinct white line on the photographic plate.

Since each of the marks on the elastic are separated by one foot when the elastic has attained its one-thousand mile length, their separation will be much less than one foot at the start of the test. Each auto turns on its camera exactly half way through the test and therefore when the elastic is stretched to five hundred miles. At this time, the separation of each of the marks is six inches. Over the time of the rest of the test, this separation of the marks will increase to one foot. The mark initially six inches in front of the line traveling at twenty miles per hour will be traveling slightly faster than the automobile. Over the duration of the test, this line will continually increase its separation until it is one foot in front of the twenty miles per hour mark. Having moved forward six inches with respect to the twenty miles per hour mark, it will therefore not expose any one point on the photographic plate long enough to produce an image. Likewise, the line initially six inches behind the twenty miles per hour mark will be traveling slightly slower than the automobile, and will also fail to expose any one point on the plate long enough to make an image. The same is true for all other lines in front of or behind the twenty miles per hour mark. Thus Alice’s photographic plate will have only one white line corresponding to the twenty mile per hour mark, and no other images. This reasoning also holds for Bob’s automobile traveling at fifty miles per hour, and considering the fifty miles per hour mark.

When the experiment is over, Alice will conclude that the event she photographed was the release of an object with a faint white line at rest from her frame of reference (traveling at twenty miles per hour). Bob will conclude the event was the release of an object with a faint white line at rest from his frame of reference (traveling at a velocity of fifty miles per hour). If the experiment is repeated with many automobiles, all traveling at different velocities, the drivers will, after a time, conclude that the event was the release of an object with a faint white line exhibiting the unique property of appearing to be at rest from all frames of reference. In reality, the event was the release of, for all intents and purposes, an infinite stream of faint white lines, traveling at all velocities from zero to one-thousand miles per hour. The problem is that, due to the nature of the observer, only that aspect of the event remaining at rest with respect to the observer can be detected.

The important point to remember in the above experiment is that the obvious conclusions to be drawn from a set of measurements are not necessarily an accurate description of the system itself. We may develop a model of a system based on a set of observations, and this model may work quite well at predicting future observations made of a similar system under similar circumstances. However, the model is not the system itself, and when future observations produce results inconsistent with the model we have developed, it is the model that must be modified or abandoned in favor of reality, not the other way around.

Arthur Eddington set out during the eclipse of 1919 to confirm certain predictions of Einstein’s general theory by measuring the degree to which starlight was deflected as it passed near the sun. We will cover this expedition in more detail later, including Eddington’s goal of "confirming" rather than "testing" Einstein’s predictions. While Eddington’s results have been confirmed more recently with better observations, it is not clear that Eddington’s results actually supported the claims he was making on general relativity’s behalf at the time. When Einstein was asked what he would have felt had the results of Eddington’s expedition not supported his general theory, Einstein remarked, "Then I would have felt sorry for the dear Lord, the theory is correct." This statement should probably not be taken as much for arrogance on Einstein’s part as for elation over his sudden fame in the eyes of the public at large. It does, however, underscore the cautionary statement above. Our models do not control nature. At best, they predict some of its behavior and give us insight into its underlying principles. Another quote attributable to Einstein is that, "If the facts don’t fit the theory, change the facts." Again, this is surely taken out of context, but the truth is just the opposite. If nature fails to behave according to our predictions and our models, then it is the predictions and models that are incorrect, not nature herself.

Suppose now we repeat the above experiment with the following changes. The light requires only one second to expose the plate. Each automobile is a train, fifty feet in length. The camera is propelled from the back of the train towards the front at a velocity of ten miles per hour (Alice and Bob’s trains are still assumed to be traveling at velocities of twenty and fifty miles per hour, respectively). The plate is exposed for the first second of the camera’s trip down the length of the train. Once again, everything the camera sees that is not stationary with respect to itself will be a blur on the photographic plate, too faint to be observed. This time, since the camera is moving at ten miles per hour with respect to the train, we have created a device that will record only objects that are moving at ten miles per hour with respect to the train. Thus, for a train moving at fifty miles per hour, an object must travel at fifty miles per hour plus ten miles per hour or sixty miles per hour in the same direction as the train in order to be recorded. In this manner, each train rider knows that the apparatus will record only objects that are traveling at ten miles per hour with respect to the velocity of the moving train. Clearly, from the above arguments, Alice will conclude the event produced a glowing object traveling at ten miles per hour as observed from her frame of reference (traveling at twenty miles per hour). Bob will conclude that the event produced a glowing object traveling at ten miles per hour with respect to his frame of reference (traveling at fifty miles per hour). If the experiment is repeated with many trains, the common conclusion will be that the event was the release of an object exhibiting the unique property of an invariant velocity of ten miles per hour for all frames of reference.

Next imagine that we replace the camera in the above examples with a device that can only detect motion at the speed of light, c, relative to itself. The fast moving end of the elastic will need to move forward at a speed not less than c plus the velocity of any potential observer. For the time being, let us agree with Einstein and state that no observer will be traveling faster than c. This being the case, the elastic must be pulled forward with a velocity of at least two times c in order for all possible experimenters to record the white-line phenomena. When the experiment is performed by many people, all traveling at different speeds, they will undoubtedly come to a common conclusion—the event appears to be the release of an object that travels at the speed of light, c, from all frames of reference.

Suppose the experiencing and photographing of elastic bands as described in the first two experiments to be a common occurrence. Then if the true nature of the elastic and markings were not known, physicists would be pressed to devise a theory for an object that remains at rest or is slowly moving when measured from all inertial frames of reference. This problem would be a little harder than the one Lorentz faced when developing his transformations, since in this case, for any observer at a given velocity, other observers can be found traveling both faster and slower than the object being observed. In Einstein’s theory, nobody and no object was found to be traveling faster than c, and so the possibility of these objects could be, and was, omitted. Our last example with the elastic band produced an event—the recording of a single white line on a photographic plate—that appears to travel at the speed of light, c, from all reference frames. We have the advance insight of knowing exactly the true nature of the stretching elastic band, so we are not fooled into thinking that the "obvious" conclusion to be drawn from the evidence on our photographs is the correct one. However, if we had not known in advance the nature of our experimental setup, what appears to us now as a far-fetched conclusion would seem very plausible indeed.

Such a situation is apparently inherent in the nature of light. We have at our disposal a limited set of observations, upon which we wish to develop a comprehensive theory of the nature of space and time. Lorentz, Maxwell and Einstein came to the same conclusion as our train riders. Having only at their disposal the experimental evidence that light has the unusual property of having the invariant velocity of c to all observers, they set out to develop models that explained this unusual characteristic.

As the next chapter will show, it is important to consider the context of Lorentz’s work. Faced with the results of the Michelson-Morley experiment and with the incredible success of Maxwell’s equations, Lorentz had to find a way to reconcile the two. The Lorentz transformations allowed the preservation of the form of Maxwell’s equations in any inertial frame of reference (a frame traveling in a straight line with a constant velocity with respect to another frame) while still supporting the results of the Michelson-Morley experiment. This experiment had shown that the "medium" of light propagation (the aether) was not dragged along by the earth in its motion about the sun. The Lorentz transformations, developed as a means to reconcile the unexpected results of the Michelson-Morley tests, predict that lengths should contract and clocks should slow down for a reference frame in motion. These transformations imply an invariant c for all inertial frames of reference, because they were developed under the assumption of an invariant value for c. However, they do not force c to be invariant. In other words, the actual motion of light is not controlled by the equations Lorentz chose to model it, any more than a red light physically stops a car from crossing an intersection. Einstein used the Lorentz transformations to formulate his second postulate—that c is a constant independent of the motion of the source. The acceptability of this postulate was improved because the required Lorentz length contraction could apparently be interpreted to apply for all electromagnetic phenomenon. Since matter is electromagnetic in nature (composed of electrons, etc.), the supposed Lorentz contraction should apply to all matter. We will later demonstrate that the Lorentz length contraction is merely a result of the particular transformations chosen to preserve the form of Maxwell’s equations, but is not a necessity for all allowable transformations of the same, nor does it represent an actual physical effect of motion.

Albert Einstein, 1905

In ancient or pre-scientific societies, light was considered predominantly as spiritual in nature. In the ninth century, the Islamic philosopher al-Kindi proposed that "everything in this world produces rays in its own manner...Everything that has actual existence in the world of the elements emits rays in every direction, which fill the whole world." From early time to the current day, the nature of light—spiritual, particle or ray—has been debated, with one idea prevailing for a time, only to fall to another. In 1611, Galileo wrote that when a substance was reduced to its most indivisible constituents, light would be created. Newton in Optiks returned to al-Kindi’s rays as fundamental units of light in his first definition "By Rays of Light I understand its least Parts." Even so, he went on "Are not the rays of light very small bodies emitted from shining substance?" Newton had thus seized upon the idea of a dual particle-ray nature of light. Michael Faraday in 1846 returned to the ray theory, giving it more of a flavor of waves on a pond "The view I am so bold as to put forth considers, therefore, radiation as a high species of vibration in the lines of force." In 1864, after unifying electric and magnetic theory and developing the equations governing the waves of electromagnetic radiation, Maxwell concluded that "light is an electromagnetic disturbance propagating through the field according to electromagnetic laws." Arthur Compton demonstrated that light "photons" can be made to bounce off electrons in the same manner as billiard balls on a pool table. Current theory holds with Newton that light exhibits both wave-like and particle-like behavior, depending to some extent on the methods chosen to observe it. In fact, under quantum theory, it is precisely the means chosen to observe light that determines whether it is in that particular instance a wave or a particle.

At about the same time that Maxwell was deriving his equations, the observable speed of light was experimentally measured to be approximately 300,000 kilometers per second (km/sec). Since this velocity was shown to be the same from all inertial frames of reference, Lorentz and Einstein proposed that the dimensions of space and time are dependent upon the relative motion between the observer and the thing being observed or measured. With this theory we instantly run into the problem of developing a model and confusing it with the reality of the thing being modeled. Lorentz and Einstein had concluded from the available observations that the speed of light itself was exactly c in all frames of reference, without adequately considering the role of the observer in making the measurements.

In quantum theory, the observer is all-important. Any book one reads on the subject raises the issue as to whether anything exists on its own accord without the presence of a conscious observer to give it substance. This hardly seems like a question for physicists. As children we all came across the question "If a tree falls in the forest and no one hears or sees it, was it ever really there?" However, in trying to understand some of the perplexing implications of quantum theory, one is often left to ask questions such as this. As the chapter on quantum theory will demonstrate, this is not a shortcoming of the theory, but is instead a result of continually trying to reconcile quantum mechanics with the theory of relativity. And at that, it is mainly relativity’s second postulate—the absolute constancy of the speed of light—that produces all the dilemmas.

The speed of light in a vacuum was determined by making physical measurements (observations) on light itself, and on the electric and magnetic properties of materials in the case of radio energy. The speed of light was not predicted from any application of first principles, nor has any analysis of the observed data yielded any explanation as to why the velocity should be strictly c instead of any other value. The role of the observer appears to be of utmost importance in the determination of any physical quantity in the realm of quantum theory. Clearly the only means by which the velocity of light has been specified is through the analysis of physical measurements, yet the velocity of light is stated as an absolute quantity, independent of any observer or any preferred frame of reference.

Based on the analysis of the previous sections, we are ready to propose what we will call the radiation continuum model (RCM) of light. In this model, light does not radiate from its source at a constant velocity of c. Rather it emanates in the same manner as a piece of elastic, anchored at the source, with one end pulled forward at a constant velocity C, with the upper case C denoting a velocity that is potentially much greater than c, and is very probably infinite. This being the case, there will be a component of the light that is traveling at any speed we pick in the range from zero to C. As important a characteristic of this model of light, and of living and electro-mechanical observers, is that only that component of light that is striking the observer at a relative velocity of c in the observer’s frame of reference will be detected. Because of this, as in the case of the "device" described earlier that detects only motion at ten miles per hour in its frame of reference, we are left with the conclusion that the observed velocity of light is invariant for all inertial frames of reference. That is to say that regardless of our velocity, any light we perceive will appear to be striking us at approximately 300,000 kilometers per second (km/sec).

As an example, choose an event such as an instantaneous burst of light from a satellite at a fixed location in space. We choose a satellite so that we may speak of distances and motion relative to the satellite and distances and motion relative to the "event" as synonymous. When one tries to discuss motion relative to an instantaneous event, the concepts of "motion", "location", and "event" become blurred in a strict interpretation of the terms. If we choose one observer, not in motion relative to the satellite, that observer will detect that component of the burst of light that is traveling at the velocity c with respect to the source. Another observer, moving away from the satellite at a velocity of 0.2c, will detect that component of the burst of light that is traveling at a velocity of c in that observer’s frame of reference. From the satellite’s frame of reference, this component of the light burst must leave at a velocity of 1.2c. To illustrate this, imagine that you want to throw a football so it passes a receiver at ten miles per hour. If the receiver is running away from you at six miles per hour, the ball must leave your hands traveling at sixteen miles per hour in your frame of reference. To reach another receiver running at only three miles per hour, you would need to release the ball at only thirteen miles per hour in your frame of reference.

One of the more significant implications of the radiation continuum model of light is that it allows a more intuitive "Galilean" structure of space and time. By Galilean, we mean that the laws of electromagnetic radiation would conform to Galilean transformations, just as Newton’s laws of motion do. Under such a transformation the concepts of space and time are absolute. This does not require that there is some preferred rest-frame against which all motion is measured. It simply means that agreements can be reached as to the simultaneous occurrence of distant events, and that transformations from one observer’s point of view to that of an observer with a different velocity are straightforward and consistent with our everyday experience. For example, consider two rockets traveling toward each other, each at a velocity of 0.4c. Following the tenets of special relativity and the Lorentz transformations, the two rockets would be approaching each other at a combined speed of only 0.7c. Under a Galilean transformation the rockets will approach each other at 0.8c, just as two cars speeding towards each other at fifty miles per hour each will collide at one-hundred miles per hour. The effect is the same as if one car were parked and the other hit it head on at one-hundred miles per hour. This is the transformation we use in our day to day experience. We are concerned only with the relative velocity between objects in physical measurements. The frame of reference of the observer is irrelevant to the outcome of the experiment and to the damage ultimately inflicted on each car.

Now, without specifying an upper limit on the speed of light C, we have developed a model of light as a rubber band anchored at its source and moving forward through space at all speeds from zero to C. There is no obvious reason to set a bound on C at any value short of infinity. In fact, we will show later on that the value of C is most likely infinite. One might argue that an upper limit of infinity on C would imply infinite energy. While this is strictly the case, it must be realized that only an observer moving away from the source with infinite velocity could detect this component, and this is a very unlikely scenario. Additionally, the frequency of the light at an infinite velocity would be shifted all the way to zero due to Doppler effects, and a zero-frequency signal contains zero, not infinite, energy. It will be shown later that the important consideration along each point on the light wave is the photon’s momentum, which remains constant for all velocity components from zero to C. From here on in this book, the meaning of c shall be taken to be a speed of 300,000 km/sec, as measured with respect to a specific source or observer, and should not be considered synonymous with the phrase "the speed of light." Instead, light is henceforth considered to travel at all speeds from zero to some as yet undetermined upper value C, such that C is much greater than c and is less than or equal to infinity.

The illustration utilized earlier of the elastic band all bunched up at one point waiting to be stretched out can not be carried too far. One shouldn’t think of a photon as being coiled up inside an electron waiting to get out. Rather, the photon is created at a point in time, according to a well behaved set of rules. The creation of this photon wave is simply (and loosely) conversion of "mass" energy into "photon" energy. Typically a photon is created when an electron in an atom drops from a high energy state to a lower one. A photon is also created during many processes of particle decay. In either case, the entire photon wave is created in an instant, in the same respect that the entire photon wave collapses in an instant, as described in the section on quantum effects. It must be noted that the photon thus produced shares many of the characteristics of the photon that was absorbed to send the electron to its higher energy state in the first place. This is what allows the well-observed group characteristics of light interaction with matter, such as angle of incidence equals angle of reflection.

Also a photon does not generally emit from a source in a straight line. It is often useful to think of a photon as emitting in a spherical volume from its source. Under special relativity, such an event produces an expanding spherical shell, with the radius increasing at a speed of c. Upon absorption of the photon by an observer, the entire spherical shell collapses in an instant. Under RCM, the photon energy expands in a spherical volume, with the outer edge of that volume expanding at C. As in special relativity, upon absorption by an observer, the entire spherical volume collapses in an instant. Special relativity envisions a spherical shell in place of a volume at the expense of Galilean concepts of the dimensions of space and time. RCM does not assume a unique velocity of light with respect to the source, and thus retains Galilean concepts of space and time.

The invariance of the speed of light was detected by Michelson and Morley (their experiment is discussed in detail later). What they discovered is that the speed of light appears to be the same whether the observer is moving toward the source, standing still, or moving away. Imagine trying to pass a truck that is moving twenty miles per hour faster than you. However, each time you speed up, the truck is still moving twenty miles per hour faster than you. If you slow down, stop or go into reverse, the truck is still moving twenty miles per hour faster than you. This is fairly easy to explain, as the truck you are following can simply adjust its speed to match yours. But what if your friend is in another car beside you and the truck is also moving twenty miles per hour faster than that car? Let us assume that you slow down while your friend speeds up. Now the truck will not be moving twenty miles per hour faster than both of you. It may be moving twenty miles per hour faster than you, but it will be moving less than twenty miles per hour faster than your friend is moving. It may even be moving slower than your friend. The speed of the truck is not invariant. It is dependent on the speed of the observer; in this case you or your friend, and you each observe a different velocity. Such is not the case with light. If the truck driver were to flash his brake lights at you and your friend, you would see the light pass you at a speed of c. Your friend would also see the light pass at a speed of c. Any theory of light has to support this unusual feature, as it was tested and confirmed by Michelson and Morley in 1887. As the previous example with the satellite showed, this is not a problem for RCM theory, though it posed all manner of problems for Maxwell and Lorentz with the assumption of a constant velocity of light. This experiment should not have posed any problem at all, except that Maxwell and Lorentz were both firm believers in the concept of the aether. The aether was postulated as a substance filling all of space that served as a carrier for electromagnetic waves. Even though in the special theory of relativity Einstein ultimately abandoned the aether concept, he retained many of the corrections to electromagnetic theory imposed by Lorentz and others in an effort to save the aether theory. We will discuss this historical path in detail later.

Despite the fact that the speed of light appears invariant under both RCM and relativity theory, there is a difference as to when and where observers in motion with respect to one another will actually see the light. In special relativity, two observers in motion with respect to each other will each observe an oncoming pulse of light at the same place and at the same time. It is this conclusion that causes problems in the analysis of the simultaneity of remote events, even though this simple concept has never been tested. This concept is a direct result of the second postulate—that the speed of light is a constant independent of the relative motion of the source and observer.

Figure 1-3 Observers in motion relative to each other will in general either be in different locations when they observe a distant event, and/or they will see it at different times.

Figure 1-3 illustrates a ray of light exhibiting the RCM property one second after its release from an explosion in space. The purpose is to illustrate when and where each of several observers will perceive the light under different conditions. We have three witnesses to the event. Alice is stationary with respect to the explosion’s source. Bob is moving toward the source of the explosion with a velocity of .5c, while Carol is moving away from the source with a velocity of .5c. Consider first the case where all three observers see the flash at the same time. We wish to determine where they must each be located for this to occur. Alice, the stationary observer, is sensitive to that component of light leaving the source at a velocity of c. One second after the explosion this light will have traveled 300,000 km, and this then must be her distance from the explosion to see the flash at that time. Bob, moving toward the source at .5c, will see only that component of light traveling away from the event at .5c with respect to its source. The .5c velocity of this component added to Bob’s .5c velocity will cause that component to have a relative velocity of c in Bob’s reference frame. This component will travel 150,000 km in one second. Bob must therefore be this far away from the source one second after the explosion in order to see the light at the same time it is seen by Alice. Carol, moving away from the source at .5c, will see only that component of light traveling at 1.5c with respect to the source (moving toward her at a relative velocity of c). After one second this light will be 450,000 km from the location of the blast, and this must also be Carol’s location at the time of interest.

Next consider the case where all three spectators see the explosion at the same location. We would like to know when each would see the flash. Let’s assume we wish all three to see the event at Alice’s location, 300,000 km from the source. We have already determined that Alice will see the light after one second. The light that Bob sees is traveling at .5c. It will take two seconds for this light to reach Alice’s location. Therefore Bob would need to make sure that he goes flying past Alice exactly two seconds after the explosion in order to observe the light flash at that point in space. The light that Carol sees is moving much faster at 1.5c. It will take only two-thirds of a second for this light to reach Alice, and Carol must plan to be passing Alice two-thirds of a second after the explosion if she wishes to observe the flash where Alice is sitting. Thus each of the observers, Alice, Bob and Carol, can observe the same event, either at the same instant and at different locations, or at the same location but at distinctly different times. This marks the first major conceptual break with the special theory of relativity. This aspect may be the most testable difference, and it forms the basis of many experiments already performed as well as new experiments proposed. Special relativity fails to predict the results obtained in existing experiments, including one by Fizeau in 1851, and the Global Positioning Satellite (GPS) system currently in use by the Government and commercial industries.

One question that comes to mind in the radiation continuum model of light is: Why is it that we perceive only that component of light that is passing us at a relative velocity of c? The "we" in the question applies to humans, cameras, radios and even objects that will reflect light (although objects that reflect light themselves act as light sources, reflecting the component that strikes them at a relative velocity of c at all speeds from zero to C).

In order for light to be seen, it must interact physically with the eye, which in turn converts this interaction into electrical activity. Similarly, a radio wave, to be detected, must interact physically with an antenna to produce an electric current in it, which is in turn interpreted by the radio electronics to produce an audible sound. A physical object that is reflecting light must physically interact with the incoming signal in such a manner that some of the "photons" are repelled from the object, in the same manner as if the object were itself a source of light. Arthur Compton reported in 1923 of elastic collisions between "photons" of light and electrons, thus demonstrating that light does indeed interact on a physical level with matter to produce detection. In fact, photons are not actually reflected from a material. What occurs instead is that one photon is absorbed by an electron, which uses the photon’s energy to make a transition to a new energy level. That electron then releases the same amount of energy in the form of a new photon, as it relaxes to a lower energy state.

Electromagnetic theory involves the mathematical description and interdependence of the following four quantities or fields: the magnetic and electric flux density, B and D respectively, and the magnetic and electric field intensity, H and E respectively. Electromagnetic theory also defines the interaction of these fields with the physical world. James Clerk Maxwell published Electricity and Magnetism in 1873, in which he unified all known electromagnetic interactions through what are now known as Maxwell’s equations. Maxwell made predictions about electromagnetic wave motion, explained light in terms of electromagnetic waves, and calculated the speed of light. While all of this work is important, only a small part of it is the subject of this section.

When one takes the units of B, D, E and H in the ratio HE/BD, the resulting units are equivalent to velocity squared. The H/B term is considered the magnetic charge, while E/D is called the electric charge. While the dimensional analysis of the above ratio yields a velocity relationship to these quantities, this analysis alone does not specify a value for that velocity. Maxwell’s equations in and of themselves say nothing about the specific velocity of propagation of an electromagnetic wave, nor of the detectable velocity or range of velocities in any particular observer’s frame of reference. Maxwell knew this when he derived the equations, but the coincidental timing of early measurements on radio waves and the determination of the velocity of light encouraged the conclusion that the velocity implied by the equations and the velocities as measured were one in the same. In the physical world of which we are a part, we can use physical devices and measuring apparatus to determine numerical values of the above four quantities in various physical settings. When the results of the values obtained from measurements of the physical interaction of electric charges with the experimental devices are combined in the above ratio, the result is always the same—the velocity implied by the measurements is 300,000 km/sec, or c.

This conclusion that the speed of all electromagnetic propagation, including light, in free space, is c appeared acceptable to everyone at the turn of the nineteenth century, but one nagging question remained. In what frame of reference is the speed of light c? A train moving at eighty miles per hour in reference to the ground is only moving at sixty miles per hour in reference to another train coming from behind at twenty miles per hour. In this example, the Earth is considered stationary for all practical purposes, and is the preferred reference frame. When determining the muzzle velocity of a rifle, we are concerned only that a bullet leaves the rifle at one hundred miles per hour with respect to the rifle. The fact that a rifle moving at one hundred miles per hour can launch a projectile at two hundred miles per hour with respect to the Earth is not important, thus, in this example, the source of the projectile’s motion is the preferred reference frame. What, then, could be the preferred reference frame for this velocity, c, of light?

As mentioned above, early theorists suggested a background "aether" in which sat and through which moved all objects in the universe. This undetectable aether was presumed to be the benchmark on which the speed of light was based. Thus, to a moving observer, the perceived velocity of light would be greater than or less than c, depending on the observer’s velocity with respect to the aether, as with the slower moving train’s velocity with respect to the Earth as described above. Since velocities of all things on Earth are slow compared to the speed of light, and given the limited capabilities of measurement at the time, this relative change due to motion could not be easily detected. However, the Michelson-Morley experiment, described more fully in another section, tested the possibility of Earth’s motion through an aether background using interferometers. This test, performed over several seasons and equipment orientations, (along with several other experiments that eliminated the possibility of the Earth "dragging" a part of the aether with it as it moved) proved conclusively that there was no aether to use as a benchmark for light velocity measurements. The speed of light appeared to be c irrespective of the relative velocity of the source and observer. Other theories suggested that it was the velocity of the source that determined the velocity of light, the most notable of these put forth by the Swiss physicist Walther Ritz in the 1908. But studies of remote stars and galaxies and the odd disturbances that this concept would produce indicated that this was likely not the case. Apparently the speed of light was c for all observers, no matter what their relative velocities, and no matter what the velocity of the source.

In the face of this experimental evidence for the invariance of the speed of light, a model had to be developed that allowed this to be possible. Beginning with the Lorentz transformation and ending with the theory of relativity, an interesting mathematical model was developed that allowed light to maintain this one, very confusing characteristic. Unfortunately, the whole structure of the universe had to change to accommodate this. Clocks in motion slowed down and rulers in motion shortened. The mass of a moving object increased without limit as its speed increased. And as objects approached each other at greater and greater speeds their combined velocities increased more slowly until, at a great enough speed (each at c), their combined velocities (measured with respect to the whole system) would still be only c, not 2c as one would intuitively suspect. Consider, for example, the case of two objects approaching each other, each with a velocity as viewed from a common rest frame of 0.9c. Their combined velocity under special relativity would be only .99c, not 1.8c as our common experience would indicate. It is interesting to note that if each of the two velocities exceeded c, then the resultant velocity under the relativistic transformation would become smaller and smaller as the component velocities increased. Of course Einstein’s theory prohibits any object traveling faster than c so this event can not occur.

All of the analysis performed by Lorentz missed an important point, alluded to earlier. Maxwell’s equations do not insist on a specific velocity of propagation. They also certainly do not insist on a velocity that is independent of the frame of reference of the observer. It is the experimental means by which we measure or observe the speed of light or the ratio of H, E, B and D that results in a frame invariant velocity of c. The distinction here is critically important. As in the case of the expanding elastic in the previous sections, the equations of motion of the elastic had little or nothing to do with the results achieved by processing the film of the moving observers. The observers came away with an experimentally verified test of an object that was at rest or moving slowly from all frames of reference. While their observations demonstrated this, the elastic itself did not actually exhibit the properties recorded. The experimenters developed a model that explained their results, but that did not reflect the reality of the situation.

The principle of equivalence tells us that if we are in a uniformly moving reference frame, then any experiment performed in that frame should produce the same results as if performed in a "stationary" frame. Clearly, therefore, the ratio of Maxwell’s four quantities in the manner above will result in a measured "velocity" of c in any uniformly moving frame of reference. Thus each of several observers in reference frames moving at different uniform velocities will each measure or observe the velocity of light from a distant source to be traveling through their apparatus at a velocity of c. As far as the speed of light is concerned, this restriction on uniformly moving frames of reference can and will be lifted as well. Clearly in relativity theory the restriction is not required, as the speed of light is absolutely invariant. In the RCM as well, the restriction is not required, as the observer simply becomes sensitive to higher and higher velocity components with acceleration away from the source. In both relativity theory and RCM, the Doppler effect (a change in frequency or detected energy due to a change in velocity) will play a role in either an accelerating or uniformly moving frame of reference.

From the above reasoning, it makes sense to state that the observed velocity of all electromagnetic propagation, in free space, is c. Thus two observers in motion relative to each other at any velocity will each see a beam of light passing them at the velocity of c. Since it is the same beam of light, that beam of light must have components of velocity (with respect to the source) of c plus the first observer’s velocity (with respect to the source), and of c plus the second observer’s velocity (with respect to the source). Since the source has no idea who its observers are, nor of their velocities, it must produce light in a radiation continuum, at all velocities from zero to C. In this manner, there is a component of that light which will pass any observer, moving at any velocity, at a relative velocity of c in the observer’s frame of reference. This is the speed at which electromagnetic radiation is capable of interacting with the physical world, as demonstrated by laboratory measurements of light and the four electromagnetic properties of Maxwell. Any component of light not at this velocity relative to the observer cannot produce any physical interaction, and is therefore undetectable by any physical observer. Stated more concisely:

Since light travels at all velocities from zero to C, no matter what our speed relative to the source, there is always a component of the radiation continuum that is passing us at a relative velocity of c. It is this component that is thus able to cause the physical interactions necessary to be detected. The end result is the appearance of light having the invariant speed of c from all frames of reference. It is interesting and comforting to note that the experimentally determined values of the fields in Maxwell’s equations predict that our observed speed of light is equal to the square root of the proportionality constant between mass and energy as derived by Einstein (denoted by c²). Of course this famous equation, E = mc², is not necessarily a consequence of relativity theory, but derives naturally from Max Planck’s observations of light emissions from a heated object, as chapter nine will show. However, given this important relation, we can gain additional insight as to why it is that we perceive light only at the velocity indicated by the c² quantity. Since the conversion of radiant energy to mass energy can occur only if the ratio of the two is given by c², it would seem obvious that c is somehow related to the velocity at which matter can absorb or release energy in its own frame of reference. The simplest relation is that light must interact with matter at a relative velocity of c in order to be detected. The same holds true for all electromagnetic energy and for gravitational effects as well. Before we close this chapter, however, we must explore briefly how the relaxed second postulate of RCM theory resolves the confusion caused by Maxwell’s equations when light was assumed to have a constant velocity of c independent of motion with respect to the source.

The key factor in the radiation continuum model of light is the relaxation of the constraints of Einstein’s second postulate. This postulate states: "The velocity of light is constant for all inertial frames of reference, and is independent of the motion of the source." The principle of equivalence tells us that the laws of physics should remain invariant under any transformation of frame of reference. For example, if we drop a ball while standing still, it will fall straight to the floor. If we are on a train moving at a constant velocity of fifty miles per hour, and drop a ball, it will again fall straight to the floor. There is no experiment we could perform wholly inside the train, without looking out the windows for example, that would make us aware of our motion at a constant velocity. We can transform the results of our experiment to the reference frame of an observer on the bank. That observer will tell us that the ball dropped vertically at the proper speed according to the laws of gravity, but also continued moving forward at fifty miles per hour with the train—precisely the velocity it had before we dropped it. This type of linear transformation of physical laws is referred to as a Galilean transformation, and it is the type of transformation we are used to dealing with and make every day without thinking about it. If we are in a car going sixty miles per hour and we are approaching a car going only forty miles per hour, we instinctively make the Galilean transformation of velocity. This transformation implies that, in our frame of reference, the other car is effectively approaching us at twenty miles per hour. In this case we hit the brakes so that the other car does not effectively run into us. If we assume the truth of the second postulate as stated by Einstein, and also wish to preserve the truth and reference frame invariance of Maxwell’s equations, then we must adopt the Lorentz transformations, in which lengths contract in the direction of motion and time slows down for the moving object. If we attempt to use Galilean transformations and also adopt Einstein’s second postulate, it will not be possible to conserve the invariance of Maxwell’s equations. The radiation continuum model simply adds one word to Einstein’s second postulate as follows: "The observed velocity of light is constant for all inertial frames of reference, and is independent of the motion of the source." Through the use of the modified second postulate, making the observed velocity of light dependent on the observer, and adopting the radiation continuum model of light, it is a simple matter to show Galilean invariance of Maxwell’s equations.

In order to illustrate the difference between Galilean and Lorentzian transformations, imagine travelers in two sets of reference frames. One of these frames, K, is stationary with respect to the source of a burst of light, while the second, K’, is moving along the X axis to the left at some fixed velocity. We place each observer at the origin of its own frame of reference. This situation is depicted in figure 1-4.

In the radiation continuum model, where light takes on all velocities from zero to C, the component of light as seen by an observer in any particular frame of reference has a relative velocity of c in that frame. Therefore, the component traveling at c in the moving system will have a velocity in the non-moving system of c plus the moving system’s velocity. In figure 1-4, consider the source of a flash of light, which is some distance from the origin of the stationary system. If at the time of the flash the two systems had their origins together, and the moving system has been traveling for a given amount of time, its origin will now be at a distance equal to its velocity times this time beyond the origin of the stationary system. The distance to the source as measured in the moving system is thus equal to the distance measured in the non-moving system plus the distance covered by the moving system itself. This is the only shift required in a Galilean transformation. Without going into its derivation, we will simply state that under a Lorentzian transformation we must slow down time and shorten lengths along the axis of motion. The formulas contracting length and slowing time are related to the square of the velocity of the moving system, and, of course, to the fixed velocity of light. In the Galilean transformation, time is absolute, and there is no transformation required in this coordinate. Thus the time as measured in the moving system is the same as the time measured in the stationary system.

We have expressed the distance to the source of the flash from the origin of the moving system in terms of the distance in the stationary system and the velocity and time of motion of the moving system. This is similar to saying that in order to measure the distance from 42^nd street to 49^th street, you would have to know how far down Broadway a person had walked, or what time of day the measurement was made. It would be helpful if we didn’t require the elapsed time to determine a value for this measurement in the moving frame. The two quantities, time and distance, should be independent. Now, in order to express the distance in the moving system in terms of measurements made in the stationary system using a Galilean transformation, consider a light signal traveling in both frames, which initially have their origins coincident, as before. We know that the moving system sees light leaving the source at a speed of c plus its own velocity. If we imagine two cars that leave from the same point and travel for one hour, one car at twenty miles per hour, the other at fifty miles per hour, the faster auto will have gone fifty miles while the slower will have gone only twenty. We can express the faster auto’s distance as equal to the slower auto’s distance times the ratio of the velocity of the faster auto to that of the slower one. This relation will obviously hold no matter how long or for what distance the autos travel. Thus if we have two components of light, we can also state that the distance traveled by the faster component in the moving system will be equal to the distance traveled by the slower component in the stationary system times the ratio of the respective component light velocities in each frame of reference. This is of course a much simpler transformation than the one proposed by Lorentz, and it is exactly the transformation we would normally use in, say, determining the distance traveled by a ball thrown on a train as measured by someone standing on the bank. It is a simple matter to show that the form of Maxwell’s equations remains invariant under the Galilean transformations above when one replaces the velocity of light in the moving system with c plus the velocity of that system. This is done explicitly in Appendix A.

The importance of the imposition of the modified light principle of RCM over the light principle of relativity theory cannot be over stressed. Lorentz was faced with the results of the Michelson-Morley interferometer experiments that were designed to test the velocity of the earth through the aether. Michelson and Morley expected to obtain different velocities of light for various orientations of their light beams, depending on whether the light was traveling parallel to or orthogonal to the motion of the earth and the aether locally. What they found instead was no variation. The velocity c was a constant, independent of the observer’s motion relative to the supposed aether. Lorentz was simply trying to obtain a method for achieving a Galilean frame invariance of Maxwell’s equations given the result that c took on a constant value in all inertial frames of reference. He discovered that by scaling the quantities of length and time in the moving system by his squared velocity relations, the desired invariance was achieved. He was not initially interested in the physical significance, if any, of these transformations, and thought of them instead as a mathematical nicety. Einstein developed the actual second postulate addressed above later, partially as a result of Lorentz’s treatment of Maxwell’s equations. Clearly, if the constraint of a constant c that is not frame dependent is relaxed, then the Lorentz treatment is not required. Einstein’s special relativity has a fixed velocity of light combined with a continuum of different standards of length and time. RCM has fixed standards of length and time with a continuum of light speeds as measured with respect to the source only. A look at the sequence of events and discoveries leading up to the idea of a fixed speed of light will aid in the understanding as to why this apparently needless restriction was imposed. This is the subject of the next two chapters.

Albert Einstein

The concept of a constant velocity of light, Einstein's second postulate, is all important in the special theory of relativity. It is therefore necessary to place the developments of Maxwell, Lorentz, Einstein and others into a historical perspective, in order to gain an insight as to why the idea of a constant c became so ingrained in the development of their theories. The advantage of placing a historical perspective on the process of the development of relativity and some of its alternatives is twofold. The first reason was mentioned above. The second reason is that science is an obviously human enterprise, which cannot be cleanly lifted out of other social, political and historical issues of the time. The issue of social factors always resides in the background of science in some way, and its influence should not be underrated or ignored. For example, one cannot help but wonder about the relationship the theories of relativity had with the great waves of skepticism and doubt in our human abilities to make sense of the world that swept through the western hemisphere at the onset of the twentieth century. However, rather than launching into a long exposition regarding the interrelationship that science shares with social factors, this chapter instead pursues a more narrow approach by documenting the development of relativity and its precursors internal to science.

As usual, development in time does not necessarily follow conceptual and logical development. Oftentimes historical patterns influence the types of questions being raised, which are better understood in a sequence that is not necessarily time ordered. Even more significantly, this latter point is usually only understood after the fact--succeeding generations have the advantage of hindsight in order to better categorize their predecessor's foresight. In this respect, relativity is definitely no exception. This chapter and the next address some of the theoretical and experimental issues that preoccupied the physics community of the time, laying special emphasis on why and how certain issues were biased in terms of the interpretation of data and the types of reasoning imposed. We will see that it was this progression of reasoning along a certain initial line of thought that resulted in the overly restrictive definition of light speed, and subsequently the conclusion that only transformations of the type attributable to Lorentz held any validity in the "real" world.

After a long history of discovering the laws of nature through interpretation of experimental results, suddenly the theories of physics would be built on a series of thought experiments with ever increasing complexity. The world of Galileo and Newton, associated with the tower of Pisa and falling apples would be replaced by hypothetical light speed trains and elevators in free-fall toward the sun. It was partially this emphasis on mathematical speculation over the checks and balances of methodical experimental research that helped lead Einstein and his contemporaries astray in the early nineteen hundreds.

What many popular expositions of relativity ignore is the necessary relationship that exists between relativity and electromagnetic phenomenon, completely dynamically described by Maxwell's wave equations. This connection is really so extensive that it is better understood that special relativity is a derived consequence of Maxwell's equations, interpreted in a certain way. As already mentioned in chapter one, RCM theory is likewise another interpretation of Maxwell's equations. Interestingly, there exists literature that seeks to reverse this relation and derive Maxwell's equations out of a context of special relativity. Many modern physicists have been so indoctrinated that they marvel at the way in which Maxwell’s equations have Lorentz invariance built in—a priori to the discovery of this particular set of transforms by Lorentz. The fact that Lorentz actually developed his transforms to try and reconcile a bad theory with good experimental data is ignored or unknown.

Maxwell's equations consist of a set of four equations relating properties known as the electric and magnetic fields and the electric and magnetic flux. The four equations are sometimes referred to as Gauss's law, the law of no magnetic monopoles, Ampere's law and Faraday's law. The three laws named after individuals are related to the precursory discoveries of those individuals in their own areas of electric and magnetic studies. It is interesting to note that these four equations are written independent of any particular system of units, which further supports that the equations in and of themselves fail to predict a specific fixed speed for electromagnetic wave propagation.

Qualitatively, each of the four Maxwell's equations refers to a separate law of electromagnetic behavior previously known by Maxwell's contemporaries. For example, Gauss's law is basically an extension of the most basic equation of electrostatics, namely Coulomb's law. Coulomb's law itself describes the electrostatic force between two charged bodies to be proportional to the products of the charges and inversely proportional to the square of their distance, just as in the case of Newton's law of gravitation. Generally speaking, Gauss's law is a convenient extension of this principle, which basically says that the strength of a radiated electric field of a charged body depends on the amount of charge present. On the other hand, the principle of no magnetic monopoles says that the behavior of magnetic fields is fundamentally different from the behavior of electric fields. Whereas the electric field can be visualized as lines of force beginning and ending at various charged sources, magnetic fields are better visualized as closed loops of force, with no beginning or end, since modern physics has yet to discover a "unit magnetic charge" from which magnetic fields might originate.

Faraday's law is a relationship for moving charges, or currents. It essentially states that current loops respond in such a way as to counteract changes in the magnetic flux through the area they enclose. The basis of this property is known as inductance. Among other things, Faraday's law led to the development of the inductor, an electronic device indispensable in radios, televisions and almost all electronic products. Last, Ampere's law is an analog to Coulomb's force law in that Ampere sought to derive the force between two closed loops of wire. This force relationship was then condensed in a manner similar to the way in which Gauss's law condenses Coulomb's law--following the magnetic field around a closed loop of wire enables one to determine the net current density around the area enclosed.

Ampere's law as discussed above deals with steady-state current phenomena, in the absence of accelerating charges, based on his experimental work. This is not the full law described in Maxwell's equations. It took the genius of Maxwell to realize this limitation, and when he subsequently generalized Ampere's law to cover the phenomena of accelerating charges, he made an astounding discovery that electricity and magnetism are integrally related. In other words, he unified the electric and magnetic fields through his realization that rapidly oscillating charges radiate both electric and magnetic field components perpendicular to one another. He deemed this phenomenon electromagnetic radiation and went on to postulate the existence of an entire spectrum of electromagnetic radiative frequencies, of which visible light is but a tiny fraction. Even more fundamentally, Maxwell was able to link this notion with what was previously known concerning the phenomenon of electricity and magnetism, showing that all these seemingly diverse physical properties fit neatly under the canopy of electromagnetism, whose dynamic relations are so elegantly specified in his four equations.

The contribution on the part of Maxwell cannot be underrated. For one thing, his insights led into an entirely new avenue of research into radiative phenomena, resulting in the discovery of X-rays and other twentieth century advances. In another sense, he introduced a new model by which magnetism and electricity, previously conceived as separate, were unified to such a grand extent as to include the laws of optics. Perhaps even most significantly, the consequences of the Maxwellian model among other things introduced field and continuum notions into theoretical physics, and the push became to explain all phenomena in terms of continuum properties. Hence, the historical debate between Newton's corpuscle or particle model of light versus Huygens' wave model of light was revived and universalized to the extent that particle and wave models gradually became imposed universally throughout physics, leading, among other things, to the development of quantum mechanics. The wave-particle duality in quantum physics seems to imply that perhaps both Newton and Huygens were right in a very universal sense--matter and energy contain both wave-like and particle-like properties. To simultaneously accept these seemingly contradictory models, however, one must abandon the notion that the microphysical world can be visualized. This controversy on the visualizability of microscopic matter persists today regarding the issue of interpreting quantum physics. Perhaps the physicist Henry Margenau puts the whole controversy in perspective when he writes that: "Trying to visualize a subatomic particle is a bit like trying to 'smell' a beam of light."

The Maxwellian continuum model of radiation foreshadowed the development of relativity in an attempt to find in what kind of medium Maxwell's electromagnetic waves must propagate. This search is discussed more fully in the following sections. In summary though, the development of Maxwell's equations can be looked upon as an example of theoretical physics at its finest.

As mentioned already in the first chapter, in a strict sense (in other words based entirely on the implications of electromagnetic theory), the value of the speed of light can be viewed as a built-in parameter rendering the units consistent among the quantities in Maxwell's equations. One can exploit the characteristics of this parameter in such a manner as to verify its value experimentally. This, of course, does not mean that the experimentally observed results are the only values that the parameter can take on, any more than one would assume that because one observes a car traveling at twenty miles per hour the car always travels at such a speed. However, as we shall see, this is just the type of assumption made by the physicists studying the velocity of light.

Recall that Maxwell's equations were written in a manner that is unit-free. While this may seem desirable in the sense of mathematical simplicity, a system of units must be attributed to these equations for them to have any physical significance. Contrary to the case of mechanics, there is no absolute standard of measuring electromagnetic quantities. The main reason for this is historic. In the eighteenth century, Newton's mechanics arose simultaneously with the great concerns to develop a universal standard for measurements of length, time, mass and other physical properties. Electromagnetic theory, on the other hand, appeared nearly a century-and-a-half later, with plenty of experts using plenty of their own systems of units. We will employ this digression on units to reveal how the value of c is derived.

First of all, as a consequence of Coulomb's law, one can straight forwardly deduce that the equation describing the electric field is given by some constant times the ratio of a charge to the square of the distance from that charge, for any arbitrary unit of charge we define. Equally fundamentally, Faraday's law defines the force experienced by two infinitely long parallel wires as another constant times a ratio of the currents to the distance between them. Due to the choice of units arbitrarily assigned to the terms in these equations, a little bit of algebra reveals that the dimensions of the first constant divided by the second is a velocity squared. So at least in principle the units match those of c². At this point, assuming we do not yet specify a numerical value for c, we can state that the ratio of the two constants is defined as c². In this respect, c² is simply a name applied to the ratio of two arbitrary constants, and is not anything like "the speed of light squared."

In addition, Ampere defined a magnetic field, induction, as being numerically proportional to Faraday's force and a third arbitrary constant. Another treatment of Faraday's law produces yet a fourth constant of proportionality. Combining these equations and insisting that all equations have consistent dimensions, it can be determined that the fourth constant is actually just the inverse of Ampere's, thus we are left with only three undetermined constants. Substituting all the previously known values for the derived interrelationships among the constants, one concludes that c as arbitrarily defined above represents the velocity of propagation in Maxwell's electromagnetic wave equations. Notice, however, at this point there has yet to be a numerical value assigned to c, even though our demonstration indicates that c is a necessary parameter for consistency in the units expressed in Maxwell's equations. Still we have the three arbitrary constants at our disposal. It turns out that depending on how one selects the particular combinations of these values, one can construct a system of units of his or her choice. Yet, note that the interrelationships among the constants are always invariant in terms of their being in multiples or ratios of the parameter c, always with c being undefined as far as a specific value within any system of units is concerned. However, we can then make physical measurements of the desired quantities and ratios, and determine as a result of these experiments, that, at least in the realm of our experiments, the value of c is consistently the same, and that that value is roughly 300,000 kilometers per second. The important point to realize when specifying the value of c is that, of necessity, it is dependent on the observer, in this case whatever experimental apparatus we use for measuring the required quantities. We can say nothing about the value of c independent of the capabilities of our equipment. If our measurements were such as to produce different inputs into the determination of the three constants, then we would as a result obtain a different value for the quantity that we named c.

At the risk of becoming redundant, the purpose of these derivations was to explicitly reveal Maxwell's thesis concerning the fundamentally electromagnetic character of light, visible and otherwise. While c is shown to be a regulative parameter keeping straight all of the units in Maxwell's equations, in the final analysis, the value of c can be derived only from the basis of directly measured electromagnetic parameters. There are many experimental approaches to deriving c from purely electromagnetic considerations, though all are ultimately limited by necessity to the perceptions of the observer. Investigating the nature of c exhaustively reveals explicitly how special relativity and similarly RCM theory originate as consequences to particular interpretations of Maxwell's equations, special relativity by treating experimental observations as fundamental truths, RCM theory by considering only first principles.

For example, one means of determining the speed of light is to measure two quantities referred to as electric permittivity (e₀) and magnetic permeability (m₀) of free space. The speed of light to be detected by any particular observer is then simply one over the square root of the product of these two quantities. Any particular observer in free space (and idealized abstraction, but useful for illustrative purposes) will measure a certain, specific value for these two quantities. When that observer takes the inverse square root of their product, the result will be the value c, roughly 300,000 km/sec. We now allow another observer to travel past the first at some arbitrarily high speed. If that observer also makes measurements of e₀ and m₀ and takes the inverse square root of their product, the result will also be the value c. Thus, two observers in relative motion, making the same set of measurements at the same point in space, will each determine that the velocity of light at that point in space is c relative to the frame of reference of each. This will hold true no matter the number of experimenters or their relative velocities with respect to one another. Any number of observers will determine that the velocity of light from some distant source as measured at a particular point in space is c with respect to their individual frames of reference. This simple statement is all that such an experiment, or a conceptually similar experiment, can say about the velocity of light. All else is interpretation. The most obvious interpretation would be that in some manner or another light must travel in a continuum of velocities, such that any observer can find a component traveling at c in its own reference frame. Special relativity, however, concludes that light has only one unique velocity (even though, interestingly enough, that velocity is not measured with respect to the source), and that time and space distort and curl up around moving observers to make the mathematics work out. But when one considers the sequence of theories and discoveries at the end of the 1800’s in perspective, some light can be shed on how this strange interpretaion came about, and, perhaps, why it has persisted for almost 100 years.

Obviously, the above derivation utilizing experimentally determined quantities revealed the observed velocity of c to be an invariant. In other words, the quantities that determine the constants are the same no matter the orientation or presumed velocity of the experimental set up. Since the quantities that determine c are always the same, then so it would seem that the value of c must be always the same. The question then became in what sense is the speed defined as c invariant? Since as Einstein discovered all matter to be essentially electromagnetic in nature, due of course to the existence of electrons, then investigating the shades of distinction regarding the invariance of c in electromagnetic theory has universal consequences. In fact, the discussion cannot really proceed unless we pick up where we left off in our story of the inception of Maxwell's equations. The fundamental pursuit during the end of the nineteenth century focused on the issue of trying to discern any "proper" frame in which Maxwell's equations reigned, if indeed such a "proper" frame could be said to exist at all. Hence we turn to the aether postulate and the subsequent experimental attempts to establish its existence.

The advent and experimental confirmation of a continuous spectrum of electromagnetic radiation predicted by Maxwell's equations arrived with its own host of puzzling questions. For one thing, Maxwell's equations implicitly predict wave phenomena, leading of course to the endlessly verified postulate of the electromagnetic spectrum. However, resulting in a great historical irony, Maxwell had actually set out to construct a Maxwellian world-view, as influential as the Newtonian world-view, of which his famous four equations would be but a consequence thereof. Yet, physics students today remember Maxwell for his four equations, and not for any foundational contributions to physics of the type Newton is usually remembered for. As mentioned earlier, some consequences of the Maxwellian "continuum" world-view did seep their way into the mainstream just the same, namely in the form of wave-particle dualism and in the issue regarding the invariance of his four equations for all inertial frames of reference. It is this latter issue that led among other things to the development of relativity theories.

Fundamentally, it is a mistake to think that Einstein was the only physicist who endowed physics with a "relativity principle." Though it is true that Einstein made such issues very explicit, based on his insights and associations, relativity is really better understood as a systematic treatment of the relationship physical laws have with frames of reference. In other words, fundamentally, relativity can be thought of as the interface between physical principles and the geometry of the observer. In this respect, two important consequences ensue. First, one can never divorce mathematical physics from geometry. Second, relativity considerations are prior to, and implicit in, every equation expressing a physical law. For example, Newtonian physics is based on Galilean assumptions of the nature of space-time and the interrelations among observers; hence it makes sense to talk about "Newtonian relativity." Though Maxwell was unsuccessful in achieving his principle goal of a Maxwellian world-view, he nonetheless unwittingly opened the door to some of the deepest speculations that arose in the history of physics.

Maxwell put the aether hypothesis forth originally, as a consequence of his electromagnetic spectrum postulate. In his time, it was thought that in order to talk about waves; one must necessarily invoke the notion of a medium, so that waves as we generally understand them can exist at all. The reason for this proposal seems both obvious and subtle. In an obvious sense, waves are only one way in which energy chooses to transmit itself through a medium. Take, for example, the case of a fluid such as the ocean. The ocean is a medium, being bombarded by all sorts of energies such as thermal energy from the heat of the sun's rays, wind energy and tidal forces. By energy conservation, the ocean must do something with all the external energy it is absorbing from its environment. It disperses thermal and mechanical energy in two macroscopic modes: convection, in the form of ocean currents, and wave propagation. Certain conditions must be met for either mode to occur.

Energy transmission in the form of waves is a very efficient means of conveyance. The reason for this is illustrated as follows: the difference between convection and wave propagation is that waves don't carry the material along with them. Though the surface of the ocean appears to move, it is only the waveforms that are propagating--the water molecules simply oscillate in their stationary positions as the mechanical energy passes through them. A macroscopic illustration of this is the "wave" spectators sometimes make at large sporting gatherings--the spectators themselves certainly don't travel around the arena with the motion of the wave. An exception to the stationarity of the medium of the waves is the phenomenon of waves breaking along the shoreline. Here, there is fluid transport occurring along with wave propagation. This phenomenon is caused by the frictional forces of the increasingly shallow shoreline interacting with the waves propagating on the surface of the water. Seen from the wave's point of view, this frictional effect of the approaching ocean floor tugs at the fluid underneath it, literally causing the wave to trip over itself. This property is known as Kelvin-Helmholtz instability and arises whenever two media with different densities interface, causing frictional effects at the boundary.

Thus we see the obvious reason why one desires to refer to a medium for wave propagation, though, as we shall see later, quantum theory dispenses with this necessity. There is also a more subtle justification, involving the notion of action at a distance, which goes back to Newton and earlier. Newtonian mechanics at the time gave a complete mathematical description of the dynamic interactions among corpuscles or bodies, consistent with a Democratean world-view of corpuscles interacting in a void. In order to explain his dynamics, Newton postulated the existence of influences, or forces, that the bodies share among each other, a most notable one being universal gravitation. Such forces form the causal link in corpuscular dynamics, and behave according to the principles laid down in Newton's three laws. Newton's first law, the principle of inertia, states that a body at rest remains at rest unless acted upon by an external force. Similarly, a body traveling in a straight line will continue its motion at a constant speed unless acted upon by an external force. The second law states that the force acting on a body is equal to the mass of the body times the acceleration experienced by that body. The third law states that for every action or force applied, there is always an equal and opposite reaction, so that the mutual actions of two bodies on each other are equal and in opposite directions.

Students of physics today usually accept Newton's idea of force at face value, and do not understand that Newton originally distinguished between two kinds of force--innate and impressed or external. Innate force is proportional to the mass of the body and should be regarded as equivalent to the bodies inertia or "laziness," a direct property of matter. On the other hand, an impressed force is better thought of as external force acting on the body, not remaining in the body once the action is over. Gravitation is an example of an external force. One reason Newton distinguished two kinds of force centered on his strenuous attempts to rescue his concept of external force from the notion of action at a distance--that a body somehow feels the instantaneous influence of another body independent of their mutual distance. The idea of action at a distance implied somewhat of a troubling occult view of nature. In a certain sense, Newton was only partially successful in achieving this purpose, but the enormous empirical success of his formula for universal gravitation somewhat immunized him from the former criticisms regarding the all-too-disturbing similarity gravitation seems to share with action at a distance. Also in his favor, other philosophers argued at the same time that "action at a contact is not a whit more intelligible than action at a distance." The net result is that the debates regarding action at a distance in Newton's theory of gravitation have become academic. That is to say they are somewhat irrelevant from the standpoint of working physicists.

Upon some reflection, one will quickly realize that the notion of an aether for waves to propagate in likewise rescues electromagnetic theory from action at a distance. For just as in the case of Newton's laws of particle mechanics, whereby wave action is explained via the local interactions among the particles of the medium, likewise this idea was extended by Maxwell to cover electromagnetic wave phenomena. Without the aether, it becomes unclear how these waves can travel through otherwise empty space, any more than a wave set up in one bowl of water can travel through the air and suddenly appear in another bowl some distance away. Maxwell's peculiar coinage of the term aether came from Descartes' original definition of aether as imponderable matter. The reason why Maxwell considered this matter imponderable had to do with the disturbing issue that this matter was undetectable, and served no useful purpose save for acting as a necessary medium for his electromagnetic waves to propagate in. And of course, this medium was necessary in order to rescue his wave mechanics from action at a distance. Maxwell also felt, however, that this aether was subject to the same laws of dynamics as any other medium. As he wrote in A Dynamical Theory of Electromagnetic Action in 1865 concerning the aether:

However, what Maxwell meant by dynamics had to do with mathematical methods attributable to Lagrange, themselves extensions of Newton's, save for the fact that in Lagrangian dynamics an explicit knowledge of the mechanism of the system is not needed. In other words, Lagrange's methods enable one to bypass listing all the impressed forces on the system of interest, in this case the aether-wave system. Thus Maxwell was also spared from having to describe explicitly the dynamical inter-workings of the aether while still claiming its ability to transport his electromagnetic waves in the customary manner.

Towards the end of the nineteenth century, Maxwell's aether hypothesis was generally accepted. In fact, the famous physicist Oliver Heavyside went so far as to suggest that the existence of the aether should be accepted as primary, and matter itself should be shown to derive its own properties from those of the aether. This formed the beginning of the electromagnetic conception of matter, culminating eventually in the discovery of electrons. However, the temporary success of the aether hypothesis raised more basic questions; namely, ones concerning invariance in all frames of reference.

A reference frame is simply a coordinate system set up "with reference" to a particular physical situation. Reference frames can obviously be specified according to any configuration. As examples, the coordinate framework can be centered with respect to an observer, the Earth's surface, the sun's center, or the "realm of the fixed stars." In this sense, the roots of relativistic considerations originate in the work of Descartes and Galileo, the architects of the modern view of physics. The usage of the term "modern" is sometimes referred to in other contexts as twentieth-century physics. Yet, historians generally agree that the modern period began roughly during the early seventeenth century. In this respect, modern physics had its basis in this period, when a purely quantitative explanation of motion was initially developed. This new quantitative explanation implicitly assumed that nature could be mathematically specified to any arbitrary degree of precision. This viewpoint reached its culmination in eighteenth-century determinism, which held that if the position and momentum of all particles in the universe were known, then the future of the universe would become absolutely determined. This determinism was not seriously questioned until the advent of quantum physics during the first part of the twentieth century.

Regarding the fundamental similarity of different reference frames, Galileo first observed that whether one was at rest or in uniform motion with respect to one’s surroundings, the observer's local experience of nature's laws of motion should be invariant. In the Diagolo he writes:

The purpose for citing Galileo directly is two-fold. For one thing, this idea of invariance was considered radical at the time, and indeed can only be effectively understood should one ignore considerations of air-resistance. Only when seeking to rewrite the laws of particle motion in a vacuum will it become "intuitively obvious" that whether a particular physical situation is in uniform motion or at rest should have absolutely no effect on the laws of dynamics experienced locally within such a situation. Secondly, upon hearing the above description, the entire concept seems within the realm of what one would call common sense. Oftentimes, it is mentioned that "genius is the art of seeing the obvious," a gift of which Galileo was certainly well endowed.

At roughly the same period, Renee Descartes developed analytic geometry, which in a more basic sense enabled one for the first time to develop and use a coordinate system specifying one-on-one the spatial relations of any conceivable object with respect to the origin of the coordinate frame. Hence, today we usually refer to such coordinate systems as "Cartesian," after Descartes. Incorporating Descartes' mathematical innovation with Galileo's aforementioned principle of invariance enables one to speak of "reference frames" and "inertial reference frames." Informally, it is best to think of the latter as an ideally rigid frame obeying Newton's first law of motion or principle of inertia--a freely moving body will describe a straight line with constant velocity unless acted upon by an outside force. Hence, an inertial reference frame is a model that embodies the Galilean-Newtonian dynamical framework. Should one choose to specify the frame of reference as inertial, then it becomes possible to use Newton's ordinary laws of dynamics within it. Inertial reference frames can be either at rest or travel along in uniform motion with respect to a third coordinate frame, the absolute frame of reference or proper rest frame. For example, we can specify an absolute frame of reference as a coordinate system centered at a train station. We can then specify an inertial frame of reference moving relative to the absolute one as a train moving at a constant speed away from the station along a straight section of track. In addition, we can specify another inertial reference frame moving relative to the first as a passenger walking in a straight line at a uniform speed inside the train.

Though Galileo's and Descartes' writings preceded those of Newton, and Newton had to obviously draw on both contributions in order to develop his three laws of motion, we will combine Galileo's insights with Newton's laws in the Galilean-Newtonian Theory (GNT), a phrase coined by Carl Neumann. One must realize that Newton was considered radical for his time, for many of his contemporaries from the Cartesian tradition accused him of invoking quantities that in a strict sense were not spatio-temporal (able to be specified on a specific coordinate point with a specific associated time). For example, the notion of force can be expressed as a vector formula given by mass times acceleration. Although this relation contains geometrical properties that can be mapped onto a Cartesian coordinate frame, it actually describes the effect of the force, not the force itself. We shall see in chapter eight when considering the use of force to measure the mass of elementary particles how important this distinction actually becomes. Forces or influences are then seen as entities that affect the spatio-temporal trajectories of corpuscles, but are themselves not describable as localizable objects, and hence cannot be specified within a Cartesian coordinate system. These attacks, coupled with Newton's unsuccessful attempts to reconcile the issue of action at a distance, ironically gave Newtonian mechanics an aura of being excessively metaphysical, vitalistic and occultish among some of his detractors. Who though, when faced with the concepts of Einstein's relativity, would fail to embrace Newton's ideas as the last vestige of concrete, common sense physics?

The Galilean-Newtonian theory, or GNT, describes a dynamical universe, capable of being mapped onto some imaginable absolute coordinate frame, hence possessing an absolute center. This proper rest frame would span a potentially infinite class of inertial reference frames that are themselves mathematically connected to one another, and to the absolute frame, via Galilean coordinate transformations. As easy as it is to visualize, this notion of an identifiable absolute center or absolute rest frame to the universe is actually paradoxical in the scope of GNT, as it comes in conflict with Newton's first law--the principle of inertia. We see this by the following argument. GNT holds that dynamical laws are invariant with respect to Galilean transformations between inertial reference frames. Hence, there is no way to detect absolute motion from any inertial frame. The reason is that, assuming an absolute frame were to exist, from the observer's point of view it would be identical to any other inertial reference frame. Since there is fundamentally no difference in the way the physical laws would behave in such an absolute frame, and from first law considerations, there would be no observable qualities for detecting one's motion with respect to this frame. Any inertial frame could therefore declare itself to be the "proper" rest frame, and there would be no mechanism to distinguish the "true" rest frame from the impostor. Thus, even though the concepts of inertial reference frames related by Galilean transformations to each other is very useful and easily understood, as soon as we try to invoke an absolutely stationary reference frame against which all others can be measured, we run into problems. Such is the fate of the aether concept as well, as we shall see.

An important point regarding Galilean coordinate transformations addressed in the first chapter is the fact that time remains unchanged when moving from one inertial frame to another. This forms the concept of inertial time, the basis of simultaneity, which enables one to form an absolute and universal time scale in GNT. Hence, GNT implies that the state of the universe is described by the set of all simultaneous events at a given instant of time. It is this notion of absolute simultaneity across all space that enables a global synchronization of clocks to take place in GNT. Einstein, on the other hand, proposed that time is nothing more than the observable measure of changes in the universe. In this respect, a clock running slow, say in the presence of a strong gravitational field, would be marking the actual slower passage of time in that arena--because the observable clock has slowed, time itself has slowed. In this respect, it becomes impossible to speak meaningfully of simultaneous events, and thus, no absolute time scale exists.

The detailed exposition regarding inertial reference frames and their unique place in GNT is necessary in order to portray the context of the types of questions that inevitably arose regarding Maxwell's aether. As mentioned, Maxwell's aether rescued his electromagnetic theory by enabling it to be subject to the general laws of dynamics, yet, at the same time, the aether itself was spared from mechanical description. The main question then subsequently centered on what kind of preferred reference frame, if any, the aether resided in, and more generally, how did the notion of the aether pan out regarding Galilean-Newtonian concepts of absolute and inertial reference frames? For some reason, such questions most preoccupied the Continental physicists of the latter half of the nineteenth century, with the work of Hertz and Lorentz standing out in particular.

Hertz was most concerned with theoretical questions pertaining to the nature of the aether, who among other things demonstrated that ponderable matter and the aether must be completely de-coupled, to the point where the aether permeates unaffected through all matter. From the outset, this may seem contradictory in the sense that two forms of matter occupy the same locality in space. However, Hertz was able to show via a narrow interpretation of electromagnetic theory that such paradoxical considerations can be avoided, at least to the extent of performing the appropriate electromagnetic calculations. Seemingly contradicting himself, Hertz then went on to choose the view that the aether was somehow "dragged along" by ponderable matter. Even so, it was the work of Hertz that influenced Lorentz to try to reconcile Galilean-Newtonian notions of absolute and inertial reference frames with the aether, in ways going beyond purely ad-hoc considerations. This meant, among other things, that for Lorentz the microscopic structure of matter must be taken into consideration. In fact, it was Lorentz who first postulated the existence of charged particles swimming around matter interatomically, particles that he dubbed ions and electrons.

The principle question surrounding the issue of the relationship between GNT and the aether had to do with seeking to find an appropriate reference frame for the aether to reside in, which can be restated as the issue of finding a preferred or proper reference frame for Maxwell's equations. There were two principle reasons for asking such a question. First and foremost was the issue of the Galilean invariance of Maxwell's equations. Invariance means that Maxwell's equations transform between coordinate systems in such a manner that the basic form of the equations remains the same before and after the transformation. Accepting, as Maxwell did, the finite speed of electromagnetic action to be the value c, and at the same time writing out his equations for a given inertial reference frame, one quickly discovers that the equations do not invariantly transform between reference frames when utilizing Galilean transformations. In other words, unlike the Newtonian equations of motion, Maxwell's equations are not Galilean invariant when one assumes a constant value of c for the speed of electromagnetic propagation. As seen in the first chapter, however, revising the constraint of absolute constancy of the speed of electromagnetic action for all inertial reference frames to read that all such frames register locally an electromagnetic action traveling at c--the modified second postulate--one can demonstrate Galilean invariance in Maxwell's equations.

Maxwell was able to overcome this defect of non-Galilean invariance by postulating that in all experimentation, electromagnetic phenomena are dependent on two separate items. These are the experimental apparatus--the behavior of the ponderable bodies, and the motion of the experimental set up through the aether--the motion of the observer. The second issue naturally follows from the first, in that if the motion of the apparatus must be taken into consideration, the question concerning what reference frame in which to situate the aether naturally arises. For Maxwell, the issue was not altogether too crucial, for he pointed to the experimental success of his equations as indicating that although they were not Galilean invariant between terrestrial inertial reference frames, the expected error arising from applying the equations in a reference frame moving slowly relative to the aether had been thus far experimentally undetectable. So for Maxwell, the error in transforming between the aether's reference frame to an approximately inertial terrestrial frame was too minuscule to cause noticeable error in writing the equations with respect to the latter.

But the experiments to detect motion with respect to the aether got better. Numerously complex theories in optics had been already put forth by physicists in the early nineteenth century in order to explain observed stellar aberration first discovered by James Bradley in 1728. Stellar aberration is an effect whereby a star's position appears displaced at an angle due to the earth's motion. One theory of interest is that of Arago put forth in 1853 which held that light corpuscles were emitted at different speeds, yet the eye is only able to detect those propagating at c, a notion similar to the modified second postulate. However, towards the latter half of the nineteenth century it was generally accepted that Fresnel's notion of the aether being partially dragged was correct, at least as an operational principle, in explaining stellar aberration. In optics, the term aether was adopted as a device to explain the medium in which light propagated. Fresnel even developed equations showing the drag coefficient for a medium moving at some velocity relative to the aether to be directly proportional to that velocity. Then came the Michelson-Morley experiments.

In a series of experiments performed by Albert Michelson and Edward Morley in 1886-7, conflicting results emerged. Michelson and Morley used an interferometer, a device that causes two beams of light traveling perpendicular to each other to create interference patterns. By carefully tuning this device, one could supposedly measure very small velocities along one path of the device with respect to the aether by a change in the pattern obtained. The device was sensitive enough to detect the motion of the earth, which travels around the sun at roughly thirty kilometers per second and changes direction every six months. By performing the experiment with several equipment orientations and over several seasons, they hoped to determine the motion of the earth with respect to the proper rest frame of the aether. The first experiment supposedly provided a direct confirmation of Fresnel's formula for aether drag. In 1887, however, the experiment was repeated, using an improved interferometer incorporating additional modifications suggested by Lorentz, and the experiment rendered null results--there was no change in the interference patterns obtained. These results indicated that there was apparently no relative motion between the Earth and the aether. Since the Earth was clearly not stationary within the aether, the entire concept was suddenly in need of help.

Lorentz sought to reconcile Michelson and Morley's null results with the former issue pertaining to a preferred reference frame for the aether, from a microphysical basis. Two important points need prior clarification, concerning Lorentz's viewpoint. First of all, Lorentz did not try to identify the aether's rest frame with the paradoxical concept of absolute space in GNT. For Lorentz, an aether frame "at rest" meant simply a medium in which no part is in relative motion with itself, and a medium against which the motion of the celestial bodies can be measured. Note that Lorentz specifies "motion" of the heavenly bodies relative to the aether, which is different from saying absolute motion of the heavenly bodies relative to some absolute center. The statements only coincide when one makes the prior assumption that absolute space exists, and that the aether's frame coincides with absolute space. Secondly, Lorentz was able to write Maxwell's equations for individual electrons such that at each point their fields induce on matter in a microscopic arena a "Lorentz force" per unit volume. This force was then related to the charge densities and the relative velocities of the particles. However, since electrons don't interact instantaneously with one another (no action at a distance), and the aether itself was postulated as not subject to ponderable forces, the Lorentz theory apparently violated Newton's third law of equal and opposite reactions. Lorentz responded to this criticism by showing that the macroscopic Maxwell equations could be recovered using appropriate integrals, and additionally that his microscopic Lorentz force was derivable via variational principles. Hence he concluded that Newton's third law is a macroscopically derived phenomenon, arising statistically from microscopic interactions not subject to such a principle. This is much the same view as is held today in quantum mechanics, whereby individual particles have no definite position and momentum, yet macroscopic objects made up of millions of such particles do.

These macroscopic and microscopic considerations led Lorentz to consider possible variants of Galilean transformations, regarding each individual inertial reference frame of each electron. After some tinkering, Lorentz concluded that the earth's relative motion with respect to the aether would not be detectable to first-order approximations, providing that a certain set of transformations of time and length related to the square of the velocity were invoked. This set of manipulations carries the name Lorentz transformations. It is from these transformations which the term gamma, g, arises, denoting the degree by which time slows and length contracts with velocity. This term is discussed in detail in the chapters dealing with the subjects of time dilation and length contraction.

These transformations were at first to Lorentz simply mathematical devices, intended to show to a first-order approximation how relative motion between the Earth and the aether frame could not be discerned. In his paper Inquiry into Electrical and Optical Phenomena in Moving Bodies in 1895, he generalized them in his so-called Theorem of Corresponding States. This theorem introduced the concept of local time, which varied with the velocity from proper time. The concept of local time described in the theorem was not given a physical interpretation by Lorentz, who saw it to be merely "an auxiliary mathematical quantity." Note also that in our prior descriptions of GNT, whereas absolute space is a contradictory notion, inertial time, implying an absolute time scale, is not. On the other hand, at a loss to explain Michelson and Morley's null results on their interferometer readings, and borrowing a notion suggested already by Fitzgerald in 1889, Lorentz chose to give a physical interpretation to the velocity dependence of length. The effect was subsequently described as "Lorentz-Fitzgerald length contraction." Lorentz-Fitzgerald contraction predicted that the length of a rod oriented perpendicular to the direction of the Earth's motion is contracted by the g factor when suddenly rotated ninety degrees so as to become aligned parallel to this motion. From a microphysical standpoint, Lorentz showed this to be somewhat of a plausible, if in-testable, notion. In his Inquiry paper he was able to show that for a frame moving through the aether, the velocity-squared relation would skew the appropriate intermolecular forces. When comparing this frame to one moving perpendicular to the aether, integrating the effects of this force over space would produce the macroscopically predicted Lorentz-Fitzgerald contraction in length.

Lorentz admitted that such effects would probably not be exactly confirmed, since the theory assumes the molecules are at rest in their respective inertial frames of reference, whereas in actuality they oscillate about their positions of equilibrium. Thus at any point in time, some molecules would have a greater velocity, while others might be moving perpendicular to or even opposite the direction of the macroscopic body. The length contraction on each of these molecules would thus be different. Nevertheless, Lorentz was able to show that such "smearing" effects due to these random motions were of such negligible size as to be incorporated into the limits of normal experimental error. In 1899 Lorentz published a second paper, the Simplified Theory of Electrical and Optical Phenomena in Moving Systems, in which he extended his theory of corresponding states to effects of second order in terms of velocity with respect to the aether. Using a somewhat complicated approach, he introduced yet another series of transformations. At this time, however, he fully incorporated his previous notion of the intermolecular skewing of force (Lorentz-Fitzgerald contraction), arguing that "as soon as translation is given to the system, the transformation really does take place, of itself, i.e. by the action of the forces acting between the particles of the system, and of the aether." Despite this treatment of length contraction, no systematic treatment is given to the role of local time and its relationship to any physical time-keeping devices, or perhaps time itself. However, Lorentz states: "It is clear that the Theorem of Corresponding States cannot be experimentally significant unless the local time bears a definite relation to the physical time keeping processes observed in the laboratory." It is again necessary to stress that much of the development of this theory lies in the frame invariance of the observed velocity of propagation of electromagnetic radiation, and the assumption that this observed characteristic represents the true nature of that propagation itself. This correlation between observed phenomena and the actual nature of electromagnetic radiation was assumed true even before Einstein framed his second postulate concerning the same.

The theory of Lorentz may on the outset appear complicated and obscure. Even so, Einstein drew heavily on this work when he formulated his special theory of relativity. Despite the alienating notions of length contraction and observer dependent local times, Lorentz's insistence to explain the phenomenological facts of electromagnetism from a microphysical standpoint led to more basic experimental research into the structure of the atom. Lorentz's electrons were confirmed experimentally in the works of Rutheford, Millikan, J.J. Thompson and others, leading to the early development of atomic theory. In short, in certain ways, Lorentz's contributions may have been as significant as Maxwell's in terms of their eventual influence in the later development of physics, even aside from the considerations of relativity theory.

Lorentz introduced his transformations in order to allow overall Galilean transformations from the reference frame of the aether to any other inertial reference frame. In a paper published in September 1905 in Annalen der Physik, Albert Einstein suggested a new interpretation of the work of Lorentz regarding the significance of his transformations, which essentially bypassed the entire aether hypothesis altogether. Einstein's paper outlined and established the foundations of his theory of special relativity.

The impetus for his intuitive leaps and groundbreaking speculation actually lay in a previous paper, published in 1902. In this paper, Einstein expanded Planck's "quantal" interpretation of electromagnetic radiation and implied even more fundamentally that classical mechanics was, except in extreme cases, only an approximation to the actual dynamical behavior of bodies. This paper formed the conceptual foundations of the later "quantum" theory of microscopic matter, one from which ironically Einstein always kept himself at arm's length. The Einstein-Bohr debates are perhaps the most significantly philosophical debates regarding the interpretation of nature since the Newton-Liebnitz debates. Basically, Einstein differed with Bohr on the interpretation of quantum theory. Bohr believed that aspects such as Heisenberg's uncertainty principle reveal fundamental limits to our understanding of nature, whereas Einstein believed that a completed theoretical structure should possess the determinacy and precision of the prior physical theories of classical mechanics and electromagnetic theory. To suggest anything less was for Einstein nothing less than to undermine the very heart of the physicist's enterprise. In a similar sense, Einstein speculated that classical electromagnetic theory was not exactly valid either. Yet he could not deny the enormous predictive success of Maxwell's equations, both for macroscopic and for microscopic matter. Rather than taking Maxwell's electromagnetic theory at face value the way Lorentz did, and further speculate on aether-molecular interactions, Einstein instead sought some kind of a universal principle enabling Maxwell's equations to be operationally correct, without having to accept a priori and literally their implications regarding the nature of matter.

Einstein was able to formulate such a universal principle by postulating two claims, the relativity principle and the light principle, sometimes referred to as the first and second postulates of special relativity. Poincare described the relativity principle as stating that the laws of nature and the results of all experiments performed in a given inertial reference frame are independent of the translational motion of the system. In other words, there exists an infinite set of inertial reference frames moving relative to one another in which all physical phenomena occur in an identical way. Note how Poincare's relativity principle coincides with Galilean-Newtonian theory, if one neglects the additional postulate Newton made regarding the existence of absolute space. Actually, Einstein's relativity principle was more specifically worded and was subsequently more restrictive than was Poincare's. Einstein explicitly mentions that the inertial reference frames are connected by the Lorentz transformations. However, Einstein's relativity principle anticipates his second postulate--the constancy of the speed of light. Thus, relaxing the constancy of this quantity enables one to remove the constraint of Lorentz transformations and recover the acceptability of Galilean transformations. Perhaps more fundamentally, Poincare's statement is more correct in its generality, for there is no a priori reason to postulate that the inertial reference frames are connected by any special set of transformations such as Lorentz's unless one first says something about the nature of light. In this respect, neither Einstein's first nor his second principle are general enough to be given the status of fundamental postulates. The second depends on observational results, and the first is worded in such a way as to anticipate the second and limit the possible classes of transformations thus available.

The second postulate states that the speed of light is independent of the motion of its source. This is Poincare's paraphrase of Einstein's light principle that states that the speed of light is constant in vacuo and independent of the motion of its source. Note also that the modified light principle of RCM theory differs little from the second postulate, save for specifying that the observed speed of light is independent of the motion of its source, and the corresponding implication of a spread of velocities of light propagation. Einstein's proposal of the second postulate, implying the constancy of the speed of light in all frames, was originally suggested by him to indicate that these two postulates "are all that are needed" to develop a full theory of electrodynamics for bodies at rest, consistent with Maxwell's equations. Hence the concept of the aether could be dropped. His conviction of the light principle only solidified when he tried to develop the equations of motion of a ballistic model for the emission of light, but was unable to come up with any solution representing a wave with a velocity depending on the motion of the source. Einstein's light principle was in actuality an extraordinary gamble for him, since at the same time he was rejecting the enormously successful Maxwellian wave-aether theory of light.

On the notion of the significance of time, in Einstein's special theory there is no correlate of absolute time as there is in GNT, due to the results of the Lorentz transformations which are themselves derived out of the second postulate. Contrary to Lorentz, for Einstein the combination of the first and second postulates forced the issue of a physical interpretation for both length and time contraction. However, debate still lingers concerning the interpretation of time in the special theory of relativity. Basically, there are some writers who maintain that since every inertial reference frame is endowed with a local time scale, based purely on the relative motion it shares with other inertial frames, it becomes meaningless to partition events into simultaneity classes. Hence talk of an absolute time-scale is a matter of pure convention. Others interpret this notion of a local time scale as an indication that there actually are infinitely many partitions of simultaneity classes, one for each reference frame. Einstein, however, believed that the primary result of special relativity was that there is no simultaneity for distant events. This point will be discussed exhaustively in subsequent chapters.

Einstein's theory of special relativity revised space-time considerations that were formerly fundamental in physics and in mathematics. In mathematics, special relativity spurred the development of new techniques in space-time geometry in the work of Minkowski. Minkowski developed the algebra for a four-dimensional space, under the assumption that a quantity related to the sum of the squares of three spatial dimensions plus time is invariant for all inertial frames of reference. Similar relations were derived for velocity, momentum, energy and other quantities. The relations for kinetic energy and total energy of a system were derived explicitly working in Minkowski space-time and applying the usual principles of conservation of momentum and energy during collisions attributable to Newton. These two relations differed from the classically expected results by implying that there is a residual energy of all particles, preventing their energy from ever reaching zero. The oft-mentioned relativistic relationship for total energy links this residual term with the inherent energy of the particle system. This relation also contains the assumed relativistic mass increase due to velocity that supposedly increases to infinity when a particle is accelerated to a speed of c. This total energy expression, which serves as a basic assumption in all nuclear reactions, also led Einstein to speculate that the inertial mass of a particle represents somehow a form of "compressed energy" when he formulated his theory of gravitation in the general theory of relativity. This equivalence of mass and energy is also put forth by RCM theory, but for different reasons, as we will see in chapter eight. Einstein's theory of general relativity, is notoriously technical, requiring a mathematical grounding in tensor analysis and differential geometry, should one wish to follow its every implication. Hence, we will give only a qualitative summary into the basic conceptual inquiries that led him to develop the theory.

The foundations of Einstein's theory of gravitation lay in the work of Lorentz and Poincare. The essential concerns of the latter dealt with the unsettling notion of instantaneous action at a distance, implicit in Newton's theory of gravitation. Due to their prior work in electrodynamics, the common knowledge of the finite velocity for the seemingly instantaneous propagation of electromagnetic action led to speculation whether c also represented a limiting value in the case of gravitational propagation. Einstein, however, appreciating the success of Newton, sought to develop a general field theory of gravitation, whose ideal limit would result in Newtonian field theory. Based on the work of Poisson, one can rewrite Newton's law of gravitation in field form, in the same manner as Maxwell's field equations were developed from Coulomb's and Ampere's force laws. Inevitably, based on suggestions made by Planck, Einstein had to analyze Newton's separate ideas of inertial mass and gravitational mass. Based on what was known concerning mass loss in radioactive decay, and on the experimental fact that one must "weigh" something in order to find its mass, Einstein concluded that the inertial and gravitational mass of a system were exactly proportional at all times. This became the impetus for his principle of equivalence.

The central thought experiments performed by Einstein, leading him to develop the principle of equivalence, concerned uniformly accelerating reference frames, such as the kind experienced by a hypothetical free-falling elevator in a uniform gravitational field. Recall that our previous discussions on Newtonian and special relativity centered on the behavior of physical laws in inertial reference frames--reference frames differing from one another by a constant velocity only. Quite succinctly the principle of equivalence says that the relativity principle, or the first postulate, can be extended to uniformly accelerated frames as well. The previously derived proportionality of inertial mass with respect to gravitational mass was extended to the equivalence principle. This principle concludes that an experiment performed with ponderable masses should yield identical results for a frame accelerated uniformly and for a frame at rest in a uniform gravitational field with an identical constant of acceleration. Sometimes this notion is subdivided into a "weak" and "strong" relativity principle. The weak principle holds that the inertial mass and gravitational mass are proportional, and the strong principle asserts the aforementioned equivalence in experimental results for uniformly accelerating frames and for rest frames subject to an equivalent uniform gravitational field. However, it was the strong principle that provided the basis for Einstein's most daring speculations in his theory of general relativity. As mentioned by Roberto Torretti in Relativity and Geometry, "No analog of the Michelson-Morley experiment was at hand to support the extension of the equivalence principle to radiation, but...Einstein was content to point out that there was no evidence against it."

In a certain sense it is misleading to think of the equivalence principle as having "generalized" the relativity principle, for really the equivalence principle does not eliminate the physical difference between inertial frames and uniformly accelerated frames, or frames in a constant gravitational field. In fact, the equivalence principle entails that a freely falling frame by definition cannot be inertial. To this extent, the equivalence principle weakened the applicability of the laws of special relativity to local regions in the universe, where gravitational fields can be assumed as uniform. More precisely, in a space-time region around an event where the gravitational field is uniform, no effect of the gravity can be discerned within a given margin of precision. Hence the laws of nature in such a region will be invariant to those laws in an equivalent frame for a space-time region where gravity is absent. RCM theory exploits this notion in determining the gravitational potential experienced by a moving observer in chapter seven, and uses this analysis to predict the change over time in the perihelion of mercury's orbit and the deflection of starlight by the sun. In the above sense, general relativity drastically reduces the range of applicability of physical law, regarding equivalence only in terms of measurable experimental results. This does not, on the other hand, necessarily imply that there are regions in the universe where natural laws do not hold, but that under general relativity, the laws, rather than encompassing the entire universe, encompass only local fragments of it at a time. Thus we see that the second postulate, which forced the adoption of Lorentz transformations between reference frames, has now gone on to limit the applicability of those transformations to very small regions of consideration. Under general relativity we cannot uniformly apply the laws of physics with the same results across reference frames separated by vast reaches of space. In the local areas where we can, we are unable to agree on the simultaneity of events, the length of a ruler or the passage of time. Things appear to be getting more confusing with each new aspect.

To Einstein's credit, his general theory of relativity functioned as a grand theoretical framework in which to place, like a mosaic, all the local regions of uniform gravitation, where in a local and approximate sense, special relativity could apply. In other words, general relativity gives one the picture of space-time which can be thought of as a full collection of Minkowskian regions where special relativity applies to an approximate degree of precision, much the way a surface is approximated by a flat-faced polyhedron. In this sense, gravity warps such a Minkowskian mosaic the way the curvature of a surface would warp the polyhedron, when mapped upon it. Though this metaphor may sound strange and outlandish, it has its original inspiration in the work of the mathematician Riemann. Riemann had argued already in 1853 that the laws of geometry cannot be derived a priori from spatial considerations alone, hence geometry itself must be an applied science of spatial relations derived from the nature of the microscopic forces holding rigid rods together.

In summary, we will borrow a thought experiment from Einstein to illustrate how special and general relativity neatly fit together according to their respective postulates, the latter derived out of the former. The central illustration of the thought experiment is, not surprisingly, light. We will also use Einstein's illustration as a means to show how his postulates, namely the second postulate, may be interpreted in a different manner, as is done in RCM theory, leading of course to vastly different theoretical structures.

Einstein's thought experiment focuses on the behavior of light in a uniform gravitational field. Arguing from the equivalence principle, he shows that the radiation of a body must be affected in such a field, if the weight of the body emitting the radiation is truly proportional to its inertial mass. From special relativity it was shown that the inertial mass of a body decreases or increases according to the total factor E = mc², based on an equivalent amount of energy radiated or absorbed. Imagine two points in space, with one point at a position higher than the other in a uniform gravitational field. Let the mass emitting the radiation be stationed at the lower point, with the radiation moving from the lower to the higher point. As mentioned above, this radiation must be affected by the uniform gravitational field in a manner equivalent to the inertial mass lost by the emitter. If this were not so, energy conservation would become violated. The reason for this is that the radiation would be lifted "at no cost" between the two points, and, by returning this energy from the higher to the lower point, an additional amount of energy would be returned, stored in the form of a falling body. Thus, the total energy of this cycle would be greater than the emitted radiation by a factor proportional to the distance between the two points. Hence, to satisfy energy conservation, the absorbed energy at the high point must be less than the energy emitted below. As shown in chapter eight, due to the contributions of Max Planck, the lost energy shows up in the form of a lower frequency being absorbed at the high point than was emitted from the mass.

Einstein concluded from this analysis that gravitational fields affect the local time of observers placed in that field, since an observer at the higher point records a periodic process, in this case the frequency of the radiation, differently than an observer at the low point. In other words, the number of "ticks" registered by one observer per unit interval of "proper" time being less than those "ticks" recorded by the other observer imply that the lower observer's time unit itself is longer than the other's. Hence, clocks must slow down as the gravitational potential increases, and gravitation is thus linked with the local time of an observer. At the same time, the second postulate in special relativity says that the speed of light is constant in an ideally gravity-free environment. But the speed of light is measured by the local time of an observer in the vicinity of the light itself. If the observer's local time is effected, then so is the speed of light, and the second postulate does not strictly hold in the presence of a gravitational field. In other words, special relativity applies only to the "Utopian" case in which no gravitational fields are present. It was this type of thought experiment that provided the motivation for Einstein's theories of gravitation, which later became known as the general theory of relativity. This line of reasoning allowed Einstein to account for the deflection of light in a gravitational field, as well as its slowing, or a change in the value of c, in the same field.

It is perhaps also fitting here to highlight the different approaches regarding this same thought experiment as handled by general relativity and RCM theory. After some reflection, one realizes that the conclusions made by Einstein in his thought experiment concerning the photon in a uniform gravitational field are derived from the assumptions made by special relativity. This is despite the fact that Einstein shows in this thought experiment that special relativity is correct only in an ideal sense. The first and second postulates give rise to the theory of special relativity, which, among other things concludes that time is "local" in each inertial reference frame, and that this time in general differs from what one might call universal "proper" time. These conclusions concerning the nature of time and the ideally constant speed of light forced Einstein to conclude in his thought experiment that the gravitational field induces different local times for clocks at different locations within the field. As subsequent sections show, RCM also agrees with Einstein that the inertial mass of a body is equivalent to its radiated energy, and that a photon is affected by a uniform gravitational field. However, the RCM model of light gives implicitly a modified interpretation of the second postulate, one that is much less restrictive, as already shown in chapter one. Light radiated under this modified postulate will also lose energy as it climbs through a gravitational field. Additionally, though time-dilation is dealt with explicitly in RCM theory, this measured effect is due to issues not resembling those predicted by either special relativity nor general relativity.

In short, it is important to realize that although three postulates are used to develop the theories of relativity--the relativity, light and equivalence principles--it is by far the second postulate, the light principle, that is most significant, and that renders coherent the special and general theories. As has already been shown, the Lorentz transformations of special relativity arise purely out of the second postulate. By the same token, relaxing the second postulate unravels the certainty of the conclusions posed by general and special relativity. When focusing on a theory, it is best to start by focusing on its axioms. It is clear that the special and general theories are an interpretation of the results of physical and thought experiments of the late nineteenth and early twentieth centuries. However, they are not the only interpretation. It is Einstein's ad-hoc restriction limiting electromagnetic propagation to the observed speed of light in his second postulate, and the subsequent limiting of reference frame transformations to those of the type of Lorentz's that requires the complex restructuring of space and time in the special and general theories. RCM theory returns to the basic axioms of the relativity principle, the modified or relaxed light principle and the equivalence principle, and is built from the ground up in the same manner as Einstein's theories. However, RCM theory maintains what feels right and intuitive about Galilean-Newtonian physics rather than overturning all common sense notions of space, time and simultaneity.

David Bohm, 1965

What is meant by the idea of simultaneous events? The most straightforward and intuitive definition would be that two events are simultaneous if they occur at the same time. Implied in this definition are the ideas of absolute space and absolute time, dating formally to the writings of Newton and conceptually to even earlier times. Under this definition, all events throughout the entire universe occurring at a specific instant of time would be considered simultaneous. The relative distances to these events are not important. Nor does the amount of time required to send a signal telling of the occurrence of an event affect its simultaneous nature. For example, consider a child born on February 26, 1993 at 2:00 p.m. EST in San Diego, California. The parents are preoccupied with the birth of their new child, and pay no attention to world events for a few days. On February 28th, the father reads that, at the moment his daughter was born, explosions rocked the World Trade Center building in New York. These two events were separated by 3000 miles, and it took two days for the "signal" from one event to reach the location of the other. However, it is clear to any outside observer that the events were simultaneous.

Not so according to the special theory of relativity. Einstein's second postulate leads to the impossibility of assigning common times and distances to widely separated events. Minkowski declared in an address in 1908 that "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." The problem in relativity theory lies in the inability to accurately synchronize clocks in different moving frames of reference. While there was no one in motion in the above example, we can imagine a vacationer flying en route to New York at the time of the birth and the explosion. In order for this observer to record the actual time of the birth and explosion, a radio or light signal from each would have to reach the plane signaling their occurrence. It is the propagation of this signal under the rules of the second postulate that causes the problems. To illustrate this explicitly, we will use Einstein's classic example of a passenger on a train that moves at a good fraction of the speed of light, and an embankment on which stationary observers may sit and watch.

Suppose in figure 3-1 that two gunpowder explosions, P and Q, occur simultaneously on the banks of the railroad tracks. That is to say that a stationary observer, Alice, seated on the bank and located midway between the two events observes the two flashes of light from the explosions at the same time. If Alice has a very long rod that stretches in both directions beyond events P and Q, and if the rod is close enough to the events that the explosions will leave marks, then Alice will be able to measure the distance to each event after it has occurred by marking off the distance from her location to each of the marks. We will define a distance of one light second as 300,000 km; the distance light travels in one second at the speed of c. We will also set up the experiment such that the distance Alice measures on her rods is six light seconds. Therefore Alice concludes that the events occurred simultaneously, six seconds before she saw the flashes.

Figure 3-1 Simultaneity in the special theory of relativity: Alice and Bob will each see a flash of light at the same place and at the same time.

In the same figure, let us look at Bob, who is traveling on a train moving at a velocity of 0.5c. Further, let's schedule his arrival at Alice's location to coincide with the time at which she observes the two flashes. According to the special theory of relativity, Bob will also see both flashes at the same time. However, since Bob has moved three light seconds in the six seconds since the explosions, he was initially three light seconds closer to explosion P and three light seconds further away from explosion Q when they occurred. Since Bob considers himself to be at rest, he does not take his movement into account when recording the distance to each of the explosions. Thus he measures on his rods a distance of three light seconds to explosion P and nine light seconds to explosion Q. From this he calculates that explosion P occurred three seconds ago and that explosion Q occurred nine seconds ago, since the light he sees traveled these distances at a speed of c in his frame of reference. According to Bob, the two events were not simultaneous. As we stated in chapter one, according to the special theory of relativity, Alice and Bob will see the event at the same time and at the same place, but will be unable to agree on a common distance to or time since the events, thus rendering the concept of simultaneity meaningless. As if this is not confusing enough, the lengths measured on each rod to the explosion marks will not be as was just indicated. In addition to the time of the events not being simultaneous, the lengths of each observer’s rods, as seen by the other observer, will be shortened, as we will see in detail later.

Now let us view the same experiment in terms of the RCM theory, as depicted in figure 3-2. The situation for Alice remains unchanged. She will observe the flashes six seconds after the explosions occur, and be left with marks on her rods indicating a distance of six light seconds to each of the events. As for Bob, we will schedule his trip slightly differently than in the previous example. Let us schedule his arrival at Alice's location to be four seconds after the explosions (an odd choice, the reason for which will be clear as the example progresses). Since Bob is traveling at 0.5c, he is sensitive to that component of light that is leaving explosion P at a velocity of 1.5c. This component will cover the six light seconds from point P to Alice in four seconds, and arrive just in time for Bob to see the flash. Since Bob has been traveling at 0.5c for four seconds since the explosion, then he was two light seconds closer to point P when the explosion occurred. Thus the mark on his rod will indicate that the explosion was four light seconds away. Once again, since Bob is unaware of his motion, he assumes that the light which he sees traveled this distance at c, and calculates that the explosion must have occurred four seconds earlier, which, of course, it did. We now see that Alice and Bob agree as to the time of explosion P, something they can't do under special relativity, but they have yet to agree on the distance to the explosion. We will address this shortly. For now though we see graphically that two observers in motion relative to each other may see the light from an event at the same place but at different times. The case of explosion Q is a little different.

Figure 3-2 Simultaneity in the Radiation Continuum Model: Alice and Bob will see the flashes at different times and locations.

We know from the preceding paragraph that at the instant of the explosions, Bob was two light seconds further away from Q than was Alice. Thus the mark on Bob's rod indicates a distance of eight light seconds to Q at the time of the explosion. Since Bob is moving toward Q, the component of light that he can see must be leaving Q with a velocity of 0.5c. With Bob and the light each traveling at 0.5c, the light will only need to travel four light seconds from Q before it and Bob are at the same location and it strikes him at a relative velocity of c. It will take this component of light eight seconds to cover a distance of four light seconds at 0.5c. Thus Bob will see the flash from Q eight seconds after the explosion occurs. Since, from the mark on his rod, he assumes the explosion occurred at a distance of eight light seconds, he concludes that the explosion must have occurred eight seconds before he saw the flash, as of course it did. This time, Alice and Bob see the light at different locations and at different times, but they once again agree as to the time of the explosion. More importantly, they each agree that explosions P and Q occurred simultaneously. If we could now get them to agree as to the location of the explosions, then the concept of simultaneity as Newton defined it would be complete.

Let's begin by assuming that Bob knows he is moving with respect to Alice and the explosions. In the case of explosion P, then, Bob knows that he has been traveling for four seconds since the explosion and has covered a distance of two light seconds by the time he reaches Alice's position. He must add this to the length of the mark on his rod, four light seconds, to get the actual location of the explosion, which is six light seconds from Alice. So far so good. Regarding explosion Q, Bob has been traveling for eight seconds, or four light seconds at his speed of 0.5c. He must subtract this value from the length of the mark on his rod, since he has been traveling toward the explosion, and concludes that the event occurred a distance of four light seconds from his current location. Since he was initially two light seconds in front of Alice and has traveled four light seconds, he must now be two light seconds past Alice. Therefore, explosion Q must have occurred a distance of six light seconds from Alice. This is, of course, the correct result.

Thus, in this example, both Alice and Bob deduced that events P and Q occurred at the same time, each at a distance of six light seconds from Alice. The important point in the above example can be seen by considering just the observations of explosion P. In RCM theory, Alice and Bob see the explosion at the same place but at different times, though they are able to agree on the location of and time since the event. In classic problems addressing the question of simultaneity in Lorentz-Einstein relativity, it is assumed that observers in motion relative to each other observe the event at the same place and time, thus leading to their inability to agree on the time and location of the event, and to the concept of relative space and time. This effect is a direct result of the second postulate, and exists only in thought experiments such as the one above. There has never been any direct experimental confirmation of this concept. One reason for this is that until very recently, the technology to perform such a test has not existed. Another and more compelling reason is that no one, until now, has ever questioned the validity of the statement. David Bohm, in describing the relativistic train experiment in his book The Special Theory Of Relativity, states "Of course [Bob] will also see the two flashes at the same time." He also goes on to address the nature of the second postulate being tied to physical measurements, even though it is treated as an absolute condition: "In fact, however, all observers must assign the same speed to light, since as we have seen, experiments show this to be the case. Therefore the train observer can no longer agree that the two flashes are simultaneous, because they cover different distances at the same speed." In his book Einstein's Theory of Relativity, Max Born repeats two statements, one from Newton's theories, the other from Einstein's:

He then makes the assertion that "Of the two statements (1) and (2), the first is purely theoretical and conceptual in character whereas the second is founded on fact." This is an interesting conclusion. The first statement is based on hundreds of years of direct experimental verification, albeit for relatively slow moving objects. The second statement is based on absolute conjecture, and has not been tested by experiment. The experiment which resulted in the statement, that of Michelson and Morley, demonstrated that the speed of light is dependent only on the observer, as opposed to the source or the background aether. In all experiments of the determination of the velocity of light, the determination has been made by the observer. Yes, the velocity of light has always the same value c, but that velocity is always dependent on the state of motion of the observer, in complete contradiction to the "fact" of statement (2).

In the above example it may appear that in order to correctly deduce the locations and velocities of Alice and Bob and explosions P and Q, we would have had to invoke additional knowledge about the locations and relative velocities not privy to the actual observers Alice and Bob. We in fact began with the assumption that we knew Bob was moving relative to the source of the explosion. Since we were using trains and embankments, this was not an unreasonable assumption. But what if we have no arbitrary "stationary" reference frame. Are we still able to determine the simultaneity of events? It will be sufficient to show that we can consistently agree as to the time of and relative distance to a single event, since this can then be extrapolated to the case of multiple events.

Consider the situation depicted in figure 3-3. Here an explosion is about to occur at some undefined point in space. Bob is in a spaceship some as yet undefined distance from the immanent explosion, traveling with an undetermined velocity relative to the event, in the direction shown. Bob is again carrying an extremely long measuring rod on which the explosion will leave a mark.

The explosion occurs and blackens Bob's ruler at a certain point. Bob is not yet aware of the explosion, as the light has not yet reached him. The light to which Bob is sensitive will have a velocity of c plus his velocity with respect to the source. At the instant the light reaches Bob, he will note the time and begin reeling in his ruler to determine how far away the explosion occurred. Suppose he records a time of one second past noon and measures a distance of 300,000 km (any time and distance will do, it is simply easier to speak of specific examples than in generalities). Bob now knows that the explosion occurred at exactly noon, since it takes light one second to travel 300,000 km. Bob is unaware of his motion relative to the source, and does not need to know it in order to determine the time of the event. He knows that, for him, light travels at c and that it traveled 300,000 km in one second to reach him. If we also allow Bob to know his velocity with respect to the explosion, he will deduce that the event occurred at a distance of 300,000 km plus the additional distance he traveled in the one second it took the light to reach him.

Figure 3-3 Observers at any velocity relative to the event or each other will be able to agree on the time of an explosion by consulting the marks on their measuring rods.

Thus we see that, from any frame of reference, Bob will correctly record that the event occurred at noon, and, if his velocity with respect to the event is known, he will deduce the correct location as well. No matter how many observers we have, traveling at any velocity and starting at any given location from the event, each will agree on the time of the explosion. If they know their velocity with respect to the event, they will also agree on its precise location relative to their current ones. The observers do not need to actually carry long measuring rods with them. If they know their distance from an event at the time of its occurrence, say from New York to San Diego, or if they can determine it from some other means, the effect is the same as carrying a measuring rod. Note that if the observers know their velocity relative to each other, but not relative to the event, they will agree on the relative distance to the event from any observer, but will be unable to pinpoint the exact location since they are unaware of any motion with respect to the event itself. Fortunately, thanks to the Doppler shift, any observer can accurately determine his velocity with respect to the source, as explained in the next section.

Doppler shift is the name assigned to the shift in frequency experienced when an observer is in motion relative to the source of the frequency. The classic example of this is the change in pitch (frequency observed) of a train whistle as it is approaching an observer and when it is moving away. During the approach, the pitch (frequency) is much higher than when the train is moving away. The explanation for this is straightforward.

Sound travels through the air at roughly 1000 ft/sec (fps). A train whistle has a frequency of about 100 cycles per second, or 100 Hertz. The wavelength of the sound is equal to the velocity of the sound divided by the frequency, or ten feet, and is a measure from the top of one peak to the next, or of one cycle. Wavelength is usually denoted by he symbol lambda, or l. Another way to look at this is that the frequency is equal to the speed of sound divided by the wavelength. That is to say that 100 peaks (cycles) of the sound wave will pass a fixed point in one second. If the train is moving toward the observer (or the observer moving towards the train) at 100 ft/sec, the observer will detect approximately ten extra cycles of the sound per second. This is because each cycle is ten feet peak-to-peak, and the observer will effectively move across ten of these peaks during one second. This is depicted in the top half of figure 3-4, with the observer moving towards the train at 100 ft/sec for clarity of illustration. The impression that the observer gets (on his ears) is that during that one second, 110 peaks went past his ears to produce a sound with a frequency of 110 Hz.

Another way to look at this example is that, since the train is approaching at 100 ft/sec, the effective wavelength of the sound passing the observer is roughly nine feet, because the train is causing the emitted waves to bunch up as it moves through the air. The frequency of this sound will be given by the speed of the sound in the air divided by the wavelength (1000fps/9ft) or roughly 111 Hz. This is illustrated in the bottom half of figure 3-4. By similar reasoning, as the train is moving away from the observer, the observer will detect approximately ten less peaks during a second, resulting in a perceived frequency of 90 Hz.

The reason the Doppler shifting of sound occurs in the above example is due to the fact that the speed of sound is not invariant for all frames of reference. In other words, a stationary observer listening to the gong of a stationary bell at 100 Hz (wavelength of ten feet) will perceive the sound passing him at 1000 ft/sec, while an observer moving towards the bell at 100 ft/sec will perceive the sound passing him at 1100 ft/sec, thus giving rise to the increased pitch described above. If another observer is moving away from the bell faster than 1000 ft/sec, he will never hear the gong of the bell, since he is moving faster than the sound from the bell, and this sound will never reach him. The classic example of this is the case of a jet breaking the sound barrier. In this case, as the jet is moving towards you, and approaching the speed of sound, the noise is being generated closer and closer to you, and the sound waves are beginning to bunch up. As the jet achieves the speed of sound, the jet is generating noise on top of the original noise, which is traveling right along with the jet. The result to you is that you are ultimately hit with a solid wall of noise, the characteristic "sonic boom". Once the jet pushes through the sound barrier, you actually hear the sound of the jet passing overhead before you hear the sound of the jet approaching. You see the jet flying one direction, yet your ears tell you it is traveling in the opposite direction.

To view the example of the bell from another perspective, if the speed of sound were invariant for all frames of reference, the observer moving towards the bell would perceive the sound as passing him at 1000 ft/sec, as would the stationary observer. The frequency of the sound heard for each observer would be given by the speed divided by the wavelength or 100 Hz. Thus each observer would hear the same pitch from the bell. We know from experience and investigation that this is not the case, and therefore, the speed of sound is not invariant from all frames of reference.

Light or radio waves observed from a point in motion relative to the source experience a shift in frequency due to this relative motion--the Doppler shift. The observed frequency of light is always given by c times one over the wavelength (l^-1). Remember that light is emitted from the source at all velocities from zero to C. In order for all light emitted from a source to have the same frequency, the wavelength of the light must therefore be proportional to the component velocity of interest. Slower components will have shorter wavelengths while faster components will have longer ones. This knowledge can be used to determine the frequency of light that an observer will see when moving away from the source at some arbitrary velocity. If we call the component velocity of light leaving the source the "initial velocity", then l^-1 is equal to the frequency of the source divided by the initial velocity. The shifted frequency of the observer is then given by c times one over the wavelength or c times the frequency divided by the initial velocity. With light, lower frequencies are redder, while higher frequencies are bluer. Thus if the source and observer are moving away from each other, we say the light is red-shifted, while if they are moving toward each other we say it is blue-shifted.

Figure 3-4 The Doppler shift causes the whistle of an oncoming train to be heard at a higher frequency proportional to the relative velocity of the train and the listener.

Now if the observer knows the expected frequency of the source (say the absorption spectrum of Hydrogen), the observer’s velocity with respect to the source can be determined by solving the above relation for frequency shift in terms of velocity. Thus the relative velocity between the source of light and the observer is equal to c times the ratio of the Doppler shift to the observed frequency. This relation is not the same formula used for determination of velocity in the classic sense involving sound. With sound, the actual velocity of the wave that the listener hears changes, while with light the velocity seen is always strictly c. However, the two formulas are accurate to within the most insignificant decimal places. The formula is also not the same as the formula derived in special relativity, though the differences become significant only when one's speed approaches c. This presents no problem for almost any experiment one can think of. In fact, the formula normally associated with the Doppler shift in sound was successfully applied to light to make one of the most important discoveries of the twentieth century.

Edwin Hubble used the light characteristic of a certain class of stars to demonstrate that they were not a part of the Milky Way galaxy. It was this discovery that led to the realization that there were indeed galaxies other than the Milky Way. By measuring the luminosity of his special stars in several galaxies, he was able to determine how far each was from the Milky Way. In 1914, Vestor Slipher had measured the spectra (characteristic frequencies common to all stars) of thirteen remote galaxies, and found that, according to the Doppler relation, they were all moving away from the Milky Way, with the single exception of the Andromeda galaxy. Hubble, armed with several additional measurements, made a graph comparing the distance to the red shift in spectra for a total of twenty-two galaxies. What he found was astonishing. He discovered that the distance was proportional to the redshift--the greater the galaxy's distance, the faster it was moving away. The universe appeared to be expanding. This conclusion was made using the Doppler shift formula for sound, as Hubble wasn't even aware of Einstein's formulas. He was also reluctant to conclude that the universe actually was expanding, or that the shift in frequencies was even due to the Doppler effect. As Dennis Overbye reflects in Lonely Hearts Of The Cosmos:

The importance of the frequency shift and velocity determination relations becomes clear if we return to the question of simultaneity and compare the relativistic model to the radiation continuum model. This is done in the next section. But first, we must review the nature of Doppler shift and see why it qualitatively prohibits the use of Galilean transformations under the assumption of an invariant velocity for light.

Earlier, it was shown how the Doppler shift arises in sound due to the fixed velocity of the sound wave's propagation through the atmosphere. Any two observers in motion with respect to each other will perceive a net velocity of sound equal to its velocity in the atmosphere plus their velocity toward it. A ball moving through the air at fifty miles per hour will hit your glove at sixty miles per hour if you are running toward it at ten miles per hour. It is precisely this lack of invariance in the speed of sound that results in the Doppler effect. There is another Doppler effect known as aberration. If the train in the previous example is moving toward you, the sound wave will be compressed since each successive peak is released closer to you than the one before. This will have the effect of an increase in frequency for an approaching train, and a decrease in frequency for a receding train. If a pitcher throws balls at you once every second, they will reach you once per second, or with a frequency of one Hertz (one cycle per second). If the pitcher is running toward you as he is throwing the balls, each successive ball will have a shorter distance to travel before reaching you, and the balls will reach you more frequently than once per second. Their frequency will increase due to aberration. The sonic boom of a jet approaching and then exceeding the speed of sound takes the aberration effect to the limit and causes a sonic boom as the speed of sound in the atmosphere in the jet's frame of reference reaches zero. Note that the speed of sound is not an unattainable velocity, even though this odd effect occurs at that speed. In the early ages of flying, it was felt that this was an impenetrable barrier. It was felt that a craft would break to pieces at the speed of sound due to the shock wave and other stresses. This myth was dispelled when Chuck Yeager flew the first aircraft to break the sound barrier.

The important point in the above discussions is again that it is the lack of invariance in the speed of sound (and of baseballs) in a Galilean universe that produces the Doppler effects. If the velocity of sound were invariant, then all sources would observe the sound wave leaving at a velocity of one thousand feet per second with respect to itself and there would be no aberration. The sound leaving the source would have the same frequency as the source. Likewise, all observers would hear the sounds approaching at one thousand feet per second, and there would be no Doppler shift. The frequency heard would be the same as the frequency emitted, for all listeners at all velocities. Additionally, there would be no sonic booms, as jets would not be able to outrace the very sound they were emitting, and they would therefore always travel at less than the speed of sound in their frame of reference.

And yet there is a Doppler shift associated with light. It is such a well documented effect that its uses range from tracking the velocity of aircraft using radar signals, to adjusting the frequency of radio receivers on ships at sea to account for their motion with respect to a satellite, to calculating the velocity of distant galaxies as they appear to stream away from us through space. The aberration effect is invoked to explain the behavior of synchrotron radiation. In a synchrotron a fast moving light-emitting electron moves at close to the speed of the emitted radiation, causing a bunching up of the wave crests in the same manner as a jet approaching the speed of sound. There is even an effect similar to a sonic boom in light called Cerenkov radiation which occurs when the speed of an electron in a certain medium exceeds the speed of light in that medium. The electron does not exceed the speed of c; the medium is one in which the speed of light is less than c, and, also less than the speed of the electron. Since the Lorentz model assumes an invariant speed of light and also accepts Doppler shifted frequencies, the familiar concept of Galilean transformations must be abandoned in relativity theory.

If we return to the general case of simultaneity presented in figure 3-3, that section concludes by illustrating that if the observer knows his velocity with respect to the source, he will be able to deduce the distance to the source. Fortunately, the Doppler shift allows us to determine our velocity with respect to the source as being equal to c times the ratio of the Doppler shift to the observed frequency.

This is a major deviation from the relativistic description of simultaneity which states that events at different points of space that are simultaneous for one observer are in general not simultaneous for any other observer moving uniformly relative to the first. To see the difference between these two models of simultaneous events, consider the following example, illustrated in figure 3-5. In the first figure, we see Alice, who is (and remains) at rest with respect to the apparatus during the entire experiment. Two flashes of light at the same frequency are radiated from a point equidistant between two mirrors, by a source stationary with respect to the mirrors. As Alice sees things, the light pulses leave the source, travel to each mirror, strike and reflect off the mirrors simultaneously, and also return to the source simultaneously. This situation is the same in the special theory and in RCM theory.

In the second figure, Bob is in motion relative to the apparatus in the direction indicated. We will first consider the relativistic viewpoint. Bob sees the right-hand mirror approaching the source, while the left-hand mirror is moving away from the source. Therefore, since light travels at c in all frames of reference under the special theory, the light will reach the right-hand mirror earlier than it will reach the left-hand mirror. After being reflected from the right-hand mirror, that light will see a receding source while the light from the left-hand mirror will see an approaching source. The effect of this will be that the total distance traveled by the light striking the left mirror equals the total distance traveled by the light striking the right mirror, and the two beams will return to the source at the same time from Bob's point of view. Thus Bob concludes that the light beams struck each mirror at different times, but returned to the source simultaneously. In the relativistic model, therefore, Alice and Bob do not agree on the simultaneity of the event of light striking each mirror, but do agree on the simultaneity of each signal's return to the source. This concept results in a slowing of clocks for observers in motion (relative to Alice) that will be explained in the section on the effect of motion on light clocks.

In the radiation continuum model, Alice sees things exactly as described above. Bob, however, sees things differently. Since he is in motion, that component of light that he sees traveling towards the right-hand mirror must have a velocity of c plus his velocity. He will see this light strike the mirror before the time at which Alice sees it strike the mirror. He will not, however, see this light immediately reflected from the surface of the mirror. This is because the stationary mirror does not "see" the same component of light that Bob sees. The mirror reflects that component of light that strikes it at a velocity of c. This component arrives at the mirror later than the component that Bob is observing. Next the light that Bob sees leaving the mirror does so at the same time in Bob’s frame of reference as it does in Alice's. Alice and Bob disagree on the time the light reaches the mirror, but agree on the time at which the reflected return pulse begins. In Bob's frame of reference, however, he can only see that component of the reflected pulse that is traveling at a speed of c minus his velocity, thus the light he sees will return to the source at a later time than that light that Alice sees. This is completely consistent with the RCM model of simultaneity presented earlier, which stated that observers in relative motion can see the light from an event at the same place, but at different times. For light traveling to the left-hand mirror, Bob observes only that component of light traveling at a velocity of c minus his velocity, slower than the component of the light that the mirror "sees" and reflects. Thus Bob will see the light reflected off the mirror before the source light he is observing reaches the mirror! (It is not actually true that Bob will "see" the reflected pulse before he "sees" the source pulse hit the mirror. It is true that the mirror will reflect that velocity component of light to which it is sensitive before the velocity component to which Bob is sensitive reaches the mirror. This is explained in the next section, where we find that in actuality, Bob will not observe any discontinuity in the reflection of either beam from the mirrors.) As in the above example, Bob will agree with Alice on the time of the reflected pulse's origin, but he will disagree on the time it took for the light to actually strike the mirror. Now the light component that Bob sees reflected from the left-hand mirror must have a velocity of c plus his velocity, and will therefore return to the source before the component observed by Alice returns. In the end, Alice and Bob will agree on the simultaneity of the origin of the light pulses, but will disagree on the simultaneity of the light pulses striking the mirrors, and disagree again on the simultaneity of the reflected pulses reaching the source. How can this conflict be resolved? The answer involves looking at the Doppler shift of the light signals.

Recall that for an observer traveling in the same direction as a light pulse, the frequency will be Doppler shifted according to the relative velocities, and that the observer can then use this shift to determine its velocity relative to the source. Bob knows that the frequency of the light that he sees moving toward the right-hand mirror is not the same as the source pulse. He can thus determine his velocity with respect to the source and mirror, and conclude that the mirror is sensitive to a slower component of light than that which he observes. Similarly for the reflected pulse, since the frequency Bob sees will be higher than the source pulse, he knows that the light that Alice sees is moving faster than that which he sees.

Suppose that Bob did not know either the frequency of the source, or his velocity with respect to the apparatus. Could he still come to the same conclusions regarding Alice's perception of the events? The answer is yes, and we will use the light pulse traveling to the left-hand mirror to analyze this. Since Bob is moving toward the source and away from the mirror, the frequency he sees leaving the source will be higher than the frequency he sees reflecting from the mirror. Knowing these two observed frequencies, the Doppler shift and velocity determination relations can be manipulated to determine the source frequency. Once he knows the unshifted frequency of the source, Bob can determine his velocity with respect to the source as before. Thus both Alice and Bob conclude that from the view of the apparatus, the emission of the two pulses was simultaneous, the reflection from each mirror was simultaneous, and the arrival of the two beams back to the source was simultaneous.

The major problem with the relativistic model is that it assumes that the time at which the light strikes the mirror in Bob's frame of reference is the same time at which the mirror itself is struck in its frame of reference. Thus, from the perspective of either Alice or Bob, the mirror reflects the same pulse of light at different times, depending on the observer's frame of reference. This is troublesome for the mirror, which has no knowledge of either observer. As with the radiation continuum model, Bob detects the Doppler shift in his observed frequencies, though he is unable to use these shifts to resolve the timing differences, and instead must use Lorentzian time and length contraction due to motion. Even armed with this information, the concept of simultaneous events in the special theory of relativity has no meaning.

Now it may seem as though we have replaced one confusing theory with a new one that is equally confusing. Light reflecting from a mirror before it arrives? How could this be? The answer lies in the way in which we actually see things, and what we can and cannot see, as the next section will demonstrate.

In previous sections, our moving observers "saw" the light pulses hit the mirror at different times than they "saw" the light pulses reflected. Additionally, a moving observer sometimes "saw" a pulse reflected from a mirror before the incident pulse struck the mirror. While we have shown that this is not actually the case, as the mirror is simply sensitive to a different component of light, there is still an apparent violation of causality from the point of view of the observer. How can this incidence-reflection time gap be resolved? First, consider a different means of "seeing," involving long rods and marks, which will support the validity of the time lag.

In previous examples, our observers carried long rods with them to detect the occurrences of explosions in space and other events, so that the times of and distances to the events could be determined in one reference frame or another. In the situations described in the previous section, we will allow our observers Alice and Bob to carry two long rods with them, sufficient to stretch to both mirrors. Since the rods are traveling at the same speed as the respective rod carrying observers, then any light-detecting device on them will detect only that component of light appropriate to its own observer's reference frame. If such a device is located so that it will be opposite the mirror at the instant when light of the appropriate speed reaches the mirror, a mark will be impressed on the plate. Since each observer knows that light travels at a speed of c in his frame of reference, he can simply measure the distance to the effected plate and thus determine the time of impact by dividing by c. We could also use a light sensitive clock that stops when a pulse of light strikes it, and each observer could simply read the time on the clock when it stopped to determine the time of impact. In this manner, each observer will determine that the time at which light in his frame of reference reached the mirror is not necessarily the same as the time at which the light was reflected by the mirror. Again this is consistent with the RCM concept of simultaneity which states that observers in motion relative to each other may see the light from an event at the same place, but will see it at different times. In this case the observers would be the stationary mirror and the moving observer's light sensitive clock.

Why go through all this trouble? The problem is, neither observer can actually "see" the light hit the mirror at all. They are able only to see that light which is reflected from the mirror and which returns to their eyes at a velocity of c in their frame of reference. This problem exists for the relativistic model as well as the radiation continuum model, as it is a fact of nature. When you shine a flashlight up into the night sky, you seem to "see" a column of light racing away from the flashlight and into space. Actually, what you are seeing is the reflection of the light from particles in the night air. If you were to shine that same light through a vacuum (say outside the space shuttle), you would not see the column of light at all. You would still be able to illuminate a distant reflector or mirror, and see the reflected light, but you would not see the characteristic "beam" of light connecting the flashlight to the mirror. The familiar trace of the "photon torpedoes" of Star Trek fame would never be seen in an actual battle.

Arthur Zajonc in Catching the Light speaks of an apparatus he has created for something called "Project Eureka." Basically the apparatus is an empty box into which a bright projector shines. The setup ensures that no part of the box itself is illuminated, and thus there is no reflection from the walls. Though the box is "full of light," one sees nothing but darkness upon looking inside. This is because you can see only that light which is aimed directly at your eye or that which reflects from an object toward your eye. One can pull a handle on Zajonc's Eureka box that moves a wand through the interior of the apparatus. When the light that defied observation before strikes the wand, the wand becomes brilliantly lit on one side. In seeing the beam of a flashlight in the night sky, the particles in the air take the place of Zajonc’s magic wand.

In previous sections, we talked of Bob "seeing" the light hit the mirror before or after he "saw" the reflected pulse begin. If he uses the rods provided above, he will draw these conclusions. But his eyes, watching the light as it is scattered (reflected) back to him off the night sky on its journey toward the mirror, will see the light hit the mirror at the exact time at which the reflection begins. The reason for this is that each scattering particle can be thought of as a mirror. It is only the light that is reflected from these "dust mirrors" at the velocity of c minus Bob's velocity (if he is moving toward the mirror) that he will see, not the actual light pulse heading toward the mirror. We can imagine a large number of these particles, all equally spaced along the path to the mirror, as in figure 3-6. Light from the source will strike these particles at regular, equal intervals, and be reflected at regular, equal intervals. Each of these reflections will be traveling toward Bob at c minus his velocity, and he will see each reflection at regular, equal intervals. Eventually, the light pulse will reach the mirror, which is one equal interval beyond the last particle. When Bob sees the light reflected from the mirror, it will occur at the same time at which he actually "saw" the beam strike the mirror. The important point here is that Bob never actually saw a beam of light racing toward the mirror. What he actually saw was effectively the reflected light from an infinite set of small mirrors along the path to the large mirror, each reflecting that component of light to which it was sensitive.

In the above figure, Bob is sensitive to that light with a velocity (with respect to the source and mirrors) of c minus his velocity. The light actually striking the particles is traveling at the much faster speed of c. Since this light covers the distance between the particles faster than Bob would expect it to, he perceives that the distance between particles is shorter than it actually is. Note that this perception of a shorter distance is not due even remotely to the Lorentz length contraction as required in special relativity. The effect here is due to the observer not knowing he is in motion with respect to the reflecting particles. If you throw a ball at a wall with a velocity of ten feet per second and it hits the wall after two seconds, you will conclude the wall is twenty feet away. However, if you are moving toward the wall at ten feet per second when you release the ball, the ball will have a velocity of twenty feet per second with respect to the wall. The ball will strike the wall after only one second and you will perceive the distance to be ten feet instead of the actual twenty. This is the "shorter distance" that Bob is experiencing.

Since the carrying of rods and clocks eliminates the distinction between "seeing" light at a distance and seeing backscattered light, we can leave the wording of the previous sections as it stands. Bear in mind, though, that when we speak of an observer "seeing" light strike a distant object, we actually mean something else. The observer is actually obtaining physical evidence at a distant point on a rod or with a distant clock synchronized to the observer’s clock, and that the rod and all clocks are stationary in the observer’s frame of reference.