The Postulates and Evidence for the Special Theory
We begin in this chapter to develop the Special Theory of Relativity. Our approach will be to first present the basic postulates for the Special Theory and comment on some of the fundamental issues either explicitly or implicitly raised by the postulates. Since the postulates are proported to refer to the physical universe, we next give a basic discussion of some of the experimental evidence for both postulates, with emphasis on some recent experimental tests. We shall find that there is excellent experimental evidence for these two postulates. This will set the stage for the treatment of several important applications of the special theory in Chapter 12. Given that some of the predictions of the theory, based on these postulates, are quite counter to our everyday experience, we believe this Chapter serves an important purpose. It presents a firm experimental base for any predictions of the theory, be they bizarre or not.
I. The Postulates
It was Albert Einstein who was able to take the cues from various streams in nineteenth century physics and distill out the fundamental basis for a new coherent picture of the universe. In this chapter we present the postulates of Einstein's 1905 Theory of Special Relativity and give an assessment of the evidence for these postulates. The theory is special insofar as it ignores the presence of gravity. In 1915 Einstein created the General Theory of Relativity, which is essentially a generalization of the Special Theory to include gravity. We turn to a discussion of the General Theory in Chapter 15.
Let us emphasize that when Einstein wrote his first paper on Special Relativity there was no overwhelming experimental support for the postulates. Although Einstein alludes to the absence of any evidence for the 'luminiferous ether' as justification for a thorough re-examination of the logical and physical foundations of physics, the Michelson-Morley experiment was apparently unknown to him. Thus it was not the case that he was initiating these revolutionary ideas in response to the results of one or more crucial experiments. In fact, it is a credit to Einstein's intuition and foresight that he realized these 'axioms' were the 'correct' ones on which to base the foundations of physical science. Below we give the postulates:
POSTULATE I: The laws of nature and the results of all experiments performed in inertial frames of reference are independent of the uniform translational motion of the system as a whole.
POSTULATE II: The speed of light is a universal constant independent of the motion of the source.
The first postulate is a kind of "relativity principle". A relativity principle says that each equivalent observer sees the same phenomena in question. Such a principle involves two components. The first is a class of equivalent reference frames and an attached set of observers. The second is the type of phenomena, e.g. mechanical phenomena or electromagnetic phenomena or gravitational phenomena or some combination of these. If one member of the class of observers attached to the reference frames could discover a difference in an experimental outcome in one frame, then it would be possible to identify a preferred frame of reference with respect to the phenomena in question. In the case of relatively moving observers, this frame, designated the "ether frame", could be identified as the absolute state of rest. All other frames would then move relative to the ether frame. If this were the case at least some of the laws of nature would necessarily possess some intrinsic dependence on the relative velocity of any member of the class with respect to the ether frame. Postulate I denies this possibility. The apparent inability to detect "absolute" velocity was discussed in Chapter 10 when we reformulated the various space and time models from the Greeks to Newton. Recall that our Galilean Spacetime had the property that all "straight" worldlines, no matter what their spacetime tilt, were deemed candidates for equivalent particles or observers. Thus, if we accept Postulate I, we might think our search for the "correct" model of spacetime is finished, because Galilean Spacetime was built just to encode the content of the contention that absolute velocity is unobservable. But Einstein's principle of relativity has a wider scope. This can be seen by considering the second of the two components of a relativity principle: The type of phenomena covered by the given relativity principle. The type of phenomena covered by the Galilean principle was that described by the laws of Newtonian mechanics. If we adopt the Galilean principle of relativity, it turns out that the laws of Newtonian mechanics are invariant with respect to equivalent observers. The laws of electricity and magnetism, however, can be shown not to be invariant with respect to the Galilean transformations. Einstein's principle of relativity, in part, is a generalization of the Galilean relativity principle in that electromagnetic phenomena as well as mechanical phenomena are to be the same for equivalent observers. Thus the content of Postulate I denies the possibility that any member of the class of inertial observers, upon performing any type of experiment involving any mechanical or electromagnetic law of nature, will be able to detect any preferred frame of reference.
It is clear from a cursory reading of the second postulate that it represents a radical departure from Newtonian and Galilean modes of thinking. Our discussion of Chapter 9 showed, however, that it is just this growing realization of the truly fundamental character of light that finally initiated the revolution in the conceptual basis of space and time. Postulate II is one of Einstein's most startling claims. We should be quite skeptical about its validity, because our everyday experience seems contrary to it. That is, it is inconsistent with the Galilean principle of relativity which suggests that light from a source moving with velocity V with respect to an observer O should be observed by O to be moving with velocity V + C. But, according to Postulate II, O sees the light emitted by the moving source to have velocity C.
Einstein apparently came to the conclusion that Postulate II was necessary because of various paradoxes that could be generated if one allowed differently moving, but otherwise equivalent observers to carry out observations and communications using light signals. If the speed at which light travelled depended on the velocity of one observer with respect to another, then it would be possible to arrange for the order of events to be changed. It must have seemed clear to Einstein that, given that light is a fundamental stuff, it would be more reasonable to endow light with a universal quality than to give up the basic causal order structure. That this line of attack was a fruitful one has been borne out by the sweeping success of the subsequent developments. In 1905, however, nothing of the sort was obvious. We shall return to Postulate II and its implications for the concepts of simultaneity, time, and length measurements in Chapter 12.
II. Evidence for the Postulates
There has been a long history of experimental tests of the two postulates listed above. It is not our purpose to give a detailed, technically complete review of all the tests. Our purpose is to present a sufficiently complete picture so that one will be able to gain a qualitatively accurate impression of the situation as it exists today. Since we will make numerous explicit and implicit uses of the postulates, it is important to have a firm phenomenal base for them. What follows in this section is a discussion of two representative experiments that directly test each postulate. There are other experiments that could be described. We have chosen the ones discussed below for their simplicity and because they are "modern" experiments, having been performed in the last fifteen years or less. The fact that these examples of successful experiments are so recent is not due to any long-standing uncertainty about the validity of the postulates. From the very first experiments to the present day it has been apparent that the postulates were valid to a high degree of accuracy in the domain of application of the phenomena, i.e. in the absence of strong gravitational fields. The only uncertainty has been due to the limit of the accuracy of the experiments. Thus the "modern" tests present no surprises. They represent the latest experiments in a long line which confirm the postulates.
It should also be pointed out that there have been literally millions of indirect tests of the two postulates. These have occurred due to various consequences of the postulates in concert with other fundamental laws of nature. The entire fabric of twentieth century physics is held together to a nontrivial extent by these two postulates. The interrelations are so finely interwoven that should there be a significant sense in which they are invalid, there would be "ripples" of inconsistency sent through all of physics.
Evidence of Postulate I
Postulate I asserts that, given two observers who perform the same experiment on some piece of nature with one observer moving at some uniform velocity relative to the other (and thus at least one observer is moving with respect to the hypothetical absolute rest frame), it shall be impossible for one observer to detect his absolute velocity by studying the outcome of the experiment. Thus Postulate I entails that the outcome of any experiment performed by any observer moving with respect to a second observer should not be dependent on the relative velocity of the two observers. Any test of this postulate must be done in some chosen "moving" frame of reference and with sufficient precision to detect any intrinsic velocity dependence that may be present in some carefully chosen mechanical or electromagnetic experiment. Note that Postulate I refers to any experimental situation wherein the contention of the postulate is false in order to establish the non-universal character of the postulate. Of course, one cannot in practice perform experiments in every conceivable frame of reference nor can one actually do every possible experiment. Rather what one does is to choose an experiment in which one's expectations of the possible outcomes are sufficiently certain so that a non-confirmatory result will be unmistakable. The choice of such "crucial" experiments requires a deep understanding of the state of a particular area in physics as well as an ability to bring together the needed technology in order to perform the test.
Mossbauer-Doppler Effect Experiment
The Michelson-Morley experiment discussed earlier in Chapter 10 was an experiment designed to test the velocity dependence of the speed of light. It yielded a null result in that it failed to find any dependence of the speed of light on the velocity of the earth with respect to the most obvious source of motion (the 30,000 meters per second orbital speed). However the outcomes of the most accurate of these kinds of experiments only showed that the velocity of the Earth around the Sun. The experiment that we are about to describe has provided a much stronger test of the velocity independence of electromagnetic radiation.
A schematic representation of the experiment performed by G. R. Isaak et al. is shown in Fig. 11.1.
Located on the opposite ends of a rigid rod are respectively an emitter and an absorber of gamma radiation. The rod is rotating about the center-point at some chosen rotation rate. The center of the apparatus is attached to the earth. Let ~e and w be the emitted and absorbed frequencies of the radiation, Q the angular rotation rate, and V the magnitude of the velocity of the center of the apparatus with respect to the ether frame. Isaak found the following relation between these parameters.
where R is the radius of the "spinning wheel" and ~ is the frequency of the radiation emitted by a source that is not moving in the laboratory. Isaak predicted that if there were any velocity dependence of the speed of light then there should be an oscillating time dependence of the fractional change in the gamma radiation frequency. The following figure illustrates the predicted effect.
The rotation rate w, the radius R, and the speed of light C are known parameters in the experiment. Thus a graph of the fractional change in frequency will have an amplitude which is directly proportional to V, namely 2 V. A measurement of the amplitude would allow V to be determined.
There are several sources of motion of the apparatus with suspect to the hypothetical ether frame. In the Figure 11.3 we show one such source due to the earth's rotation.
As mentioned above, in order to make an accurate determination of the velocity of the apparatus with respect to the ether frame, it is crucial to be able to make precise measurements of the frequency difference between emitted and absorbed radiation. The frequency of the emitted radiation is determined by the type of emitter, some giving very sharply defined frequencies, others emitting a broad band of frequencies. Thus the choice of emitter and absorber is a crucial element in the experiment. The gamma radiation used in the Isaak experiment is that emitted by a particular isotope of Iron ( Fe). In 1958 R. Mossbauer discovered that gamma radiation was emitted by this isotope of iron without imparting a "kick" to the atoms in the solid Iron. Similarly the radiation could be absorbed by the same isotope without any of the energy being taken up by the solid.
Thus if a given emitter of 5 Fe gamma rays is placed adjacent to an absorber of the gamma rays with both emitter and absorber relatively at rest there should be absorbed a gamma ray at the same frequency as emitted. If, however there is relative motion between the two there could be a detectable difference between the emitted and absorbed frequencies if the frequencies are measurable with sufficient accuracy. This motion dependence on the detected frequency comes from the so-called Doppler effect, which states that a source of radiation moving away from an absorber will have a lower frequency than when there is no relative motion. Similarly when the source of radiation moves toward the absorber the frequency will be higher than when there is no relative motion. If the frequency is measurable to high accuracy differences in frequency can be detected with higher accuracy. This, in part, is the reason for the choice of the gamma radiation emitting Fe isotope. Its gamma radiation frequency is very sharply defined.
Thus in Fig. 11.1 at either end of the rotating rod are attached Fe. The experiment was performed under various rotation rates and at various times of day. If there should be any shift in the frequency of the emitted versus absorbed radiation, the experimenters could expect this apparatus to detect it to high accuracy. However they tried, no discernable change in frequency was observed, within the experimental accuracy of the method. Isaak assessed the sources of experimental uncertainty and concluded that the data indicated the maximum velocity of the apparatus with respect to the hypothetical ether frame was less than 5 centimeters per second! This is an upper limit to the velocity dependence that is quite small compared to the magnitude of the Earth's rotational velocity of 46,300 centimeters per second. The results of the Isaak experiment are more accurate than the Michelson-Morley experiment by a factor of several thousand. Our conclusion is that Postulate I is confirmed with ever-increasing accuracy.
Evidence for Postulate II
This postulate is surely the most revolutionary in tone and, as we shall discuss in Chapter 12, it compels us to reassess the meaning of time measurements made by pairs of observers located at spatially separated points. The acceptance of Postulate II entails the demise of the Newtonian concept of a universal, observer-independent time function. A related development of extreme importance is that the central conceptual position held by light forces us to re-examine the causal structure of events in space and time. Every point of space at every moment of time becomes endowed with additional structure induced by the fact that the speed of light is a universal constant. The total set of events with respect to any given event e is divided into three distinct sets of space-time points. These are the "absolute future, absolute past, and absolute elsewhere of the given event". The first set is the set of events which can be reached from e by a causal signal. The second set is the set of events which can reach earth by a causal signal. The third set is the collection of events that can not be reached with a causal signal which travels out from the event e at or slower than the speed of light. Light signals to and from e separate the set of events into three distinct regions with different causal properties as viewed from e (cf. Chapter 13).
Postulate II asserts that the speed of light is independent of the motion of the source of the light. As mentioned above, Einstein did not have any experimental basis for assuming Postulate II. No test of Postulate II was performed in the first few years after his 1905 paper. However, since 1913 many tests of Postulate II have been performed. In 1913, W. de Sitter pointed out that if the velocity of light did depend on the motion of the source, then there should be peculiarities in the properties of the light received at Earth from binary star systems.
We show in Fig. 11.4 a schematic representation of this type of observational situation. In these systems the two stars are rotating about a common center of mass. If one takes careful telescope photographs of the system over some extended period of time, one notices a regular repeating motion which possesses characteristics of the orbital pattern expected of any pair of gravitationally attracting bodies locked into elliptical orbit. de Sitter showed that a velocity dependence ~f the speed of light could produce, for an observer situated on the Earth, anomalies in the observations. There would be multiple images of the stars due to the arrival of light at the Earth at the same time from different points in the orbits of the stars. The multiple images would result from the fact that the component of the orbital velocity in the direction of the line of sight changes as the star traces out its orbit. At one point there may be no component in the line of sight, at some later point in the revolution the component of the velocity in the line of sight is non-zero. Since the velocity of light and the velocity of the star add as vectors according to Newtonian mechanics, the effective velocity of light from the star will be different at different points in its orbit. Thus, the light leaving the star system from two different orbital points may arrive at the Earth at the same time. This effect is illustrated in the
Here we have illustrated only two of the light signals emitted by the binary stars, one approaching the earth with maximum velocity and the other receding when the line of sight orbital component is zero. The velocity of light emitted in the former case is larger and thus the slope of the worldline of the light is larger. This faster moving light signal will arrive at the earth at the same time as the light emitted by the star from position 2. If we consider that the stars are emitting light continuously, we see that there should be a jumble of signals arriving at the earth at a given time, some of which could have emitted at quite different points in the orbit of the two stars. This would present the observer with a confusing state of affairs.
If, on the other hand, the velocity of light is independent of the motion of the source (the stars in this case), then there would be no additional
images received at the earth.
In 1913, W. deSitter analyzed some photographic plates of binary star systems in our galaxy and concluded that there was no available evidence of any "ghost" images. His conclusion was that the second postulate was valid.
However, the story does not end at this point. The situation envisioned by deSitter and others is clouded by the possibility of the intervention of matter between the place where the binary star system resides and the Earth. We illustrate the situation in Fig. 11.6.
The light must, in general, travel through a tenuous, possibly ionized, mostly hydrogen gas that makes up the interplanetary medium. It is a property of the way that electromagnetic radiation and matter interact that the electric and magnetic fields which constitute the light waves, initially travelling at some speed, are cancelled by the fields produced by the medium in response to the incident waves from the binary star system and are replaced by re-radiated electromagnetic fields which travel at the speed of light characteristic of the particular medium. Actually the original speed does not persist for a distance inside the medium larger than that termed the "extinction distance". This effect is illustrated in Fig. 11.7.
The extinction distance X can be shown to be given by the formula
where lambda is the wavelength of the radiation, rO is a constant with the dimensions of length, and N is the number of electrons per unit volume with which the electromagnetic wave can interact. For visible light propagating through the tenuous, mostly hydrogen gas-filled medium between the stars in our galaxy, X turns out to be around two light-years. Thus if a binary star system is more than two light years distant from the earth the velocity of a light signal would be somewhat "extinguished" and replaced by a signal travelling at the speed of light characteristic of the medium. After several of the extinction lengths, most of the velocity dependence contained in the light signals which entered the medium at a given velocity would have been "washed out".
Now, as long as the object of the observer is to measure the velocity of light, this extinction effect is of potential trouble due to the fact that the closest binary star system to the earth is the Sirius system. Its distance is known to be 8 light years, or four visible light extinction lengths away. The extinction effect thus seems to severely limit any direct measurement of the velocity of light and renders questionable many of the conclusions drawn from the study of the early photographic plates of binary star systems.
Interestingly enough, de Sitter's technique of probing for a velocity dependence of the vacuum speed of light has recently yielded meaningful results which do not suffer from the criticism that invalidates de Sitter's conclusions. During the last ten years a number of binary star systems have been catalogued in which one of the partners is pulsing radiation with astounding regularity. Among these sources there are some that emit X-ray pulses. An X-ray pulsar is a collapsed 'neutron star' which is orbiting about another star. The other star is the one that can be seen in visible light. While the X-ray pulsar is small and very dim, so that it can not be seen in the telescope. We show a schematic picture of these kinds of binary system in Fig. 11.8.
It is thought that gas from the ordinary star is pulled toward the orbiting neutron star. As the gas crashes onto the surface X-ray are produced. Also, neutron stars apparently have 'hot spots' on their surface. Neutron stars are known to be rapidly rotating. When the 'hot spot' rotates around in the direction of the earth the radiation emitted at the surface will be beamed to the earth. In the case of the X-ray pulsars there is a significant amount of X-radiation emitted. Since the neutron star rotates on its axis faster than once per second we expect it to be pulsing X-rays at a regular rate as it orbits around the ordinary star.
Recall that the extinction distance depends upon the wavelength of the radiation. For X-rays the extinction distance will be much longer than that for visible light because the wavelength is much shorter. Thus, if one could find sources whose X-radiation has a sufficiently long extinction distance compared to the distance between the earth and the X-ray source, the properties of the light which are characteristic of the motion of the source could be observed here on Earth. Then the basic deSitter argument could be safely used. In 1975 this was done by K. Brecher in his analysis of the pulses of several X-ray pulsars. He utilized her X-l and Cen X-l, which are two X-ray sources located in our galaxy some 18,000 light-years and 24,000 light-years respectively from our solar system. Another source studied was SMC X-l, a binary X-ray pulsar situated in a small satellite galaxy to our galaxy, called the Small Magellanic Cloud. Its distance is around 180,000 light-years.
In order to get a feel for the distance scales involved we show, in Fig. ll.q, a rough picture of the positions of the X-ray sources relative to the position of the earth in our galaxy.
In all cases studied by Brecher, the extinction distance for X-rays emitted by all these sources is safely longer than the distance to the source in question. Brecher found that there was indeed no velocity dependence of these pulses on the motion of the source. The limit set by the uncertainties in the observed quantities is less than two parts in one billion. We thus take Postulate II to be confirmed.
As with Postulate I we emphasize that these recent results present us with no surprises. As a further example, we mention the following experiment performed in 1964 at CERN in Geneva. In this experiment a beam of elementary particles is produced by colliding a proton beam from a high-energy accelerator with a target material. In the CERN experiment a beam of neutral pi-mesons, which are a very short-lived particle produced in the original proton-target collision, was utilized. Now it is well known to physicists that when the pi-meson decays in flight it decays into electromagnetic radiation. In Fig. ll.10 we sketch the set-up in its barest essentials.
Basically the experimenters measured the speed of the radiation emitted by the pi-meson. The pi-meson was moving with a speed of .99995 of the vacuum speed of light. It undergoes a decay into two packets of radiation (two photons) after very short time intervals. The apparatus was designed to detect the arrival of radiation at two points a fixed distance apart down stream from the region where the radiation was produced. The ratio of the distance travelled to the time required for light to travel between the two points constitutes a direct measurement of the speed of light. The extinction distance was calculated for this type of radiation and found to be comfortably longer than any important lengths in the experiment. Because of experimental accuracy limits, the experiment was able to set a less stringent limit than the X-ray pulsar analysis. Even so, the CERN experiment found no dependence of the velocity of the source to one part in 10,000. Thus, on the scale of the Earth-bound laboratory this result confirms Postulate II rather conclusively. We can take the X-ray pulsar "experiment" to demonstrate that Postulate II is valid at distant points in, as well as outside, our galaxy.
Postulates I and II have subjected to a continual series of tests of ever-increasing accuracy since 1905. The experimental evidence overwhelmingly confirms the validity of the two postulates in the absence of strong gravitational effects. It turns out that the postulates will always hold true if observations are restricted to a small enough region of space time. After looking at some applications of the special theory in chapters 12 and 13, we turn to the task of incorporating gravitational phenomena in such that the special theory remains valid in small regions (as the evidence demands). It was, again, Albert Einstein who lays the way to generalize the special theory in the appropriate way. This is the topic of Chapter 14 and 15, so we will leave further comment to those chapters.