Postulates of the Special Theory of Relativity

The problems that existed at the start of the twentieth century with regard to electromagnetic theory and Newtonian mechanics were beautifully resolved by Einstein’s introduction of the special theory of relativity in 1905. Unaware of the Michelson–Morley null result, Einstein was motivated by certain questions regarding electromagnetic theory and light waves. For example, he asked himself: “What would I see if I rode a light beam?” The answer was that instead of a traveling electromagnetic wave, he would see alternating electric and magnetic fields at rest whose magnitude changed in space, but did not change in time. Such fields, he realized, had never been detected and indeed were not consistent with Maxwell’s electromagnetic theory. He argued, therefore, that it was unreasonable to think that the speed of light relative to any observer could be reduced to zero, or in fact reduced at all. This idea became the second postulate of his theory of relativity. In his famous 1905 paper, Einstein proposed doing away with the idea of the ether and the accompanying assumption of a preferred or absolute reference frame at rest. This proposal was embodied in two postulates. The first was an extension of the Galilean–Newtonian relativity principle to include not only the laws of mechanics but also those of the rest of physics, including electricity and magnetism:

First postulate (the relati£ity principle): The laws of physics have the same form in all inertial reference frames. The first postulate can also be stated as: there is no experiment you can do in an inertial reference frame to determine if you are at rest or moving uniformly at constant velocity. 

The second postulate is consistent with the first: Second postulate (constancy of the speed of light): Light propagates through empty space with a definite speed c (3.00Χ10^8 m/s) independent of the speed of the source or observer. 

These two postulates form the foundation of Einstein’s special theory of relativity. It is called “special” to distinguish it from his later “general theory of relativity,” which deals with noninertial (accelerating) reference frames. The special theory, which is what we discuss here, deals only with inertial frames. The second postulate may seem hard to accept, for it seems to violate common sense. First of all, we have to think of light traveling through empty space. Giving up the ether is not too hard, however, since it had never been detected. But the second postulate also tells us that the speed of light in vacuum is always the same, no matter what the speed of the observer or the source. Thus, a person traveling toward or away from a source of light will measure the same speed for that light as someone at rest with respect to the source. This conflicts with our everyday experience: we would expect to have to add in the velocity of the observer. On the other hand, perhaps we can’t expect our everyday experience to be helpful when dealing with the high velocity of light. Furthermore, the null result of the Michelson–Morley experiment is fully consistent with the second postulate.

Einstein’s proposal has a certain beauty. By doing away with the idea of an absolute reference frame, it was possible to reconcile classical mechanics with Maxwell’s electromagnetic theory. The speed of light predicted by Maxwell’s equations is the speed of light in vacuum in any reference frame. Einstein’s theory required us to give up common sense notions of space and time, and in the following Sections we will examine some strange but interesting consequences of special relativity. Our arguments for the most part will be simple ones. 

Simultaneity

An important consequence of the theory of relativity is that we can no longer regard time as an absolute quantity. No one doubts that time flows onward and never turns back. But according to relativity, the time interval between two events, and even whether or not two events are simultaneous, depends on the observer’s reference frame. By an event, which we use a lot here, we mean something that happens at a particular place and at a particular time. Two events are said to occur simultaneously if they occur at exactly the same time. But how do we know if two events occur precisely at the same time? If they occur at the same point in space—such as two apples falling on your head at the same time—it is easy. But if the two events occur at widely separated places, it is more difficult to know whether the events are simultaneous since we have to take into account the time it takes for the light from them to reach us. Because light travels at finite speed, a person who sees two events must calculate back to find out when they actually occurred. For example, if two events are observed to occur at the same time, but one actually took place farther from the observer than the other, then the more distant one must have occurred earlier, and the two events were not simultaneous.

We now imagine a simple thought experiment. Assume an observer, called O, is located exactly halfway between points A and B where two events occur, Fig. 26–3. Suppose the two events are lightning that strikes the points A and B, as shown. For brief events like lightning, only short pulses of light (blue in Fig. 26–3) will travel outward from A and B and reach O. Observer O “sees” the events when the pulses of light reach point O. If the two pulses reach O at the same time, then the two events had to be simultaneous. This is because (i) the two light pulses travel at the same speed (postulate 2), and (ii) the distance OA equals OB, so the time for the light to travel from A to O and from B to O must be the same. Observer O can then definitely state that the two events occurred simultaneously. On the other hand, if O sees the light from one event before that from the other, then the former event occurred first. The question we really want to examine is this: if two events are simultaneous to an observer in one reference frame, are they also simultaneous to another observer moving with respect to the first? Let us call the observers O1 and O2 and assume they are fixed in reference frames 1 and 2 that move with speed v relative to one another. These two reference frames can be thought of as two rockets or two trains (Fig. 26–4). O2says that O1 is moving to the right with speed v, as in Fig. 26–4a; and O1 saysO2 is moving to the left with speed v, as in Fig. 26–4b. Both viewpoints are legitimate according to the relativity principle. [There is no third point of view that will tell us which one is “really” moving.]

Now suppose that observers O1 and O2 observe and measure two lightning strikes. The lightning bolts mark both trains where they strike: at A1 and B1 on O1's train, and at A2 and B2 on O2's train, Fig. 26–5a. For simplicity, we assume that O1 is exactly halfway between A1 and B1 and O2 is halfway between A2 and B2. Let us first put ourselves in O2's reference frame, so we observe O1 moving to the right with speed v. Let us also assume that the two events occur simultaneously in O2's frame, and just at the instant when O1 and O2 are opposite each other, Fig. 26–5a. A short time later, Fig. 26–5b, light from A2 and from B2 reach O2 at the same time (we assumed this). Since O2 knows (or measures) the distances) O2A2 and O2B2 as equal, knows the two events are simultaneous in the reference frame.

But what does O1 observer observe and measure? From our (O2) reference frame, we can predict what O1 will observe. We see that  O1 moves to the right during the time the light is traveling to O1 from A1 and B1. As shown in Fig. 26–5b, we can see from O2 our reference frame that the light from B1 has already passed O1 whereas the light from A1 has not yet reached O1. That is, O1 observes the light coming from B1 before observing the light coming from A1. Given (i) that light travels at the same speed c in any direction and in any reference frame, and (ii) that the distance O1A1 equals O1B1 then observer  O1 can only conclude that the event  at B1 occurred before the event at A1 The two events are not simultaneous for O1 even though they are for O2. We thus find that two events which take place at different locations and are simultaneous to one observer, are actually not  simultaneous to a second observer who moves relative to the first. It may be tempting to ask: “Which observer is O1 right, or O2 ” The answer, according to relativity, is that they are both right. There is no “best” reference frame we can choose to determine which observer is right. Both frames are equally good. We can only conclude that simultaneity is not an absolute concept, but is relative. We are not aware of this lack of agreement on simultaneity in everyday life because the effect is noticeable only when the relative speed of the two reference frames is very large (near c), or the distances involved are very large.

from: Giancoli 7th ed Physics Chapters, Ch 26 The special theory of relativity

Giannis Simantirakis Reczko