CHAPTER 5

SPECIAL RELATIVITY

Is there any point to which you would wish to draw my attention?

To the curious incident of the dog in the night-time.

The dog did nothing in the night-time.

That was the curious incident, remarked Sherlock Holmes.

The Memoirs of Sherlock Holmes (1893)

5.1 What Relativity Means

The discovery of Maxwell’s equations elevated the description of electromagnetic phenomena to the same level that the discovery of Newton’s laws of motion had accomplished for mechanical phenomena. However, there was a profound difference between the two theories that would soon completely revamp our notions of space and time and our place in it. Newton’s laws of mechanics look the same in all inertial frames of reference. However, Maxwell’s equations do not.

We will use the term frame of reference a lot in the ensuing discussion. It typically refers to a set of space and time coordinates that a particular observer uses to depict the occurrence of events. You can think of an observer’s frame of reference as his or her laboratory, equipped with rulers, clocks and other measuring devices moving along with the observer and providing him or her with all the tools necessary to make a measurement and a framework within which the results of those measurements may be described. An inertial frame of reference is one that is not accelerated. At most, it moves at constant velocity relative to another, non-accelerated frame of reference.

Confused?

Do not be confused. Obviously, the behavior of mechanical phenomena looks different when observed from different frames of reference, but the underlying laws that describe the behavior do not. Aristotle was wrong when he argued that a spear, thrown vertically upward, would get left behind if the Earth were moving. He concluded that, since it doesn’t, the Earth must be at rest. Galileo destroyed that conclusion with the following argument that gets to the heart of the matter … “Shut yourself up with some friend in the main cabin below decks on some large ship, and have you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully . . . have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.”

Galileo Galilei

What is Galileo saying here? Imagine being inside a train that moving smoothly, free of all vibration, along a track at 60 mph. You flip a coin up in the air. You observe that it leaves your hand, goes straight up and then falls back down and you catch it. Note that, from your perspective, the coin has moved vertically but not horizontally. The behavior is exactly what you would observe if you performed the experiment outside the train on the Earth. In each case, you would use the same laws of physics to calculate the trajectory of the coin. In each case, the calculation would predict the same outcome in agreement with your observation.

Galileo’s words represent the first statement of a relativity principle. What he is saying is that you cannot perform an experiment with moving objects “inside” a particular frame of reference that can tell you whether or not the frame is in a state of constant motion. You can only detect that one frame of reference is in constant motion relative to another frame of reference. You cannot tell which frame is moving in some absolute sense. As we will discuss shortly in more detail, Maxwell’s equations describing electromagnetic phenomena seemed to imply otherwise. Einstein, struggling with this implication, was convinced that something was amiss. He firmly believed that Galileo’s notion was unequivocally true for all physical phenomena including electromagnetic and mechanical ones. With suitable modification, he turned Galileo’s observation into one of the fundamental premises of his theory of special relativity. If one could say that Galileo’s idea was an overture then Einstein’s work was the completion of the entire symphony. The modern basis of special relativity is not just the idea that the appearance of phenomena in different inertial frames of reference is identical, but that all laws of physics that describe the behavior of any natural phenomena are invariant, that is, the laws of physics take on the same mathematical form in different inertial frames of reference.

Let’s extend Galileo’s idea to see what the phrase, invariance of the laws of physics, means. For example, what would the “Earth” observer in the above example see when looking at the experiment of flipping the coin performed by the “train” observer? The Earth observer would see the coin travel not only up and down but horizontally as well. Since the coin would leave the train observer’s hand and return to it after the train had traveled some horizontal distance, the coin would have to travel this same horizontal distance. The coin, seen to travel only up and down by the observer in the train, would travel along a parabolic trajectory as seen from by the observer attached to the Earth. The behavior of the coin would look different to the two observers. The appearance of the outcome of the experiment depends on the relative motion of the two observers. But would they agree that its motion is still describable in each frame of reference in terms of Newton’s laws of motion and the law of gravitation? The answer is --- they would, given certain assumptions about the nature of space and time.

5.2 Galilean Transformations

… Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external."

Sir Isaac Newton, Principia

Newton’s laws of motion exhibit this invariance in a precise, mathematical way. If we express Newton’s laws mathematically in terms of the space and time variables indigenous to one inertial frame of reference and then transform them so that they are expressed in terms of the space and time variables indigenous to a second inertial frame of reference moving at constant velocity with respect to the first, Newton’s laws exhibit the same mathematical form. This is what we mean when we say that the Newton’s laws of motion are invariant

We need to clarify what we mean by the process of transforming Newton’s laws from one inertial frame to another.

· Insert figure of coordinate system at rest and one moving at constant speed

The accompanying figure shows two frames of reference, one at “rest,” like the Earth in the above example, and one moving at constant speed u along the +x-direction, like the train. We denote position and time in the rest frame by the variables x and t and in the moving frame by x’ and t’. For the sake of simplicity, we will assume that all motion is restricted to just a single dimension --- along the x-axis. Suppose an object is moving in the +x direction. Its position and time at that position would be given by the values of x and t in the rest frame. Its position and time in the moving frame would be given by the corresponding values of x’ and t’. How would these observers “connect” their observations, that is, how would they tell each other where the object would be at any particular time?

Let’s construct a simple example. Suppose two observers, each with some kind of ruler and clock that allows each of them to measure positions and the time of occurrence of events in their own frame of reference, are located on “Main Street” in some city. Suppose Main Street runs east and west and the blocks in the city are laid out on a rectangular grid. Numbered streets run north and south and the origin of the city’s fixed coordinate system is the intersection of Main and 0^th Streets. Suppose that one observer remains at rest at the origin and the other walks due East along Main street, at a speed v = one block per minute. Further suppose, that a boy on a skateboard is traveling due East at the rate of u = two blocks per minute, as measured by the observer at rest. Assume that both observers and the skateboarder are located at Main and 0^th streets at time zero and that the two observers set their clocks to zero at this instant, thus insuring that their clocks are synchronized. We will denote positions of the skateboarder in the rest frame by numbers of blocks away from 0^th Street and his positions in the moving frame by numbers of blocks away from the moving observer. At the end of one minute, the skateboarder is 2 blocks east of the “rest” observer, but only one block east of the walking observer. At the end of two minutes, these values are 4 blocks and 2 blocks respectively, etc. In the rest frame, the speed of the skateboarder is u and his position is x = u t. In the moving frame of reference, the speed of the skateboarder is u’ = u – v = 1 block/minute and his position relative to the moving observer is x’ = u’t’.

The equations that relate these two different positions and times are called Galilean transformations. Expressed mathematically, the Galilean transformations that relate position and time in the two frames of reference are

The transformation that relates the speeds is given by

(Obviously the above – sign would be + if the relative direction of motion were changed.)

For example, pick a position x of the skateboarder at any time t as measured in the rest frame, say x = 10 blocks at t = 5 minutes. The transformations tell us that u’ = 1 block/minute, t’ = 5 minutes and x’ = 5 blocks. Note that the form of the Galilean transformations hinge on the assumption that both spatial and time intervals appear to be identical to observers in different inertial frames of reference. In other words, the length of rulers can be compared and clocks can be synchronized when two observers are together and then when one of those observers moves relative to the other, the two different sets of measuring instruments will continue to register correctly any ensuing length and time interval. It all seems reasonable.

The above simple example deals with the limited situation in which the velocity of the moving object, the skateboarder, is a constant. We now address situations in which the object might be accelerating, or changing its velocity. Suppose for example that the skateboarder has some way of gaining speed. Acceleration along a straight path is the rate of change of speed. In the rest frame, the acceleration of the skateboarder is a = Δu/Δt.[1] In the moving frame it is a’ = Δu’/Δt’. Since a time interval Δt’ in the moving frame is equal to a time interval Δt in the rest frame and u’ = u – v (remember, v is constant --- the moving frame of reference travels at a fixed velocity) a small change in the velocity u’ (Δu’) is the same as a small change in the velocity u (Δu) during any interval of time. Thus, accelerations in the two frames of reference are equal

For example, our fearless skateboarder could be picking up speed at the rate of 1 block/minute --- each minute. His acceleration would be a = a’ = 1 block/min² --- the same in both frames of reference.

In order to complete the exercise of transforming Newton’s laws of motion, we need to know how the mathematical description of a force depends on the state of the observer’s motion. No force was included in the above example. Let’s introduce Newton’s law of gravitation. Assume that a mini black hole is located at the eastern end of Main Street. It exerts a gravitational force on all surrounding objects. Since it is extremely massive it does not move. It is attracting our intrepid skateboarder and that is why he is accelerating towards it. In the rest frame, let the fixed position of the black hole be x_b. x is the position of the skateboarder as before. In the moving frame of reference, these positions are x_b’ and x’ respectively. The distance between the black hole and the skateboarder can be calculated from these positions in each frame of reference. They are

Now, we transform the positions in the moving frame to those of the rest frame, obtaining

The distance between the two objects is the same in each frame of reference. Newton’s law of gravitation is thus invariant, i.e.,

… assuming that the masses of the mini black hole M and skateboarder m and the universal constant of gravitation G do not depend on any frame of reference.[2]

Any law of force more complicated than Newton’s law of gravitation would also be invariant as long as it depended only upon the relative positions (and/or velocities) of two interacting bodies. If, however, the law of force depended on absolute positions, i.e., if in the above example, the force between the mini black hole and the skateboarder contained terms like x_b² – x², the force law would no longer be invariant! Nothing like this has ever been discovered. We know of no forces that would make the laws of physics different for an observer in the laboratory and one, say, in a moving train.

We conclude that the laws of motion are indeed invariant under the Galilean transformation. They take on the same mathematical form, F = ma (F’ = ma’) in all inertial frames of reference. Each observer uses the same laws of physics in order to calculate the resultant motion of objects subject to some force.

However, Maxwell’s equations did not exhibit this feature of invariance. When using the Galilean transformations to transform them from one inertial frame of reference to another, they changed form! Maxwell’s theory led to the unequivocal conclusion that light was a wave that traveled at a very specific velocity whose value could be calculated from two constants that related directly to the phenomena of electricity and magnetism.[3] This result implied that the equations of electrodynamics as written down by Maxwell, unlike Newton’s, worked correctly only in a certain absolute frame of reference. Physicists thought that a “luminiferous ether” formed the fabric of this absolute frame and served as the medium through which light waves propagated.

Thus Maxwell’s theory of electromagnetic phenomena implied that different inertial frames of reference were not equivalent --- that all motion was not merely “relative” but that absolute motion did exist and that one ought to be able to find the preferred frame of reference in which the laws of physics were valid by detecting absolute motion through it. For example, an observer traveling in the same direction as a light wave ought to see it slow down, rather like an observer in a car moving at 55 mph looking at another traveling at 65 mph and seeing it recede at only 10 mph. Similarly, the measured speed of the receding light wave would be slower if an observer was trying to catch up with it and the reduction of its speed would determine one’s own absolute speed through the “luminiferous ether.” So it ought to be possible to devise an experiment that could detect changes in the velocity of light, in other words, build an “absolute speedometer,” but one would have to travel fast enough to see an effect and given the enormity of the speed of light, that would not be so easy to do. Nonetheless, in 1886, A. A. Michelson and Edward Morley set out to do just that.

5.3 There Is No Luminiferous Ether!

"… Absolute space, in its own nature, without relation to anything external, remains always similar and immovable …"

Sir Isaac Newton, Principia

By the late 19^th century it had become evident that the ether would have to have very bizarre properties. First, it would have to be extremely rigid in order to propagate a disturbance as fast as the speed of light. On the other hand, it would have to permeate all of space and the material in that space as well, if the astronomer, James Bradley’s observation of stellar aberration, carried out in 1725, could be explained in terms of a wave theory of light. Bradley had measured a periodic shift in the apparent position of stars as the Earth orbited the Sun. This shift is rather like a runner observing a shift in the angle of rainfall as he or she runs though it. If the Earth dragged the ether along with it, or shoved it aside, in its motion around the Sun, this angle would not continually change as in fact it did. If the ether existed, it would have to pass right through the moving Earth, rather like a ghost passing through the walls of a house --- yet this ghost was an extremely rigid one.

On the other hand, scientists knew that light traveled through transparent media like water --- albeit at a speed slower than its speed in a vacuum. In 1818 the brilliant scientist J.A. Fresnel had suggested that flowing water might partially drag the ether along with it so that the speed of light through the water might depend on its speed of flow. H.L. Fizeau demonstrated the truth of this conjecture in an experiment he performed in 1851.

However, the idea of ether drag was difficult to reconcile with the measurement of the aberration of starlight. On the one hand, the ether is partially dragged along inside a moving transparent fluid if we are to understand Fizeau’s experiment, but that drag cannot be communicated to the ether that lies outside an opaque moving body like the Earth if we are to understand Bradley’s stellar aberration. The only way around this dilemma was to assume that the ether remains at rest in absolute space and that it passes completely through moving bodies. In the case of flowing transparent media, it must be the light wave that is partially dragged by flowing material and not the ether within it.

Thus, no matter how a body moves, the ether itself always remains at rest in the “absolute space” that Newton had introduced in his Principia. It was therefore assumed that the ether constituted an absolute frame of reference in which the absolute motion of bodies such as the Earth could be measured. In 1887 Michelson and Morley devised their famous experiment in order to test this hypothesis.

The device they built is one of the most famous pieces of experimental apparatus ever made. Its purpose was to enable Michelson and Morley to see whether a beam of light moving parallel to the Earth’s motion through the ether traveled at a speed different than that of a beam of light moving perpendicular to it. In essence, the device was designed to serve as a speedometer for the motion of the Earth through the ether. The device is called an interferometer and it was capable of making measurement of the highest precision ever achieved.

· Insert figure of interferometer

In the device, a single beam of light is split in two and each half is directed along one of two mutually perpendicular arms. One arm is oriented parallel to the Earth’s motion and the other perpendicular to it. At the end of each arm is a fixed mirror that reflects the two beams back to a common point where they can be observed with an eyepiece. The time of travel for a round trip is different for each beam if the speed of light depends on the Earth’s motion through the ether. Thus, the light waves will arrive somewhat out of phase and will interfere with each other. If the apparatus is rotated by 90 degrees, the interference pattern should change since each beam switches transit times with each other. The result of the Michelson-Morley experiment can be summed up with a quote from their first publication of the results of their experiment

“The interpretation of these results is that there is no displacement of the interference of the interference bands …

The result of the hypothesis of the stationary ether is thus shown to be incorrect.”

It was, and perhaps still is, the most famous “null result” experiment ever performed. The Earth’s motion through the luminiferous ether could not be detected by looking for observed changes in the speed of light. There was none! Classical physics was in crisis!

5.4 The Postulates of Special Relativity

Even though the Michelson-Morley experiment was a classic and had been performed 18 years prior to Einstein’s publication of the special theory of relativity in 1905, there is great controversy surrounding the role it played in shaping Einstein’s resolution of the problems it created for the classical theory of light. He did not mention the Michelson-Morley experiment at all in his publication and when asked many years later about the basis upon which he built his theory, he claimed that he had no knowledge of Michelson-Morley prior to his publication. But there can be no doubt that he knew about it.[4] Indeed, in 1931 he said … the Michelson-Morley experiment …

“… uncovered a serious defect in the ether theory of light, as it then existed, and stimulated the ideas of H.A. Lorentz and Fitzgerald out of which the special theory of relativity developed.” [5]

Nonetheless, it is clear that he was primarily driven to the theory of special relativity not merely by worries about the outcome of any particular experiment but more so by fundamental considerations about the nature of space and time. Einstein had long worried about what he would see if he could run as fast as a beam of light and observe its electric and magnetic fields. Wouldn’t they appear to stand still as though frozen in space? Yet there was no such thing based either on experience or Maxwell’s equations. Thus, Einstein had come to believe that the concept of absolute motion through absolute space was a meaningless concept and that Galileo’s ruminations about the outcome of experiments in mechanics carried out in different frames of reference also applied to electromagnetic phenomena as well. He was thoroughly convinced that all laws of physics, including those of electrodynamics, ought to take on identical forms in different inertial frames of reference. This conviction formed the basis of his first postulate of the special theory of relativity:

1. All inertial frames of reference are equivalent, that is, all laws of physics are invariant under transformations between inertial frames of reference.

Indeed, in the beginning of his 1905 paper he makes this point clear when he notes that the mutual interaction of a magnet and a conductor depends only on their relative velocity --- that the absolute velocity of either is a meaningless concept.

“Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that … the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture … to the status of a postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.”

Moreover, as indicated by the latter part of this quote, Einstein concluded that if Maxwell’s equations were to remain valid in all inertial frames of reference then the speed c that they predicted for the propagation of electromagnetic waves in a vacuum had to be a fundamental constant of nature. In other words, any observer would always obtain the value c upon measuring the speed of light no matter his or her state of motion. Einstein raised this conclusion to the status of a second postulate of special relativity --- a necessary position to take if Maxwell’s equations, previously thought to apply only in some absolute space, were in fact to apply in all inertial frames of reference.

2. The speed of light propagates through empty space at the same speed c in all inertial frames of reference.

5.5 Measuring Intervals of Time and Space

The constancy of the speed of light in nature convinced Einstein that that this unique feature could only be understood by completely revising the concept of space and time. The failure of the Galilean transformations to leave Maxwell’s equations unchanged when transforming them from one reference frame to another convinced Einstein to pursue such a radical course of action. The Galilean transformations changed the form of Maxwell’s equations and led to the prediction that the speed of light depended upon one’s frame of reference. Clearly, another transformation would have to be found that left the speed of light unchanged. However, the Galilean transformations were based on centuries of experience with rulers and clocks employed by different inertial observes to measure space and time intervals. Neither the length of a ruler nor the interval of time depended on their state of motion. These ideas were universally accepted as “common sense.” But Einstein wondered if they were really true. Had these assumptions ever been put to rigorous experimental test?

Moreover, what would happen to the invariance of Newton’s laws of motion if one replaced the Galilean transformations with a different set? Wouldn’t Newton’s laws then suffer the same problem that afflicted Maxwell’s equations under the Galilean transformations? If invariance of all physical laws under the “correct transformations” was a fundamental rule of nature as Einstein claimed, then there had to be something wrong with Newton’s laws! That is the road down which Einstein traveled --- one that led to a formulation of classical dynamics that differed from Newton’s based on a new concept of the nature of space and time.

5.6 Simultaneity

Einstein had come to the realization that all attempts to resolve the relativity paradox were doomed to failure as long as the absolute character of time, in particular of simultaneity, remained anchored in the unconscious. Before Einstein, no one had questioned the validity of the concept of absolute time. Once Einstein recognized the arbitrariness of this concept, he was able to subject space-time concepts to an unprejudiced analysis. During his tenure as an unknown patent clerk in Bern, Switzerland, he began to see difficulties with the measurement of time intervals. While riding home in a streetcar one day, he noticed a clock behind him on a church steeple and wondered how it would appear if his streetcar started moving faster and faster. Wouldn’t the rate at which the clock ticked the seconds begin to slow? When the streetcar neared the speed of light, wouldn’t the clock seem to almost stop? The image of the clock is carried to the observer by light waves and since the observer was moving along with the waves, the hands of the clock would appear frozen.

But what about a clock mounted inside the streetcar? That clock would tick away the seconds exactly like the clock on the church steeple would tick away the seconds as far as an outside “rest” observer was concerned. Maybe time, itself, in a frame of reference in motion relative to an observer outside that frame of reference flows at a different rate?

We can illustrate why Einstein’s postulates of relativity forced him to question whether time behaves the same or differently for different inertial observers. Consider a train that contains an observer O’ with a clock and let him position himself at the exact center of the train. The observer could do this by measuring the length L’ of the train with a ruler that he also carries with him and then placing himself at the position defined by L’/2. However, the observer would have to move relative to the train in order to measure its length this way and since we are calling such measurements into question, we need be careful. In principle, the observer could eliminate any problems that might be introduced by this motion by making certain that the ruler is not moving relative to the train when he measures off any segment of its length.

According to Einstein, there is a better way to make the measurement. The observer O’ could mount two reflective mirrors at each end of the train, walk to the center of the train, stop and emit a flash of light by suddenly turning on a bright light bulb. The light flash would spread outward from its point of origin with two of its beams heading toward each end of the train. Each of these beams would reflect off the mirrors and return to the observer O’. The observer O’ would find himself positioned precisely at the center of the train when he received both reflected light beams at the same time as measured by his clock (see accompanying figure b). He knows this to be true for two essential reasons. (i) Light travels at the same speed c inside the train regardless of its state of motion and therefore if the two beams of light return to the same point in space at the same time, they must have traveled equal distances. Einstein has convinced the observer O’ of this fact (and so has Michelson-Morley). (ii) There can be no question about the simultaneity of the arrival of the two reflected light beams; they each arrive together at the same spatial point. The arrival of the two light beams constitutes two events that are not physically separated. Their times of arrival are measured with the same clock fixed inside the train within which the light moves. These two different methods by which the observer O’ locates the center of the train in fact yield identical results.

· Insert figure of observer O’ locating the center of a train

Let’s now pursue the issue regarding the simultaneity of two spatially separated events. Suppose that the train carrying the observer O’ at its center is traveling at some velocity v down a track and that a “rest” observer O with her own clock, identical to that of observer O’, is positioned outside somewhere along the track downstream of the onrushing train. Further suppose that each observer carries a piece of wire hooked up to their respective clocks such that if the two ends of the wire touch, a circuit is completed that resets or “zeroes” each observer’s clock. Now at the exact instant that the train passes by such that observers O and O’ coincide, the two wires touch, completing the circuit, and the clocks are now zeroed at the same time. Presumably, they are now precisely synchronized and should remain synchronized if they are identical.

· Insert figure of train speeding along and observer O and O’ coinciding

Now suppose, at exactly the instant the two clocks are initially synchronized, two lightening bolts strike each end of the train, creating two flashes of light and scorching each point of impact on the ground and the two ends of the train as well. The light from each flash arrives at O at the same time. Furthermore, she measures the distances to each scorch mark on the ground and determines that they are identical. Thus, observer O concludes that the bolts struck each end of the train simultaneously. She knows this because light from the bolts traveled to him at the same speed c and therefore covered equal distances in the same time.

What about the observer O’? If the events did indeed occur simultaneously as claimed by observer O, then O’ in the moving train must see the light flash arrive from the front end before seeing the one from the back end since the train is moving forward towards the light wave approaching from the front end and away from the one approaching him from the back.

· Insert figures of observer O’ encountering pulse from front (a) then back (b).

But the events are separated by equal spatial intervals in his frame of reference too! Therefore, the two events cannot have occurred simultaneously in the O’ frame of reference if they did occur simultaneously in the O-frame! They occur simultaneously to O’ only if he sees the light flashes arrive at the same time. Why? The two events are equidistant from O’ in his frame of reference and the speed of light carrying the information of the lightening strikes is the same for the light approaching him from both the front and rear end of the train. Therefore if he fails to see the two flashes at the same time, he has no alternative but to conclude that in his frame of reference the two events did not occur simultaneously. A lightening bolt struck the front end of the train before a second bolt struck the rear end! Thus, two spatially separated events that occur simultaneously in one inertial frame do not occur simultaneously in another.

5.7 Time Dilation

The argument just presented implies that elapsed time intervals are not the same in different inertial frames of reference. We are now ready to demonstrate this quantitatively. Let’s consider our train again. This time suppose that observer O’ has a “light clock” as indicated in the accompanying figure.

· Insert figure of light clock at rest in train car.

The light clock consists of two mirrors, one mounted on the ceiling of the train and one mounted on the floor. We imagine a pulse of light bouncing back and forth between the two mirrors such that it registers a “tick” every time it strikes the bottom mirror. The time between ticks is determined by the distance h between the mirrors and the speed c of light, each of which is a constant. The time of one clock tick or the time it takes for a pulse of light to travel from the bottom mirror to the top and back again is

The “ticking” of this clock is now shown from the point of view of the stationary observer O.

· Insert figure of light clock seen by observer positioned outside moving train.

Let Δt be the time between clock ticks as measured by O. As seen by O, the light pulse travels from A to B during a ½ clock tick or Δt / 2. During this time, the light pulse has traveled a distance equal to cΔt / 2 while the train has traveled a distance vΔt / 2. (Here is the root cause of the surprising predictions of relativity. We have used the fact the speed of light is c in both frames of reference, that is, both O and O’ measure the same speed for light.) Now using the Pythagorean theorem, we see that

and solving for Δt

in which we have introduced two dimensionless factors β and γ commonly employed in equations dealing with special relativity. Each factor depends on the relative speed v between two different inertial observers.

β is the ratio of the relative speed of the two frames of reference and γ is a factor that depends only on this ratio. As we will soon see, 0 ≤ β < 1 and 1 ≤ γ < ∞.

What Do You Think About That?

If you look at the above equations that relate the time intervals measured in the two frames of reference, you will note that as the train speeds up such that it approaches the speed of light (v → c), the factor β approaches one (β →1) and the factor γ approaches infinity (γ → ∞). Thus, Δt → ∞. The clock ticking inside the train appears to “freeze” according to the observer located outside the train. This is precisely what Einstein envisioned when riding home from the patent office and looking at the clock on the tower outside. What do you think would happen if the train sped up even more so that it was gong faster than the speed of light c?

The astonishing thing about this result is that the two time intervals are not the same. Let’s be very clear about what is going on here. Suppose the observer outside the train has a light clock that is exactly identical to the one inside the train. (At least they were identical when the two observers set them up in a common laboratory, before observer O’ took off with his on the train.) The observer outside the train observes does not notice anything funny about her clock. It ticks off intervals of time equal to 2h/c as always. The observer inside the train sees nothing funny going on with his clock either. It also ticks off time intervals equal to 2h/c. The problem arises when either observer looks at the other observer’s clock when there is relative motion between the two of them. The other clock appears to be running slow relative to each observer’s own clock. For example, suppose β = 3/5, that is, the train is traveling at 60% of the speed of light. Then γ = 5/4 and observer O positioned along the track sees his clock tick 5 times while the clock on the train ticks 4 times! Clocks moving relative to any observer appear to tick more slowly than his or her own clock ticks, even though they were synchronized when there was no relative velocity between the two and their rate of ticking under those circumstances was measured to be precisely the same. This effect is called time dilation.

The difference between the time intervals Δt and Δt’ is a direct consequence of the second postulate of relative, essentially that the speed of light c in a vacuum is a fundamental constant. Do not think that one of the clocks in one of the frames of reference is somehow “ticking wrong.” It is not. Each observer can measure the rate of ticking of their own clock at any time by checking it against any accurate standard as long as the standard is not in relative motion --- and when they do so, they discover that their clock is ok. The time difference between Δt and Δt’ applies to all clocks --- be they mechanically based like a swinging pendulum, an oscillating spring or a rotating neutron star --- or electrodymamically based like microwaves resonating in a maser. In other words, time itself, flows differently in one frame of reference than it does in another moving relative to it --- as seen by any single observer.

What Do You Think About That?

What a minute! How could each observer think that the other one’s clock is ticking more slowly than his or hers? That doesn’t make sense. Couldn’t they bring them together and compare them to see which one is running slow? Either one of them or the other might run slow if this relativity stuff is right but they both couldn’t run slow. This is clearly impossible.

5.8 Length Contraction

We have just concluded that the duration of elapsed time intervals as measured by two different observers depends on their relative state of motion. We will now see that we have no choice but to conclude the same thing about distance intervals between two points in space. We can demonstrate this by means of a simple thought experiment. We set up a scenario in which an observer O measures the length of an object that is moving with a velocity v relative to her and compares the result that she obtains with the length of the object measured by an observer O’ at rest with respect to it. For example, suppose that observer O’ is inside a space ship and that he knows the value of its length. He can measure this length at any time with a ruler that he carries on board. Since there is no relative motion between his ruler and the spaceship during the measurement process, the length that he obtains is unaffected by its motion through space. Any length measured this way is called a proper length L₀.

Now suppose that the spaceship, carrying observer O’, is traveling through space with a velocity v relative to another observer O, who, for the sake of simplicity, we will say is at “rest” in that space (see accompanying figure).

· Insert figure of rocket flying by rest observer

The observer O’ does not notice anything “funny” about the clocks and rulers that he carries along on board with him. They continue to behave as they always have. However, we know that if the observer O’ uses his clock to measure the time interval between two events that occur at the same spatial point in O’s frame of reference, he will obtain a longer interval than the one measured by observer O. The relation between these two measured time intervals is given by the time dilation formula, Δt’ = γΔt. We will now see what this difference implies about the length of the moving space ship as measured by observer O.

Observer O, knowing that the velocity of the approaching spaceship is v, can measure its length by measuring how long it takes for it to pass by. She does this by starting and stopping her clock when the front and then the back of the ship pass by her location in space. The time interval between these two events, which occur at the same spatial point in his frame of reference, is Δt and she therefore concludes that the length of the ship is

Observer O’ concludes that the elapsed time between these same two events is Δt’ = γΔt. Therefore, he concludes that the length of her spaceship is

But observer O’ knows that this length L’ must be equal to the proper length L₀ of his ship since he is at rest with respect to it and he has already measured its length with his ruler. The length of the spaceship as measured by O is different than the length as measured by O’ as can be seen from the ratio of the above equations

Since γ is always greater than one, the length L of the moving ship appears shorter to the rest observer. This effect is called the length, or Lorentz, contraction, after Henrik A. Lorentz, the physicist who proposed it in 1882, in part to explain the null result of the Michelson-Morley experiment as due to shrinkage of the arm of the interferometer that is parallel to the motion of the Earth.

Like time dilation, length contraction is an effect that is completely symmetrical for the two observers. If the observer O’ were to make a measurement of a similar spaceship whose proper length was L₀, at rest with respect to observer O, he would conclude that its length was shortened by precisely the same factor. In other words, each observer concludes that the length of objects in a frame of reference in motion with respect to him or her, shortens --- or suffers a length contraction.

Focus Box 1 A Trip to Tau Ceti

τ-Ceti, a class G8V star that resembles the Sun and (some think) could serve as the home star for a planet carrying intelligent life, is about 11 LY from Earth. The space craft, Voyager 2, launched back in 1977, is now entering the outskirts of the solar system at a speed of 30 km/s, or about 10^–4 c, the speed of the Earth in orbit about the Sun. Such speeds are the greatest that have yet been attained by any man-made vehicles and, in part, they were attained by using Jupiter and Venus as gravitational slingshots. At such a speed, Voyager 2 would reach τ-Ceti in about 113,000 years if it were traveling in that direction --- which it is not. Even traveling at a speed of 0.1c would require a trip of more than 100 years, still long by the standard of a human lifetime. Clearly, space travelers would need to travel at speeds close to c if they hope to reach τ-Ceti --- with enough vigor remaining in their bodies to allow them to explore its solar system. The problem is even more pronounced if they wish to return to Earth before expiring!

But wait! Time dilation offers a way out. Suppose, we build a spaceship that can accelerate to a speed v = 0.995c and our space travelers travel at that speed to τ-Ceti. According to an observer who stays behind on Earth, it will take our intrepid travelers just a little bit longer than Δt = 11 years to reach their destination (Δt = D/v = 11 LY/0.995 LY/Y ≈ 11.06Y). That’s not so bad. But this is “Earth time” --- the time of travel according to Earth-based clocks. Things get even better for the travelers. Clocks on the spaceship are moving relative to Earth. Therefore, Earth observers will conclude that those moving clocks are ticking more slowly than Earth-based clocks --- that time intervals measured by those clocks are time dilated. Since β = v/c = 0.995, γ = 10 and the time that elapses during the trip for the space travelers is given by Δt’ = Δt / γ = 1.1 years. This, in fact, is the time that elapses for the space travelers according to their clocks. It is their proper time. During the trip, they age only 1.1 years and thus could even return to pick up groceries if they so desire.

But this analysis raises a question. The space travelers know that their speed is v = 0.995c. They can measure their speed by noting how fast they move past any nearby stars. So how can they reach a star that is 11 LY away in only 1.1 years if they travel at a speed just under the speed of light? Can you solve this problem? (Problem 5.5)

· Lengths Perpendicular to the direction of the Relative Velocity

What about lengths perpendicular to the relative velocity between two observers? Do they contract --- or do they maybe expand --- or do they stay the same? Let’s again imagine two observers, O and O’, in motion relative to each other with velocity v directed along common x-x’ axes (see accompanying figure).

· Insert figure of O-O’ moving in x-dir with meter sticks along y-y’ axes.

Each observer has a meter stick oriented vertically along the y-y’ axes. Each observer has carefully positioned their own stick such that one end is fixed to its respective origin. The other is exactly 1-meter away. The question is --- what happens when the origins of the two coordinate systems, in motion relative to each other, coincide? Do the ends at y = 1 meter and y’ = 1 meter also coincide --- or does each observer measure a change in the position of the other observer’s meter stick? Suppose we attach a very thin paintbrush to each of these ends and offset the coordinate systems along the z-z’ axes just a little so that when the meter sticks pass each other they won’t collide and the paintbrush attached to each meter stick will make a mark on the other. Notice that the scenario seen by each observer is perfectly symmetrical. Each observer claims that his or her meter stick is 1 meter long and each claims it is oriented perpendicular to the direction of motion of the approaching meter stick. The two observers agree on the result of the measurement because they agree that the measurements (the marking of each other’s meter stick) occur simultaneously --- the measurements occur at the instant the origins coincide --- and this event takes place at the same point in both frames of reference! After the measurement, each observer will find that either his or her own paintbrush has been marked or that his or her meter stick has been marked below his or her own paintbrush while the other observer finds no mark at all. Both meter sticks cannot shrink or expand. Each meter stick moves with each observer and that meter stick does not change. There is no other logical alternative; either the two meter sticks have the same length or one is absolutely shorter than the other!

The latter possibility is not consistent with the principle of relativity. Suppose, for example, that observer O finds a mark on her meter stick below the paint brush and that observer O’ finds no mark. Each observer concludes that the meter stick of observer O is shorter which therefore implies that it is a special frame of reference. The same argument holds if O’ finds a mark and O does not. Either of these scenarios would therefore give us a basis for preferring one frame of reference over another --- which contradicts the principle of relativity. The laws of physics could not be the same in all inertial frames of reference and we would have a way of detecting absolute motion --- the frame of reference with the meter stick that shrinks in a direction perpendicular to the motion is moving! Thus, length intervals along axes perpendicular to the motion must be the same.

5.9 The Lorentz Transformations

In Section 5.2, we discussed how position and time of occurrence of events could be transformed via the Galilean transformations from one inertial frame of reference to another. However, the Galilean transformations were based on the concept of absolute space and time. If these concepts are incorrect then the Galilean transformations must be incorrect as well. So how do we construct transformations that are consistent with the postulates of special relativity? That is the issue we now address.

We can derive the correct transformations by considering how two observers O and O’, moving relative to each other, measure the position of an event that occurs in the other’s frame of reference. We again use the example of the train carrying the observer O’ along the positive x-direction with velocity v and the observer O at “rest.” along the track. We assume that the observer O’ is positioned at the rear end of the train which serves as the origin of his coordinate system. Shown in the accompanying figure (a) are the two observers O and O’ at the moment they coincide. They each have a wire that connects to a circuit that is energized when the two wires touch at the moment they coincide and starts their clocks ticking so that t = t’ = 0 when x = x’ = 0.

· Figure (a) end of train coincides with rest origin; observer O’ pushes plunger to trigger light pulse in center of train.

· Figure (b) Light pulses in moving train and hit front and back, detonating bombs marking positions x_0’ and x.

Suppose that there are explosives located at the front and back ends of the train that detonate upon receipt of a light pulse. At the moment observers O and O’ coincide and their clocks start ticking, an electric pulse is also generated that travels along a wire toward the center of the train where a light bulb is located. When this pulse reaches the bulb it turns it on, generating a light flash that spreads outward from the center of the train at the speed of light according to observer O’. When the light strikes the front and back of the train (simultaneously according to O’) it detonates the two explosives. The resultant explosions blacken the front and back ends of the train and their positions on the track. According to O’, the position of the back end of the train is x’_0’ = 0 since that is the origin of his coordinate system. The position of the front end of the train is x’ and this value is also the proper length of the train L’ = x’ - x’_0’.

What are the positions of the ends of the train according to observer O? The two marks on the track denote: (i) x_0’, the back of the train which is the origin of the O’ frame of reference and (ii) x, the front of the train. The position of the back of the train is given by

Now the length of the train according to O’ is L’ = x’ while the length according to observer O is

since it is moving relative to him and is Lorentz contracted. Hence, the position of the front of the train must be

Therefore, the transformation that relates positions measured by observers moving relative to each other is given by the equation

Notice that, at slow speeds, v is small compared to c and γ ≈ 1. Transformation equation 5.9.1 reduces to the familiar Galilean transformation given by the first of Equations 5.2.

We can find the transformation that relates time in the two frames of reference by inverting the argument presented above. We place observer O’ at rest outside the train and we place observer O in a train moving in the –x direction with speed v. We then examine how observer O’ interprets lengths measured by O in her frame of reference. Clearly, we will obtain a transformation in which x and t exchange places with x’and t’ and v is replaced by –v in the above equation obtaining

We can now obtain the transformation equation for time by inserting (5.9.1) into (5.9.2) obtaining

Solving for t’, we find

After a little algebra, we get

As before, we can obtain the inverse transformation for time by exchanging x and t with x’ and t’ and replacing v by –v.

Again notice that at slow speeds, v is small compared to c and γ ≈ 1. Transformation equations 5.9.3 and .4 reduce to the familiar Galilean transformation t = t’. The relationship between the spatial coordinates and times of occurrence of events seen in two different inertial frames of reference deviates from that based on common sense experience only at very large velocities. Typically, v must approach speeds ≥ 0.1c in order for relativistic effects to become readily apparent.

In view of the discussion at the end of Section 5.8 regarding lengths perpendicular to the direction of the relative motion, we note that the y-y’ and z-z’ coordinates of any event must be the same in each frame of reference. Thus, the transformation equations for these coordinates are

The above transformations (5.9.1 - .5) are known as the Lorentz transformations. H. A. Lorentz first derived them in 1904 as a way to preserve the form of Maxwell’s equations when transforming them from one inertial frame of reference to another. He was driven to this in hopes of reconciling the null result of the Michelson-Morley experiment with the existence of a unique inertial frame of reference provided by the luminiferous ether. Einstein derived the transformations independently of Lorentz a year later. Their derivation by Lorentz did not imply a breakdown of the concept of absolute space and time as Einstein’s did.

5.10 The Addition of Velocities

After doing battle with --- and almost completely destroying --- the Klingon war bird Ramanu, the starship Enterprise is limping along on impulse power at a speed of v = 0.6c towards the nearest Federation outpost in order to repair its disabled warp drive engines and damaged phaser and photon torpedo launchers. Marduk, Captain of the Ramanu, his disabled war bird now at rest, sends a radio message to another Klingon war bird, the Nergal, patrolling the area just ahead of the oncoming Enterprise, alerting it that the severely damaged Enterprise will soon pass nearby and to intercept and destroy it. Captain Kirk of the Enterprise picks up the message and, in an attempt to thwart the request of Captain Marduk, instantly launches an impulse-driven missile, whose speed is u’ = 0.6c relative to the Enterprise. Just before launch, Kirk reset the course of the Enterprise directly towards the Klingon war bird Nergal, reasoning that, boosted by his ship’s speed, the speed of the missile in the rest frame will be u = u’ + v = 1.2c and thus it will reach the Nergal well ahead of the Klingon message which is traveling at the speed of light. If Kirk is right, then he has succeeded in making something travel faster than light speed and nothing can supposedly do that. What’s going on?

The problem with this scenario is that the observer O’ in the moving ship (Captain Kirk), calculated what the resultant velocity of the missile should be in the observer O frame of reference (the “rest” frame of Klingon Captain Marduk) using the addition of velocities formula given by the Galilean transformation of Equation (5.2.2). This transformation is the “common sense” one based on the idea of an absolute space and time that are not connected in any way. In fact they are connected, according to the special theory of relativity. Space and time intervals between events do not have the same values in different frames of reference.

Kirk should have used the Lorentz transformations to derive the correct formula for the addition of velocities. We can do this by calculating the spatial and time intervals that separate the following two events in the rest frame of reference, given Kirk’s observations in the “moving” frame:

(i) the missile is launched from the space ship

(ii) the missile strikes its target.

Suppose, according to observer O’, Captain Kirk, the target is x’ = 0.3 LY ahead of the position from which the missile is launched. Kirk knows that the launch speed of his missile relative to the moving Enterprise is u’ = 0.6c. Therefore he correctly concludes that, according to his ship’s clock, the missile will strike the target at t’ = x’/u’ = 0.3 LY/(0.6 LY/Y) = 0.5 Y (note, we are using units in which the speed of light is c = 1 LY/Y).

· Insert figure (a) Enterprise launching missile (show coordinates)

· … figure (b) missile reaching target (show coordinates)

Now what does observer O, Captain Marduk, measure for the space and time intervals between these two events?

First we calculate the γ-factor that is needed in the Lorentz transformations. Since β = v/c = 0.6, we obtain γ² = 1/(1 – β²) = 25/16 or γ = 5/4. Now we calculate the distance separating the position where the missile is launched and the position of the target, (Equation 5.9.2), i.e.

The time it takes the missile to travel between those points, (Equation 5.9.4), is