Introduction:
 

        We always think of Einstein when we think of the term relativity. And this is rightly so because Einstein's contributions to the understanding of the world through his theories of relativity were highly significant. We should recognize that there are two theories of Relativity, Einstein's Special Theory, which we explore in this chapter, and the General Theory, which we shall investigate in Chapter Ten.

        We should first of all recognize that the term relativity is one that applies to older perspectives than those developed in this century. Aristotle talked about relativity and Galileo and Newton actually developed some perspectives on the relativity of systems and analyzed motion as viewed from different reference frames. Many mathematicians considered the concepts of relativity in the transformation of refernce frames. Yet their perspectives depended upon some absolute coordinate reference frame.

        Einstein proposed Special Relativity in 1905, one of his three great papers of that year. Since then we have seen how relativity naturally explains phenomena that Newtonian physics cannot begin to address. And the basic ideas of relativity do not require complex calculus and advanced math. A reasonably literate person can do the mathematics needed for relativity so it is accessible to the world. This is not the case with Quantum Mechanics, for example, which requires partial differential equations and other advanced mathematical ability. One should wonder why we spend so much time in the middle schools and high schools teaching Newtonian Mechanics (which besides being very detailed rigor, is very boring). Perhaps a logical conclusion is that laying the groundwork for relativity would not only broaden the perspectives of more people, but take the mystery out of some areas of science.

Galilean Invariance:

        In Galilean Relativity we consider the effects from two different perspectives. We compare for example a ball dropped from the mast of a ship. To the person who drops the ball, it appears that the ball drops directy down to the deck below; i.e., the path is a straight line. Consider an observer who stands on the shore as the ship moves by with some velocity v. The observer sees the ball with an initial velocity in the horizontal direction, relative to the shore reference frame, so the trajectory is not a straight line as perceived by the dropper, but a parabola! The outcome is that the result is the same in both reference frames. The ball hits where both observers  would predict that it would hit.
 

Figure 9-1 Paths seen by stationary observer and moving observer are different.

         We can do the mathematics to show that the motion is equivalent in a transformation from one reference frame to another. In fact, if the ship moves at a constant speed v relative to the shore, then every point x in the stationary reference frame has moved a distance vt in the moving system.  For all three dimesions, with motion just parallel to the x direction only, we can write:

        These are called the Galilean Transformations. Even in the time of Newton, it was recognized that the laws of mechanics worked in any inertial reference frame, so that any observer could predict the outcome of a mechanical experiement. Actually, these work quite well for our normal, classical world. We mean here by inertial reference frame, one that is moving at a constant speed relative to a stationary reference frame. Einstein went a step further to say that all laws of physics, not just mechanics, are valid in inertial reference frames. This becomes one of the fundamental postulates of relativity.

        From this perspective, speeds are also relative. Since time is measured the same, and speed is distance over time, we get:

where u is the speed of one reference frame relative to the other. So if the ball on the ship is given an initial horizontal speed of say 10 m/s relative to the ship, and the ship is moving in the same direction at 10 m/s, the stationary observer on the shore sees a horizontal velocity for the ball of 20 m/s. Similarly, suppose you travel at 30 mph heading east on a city street. A car heading west approaches you. It is also going at 30 mph relative to the street. You see the car approaching you at 30 + 30 or 60 mph. In Galilean Relativity, velocities merely add. This is not simply the case for Einstein's theory. The results do not work, however, when the speed of light is approached. It works fine for low speeds like airplanes and rocket ships which move at only a few thousand miles an hour. The speed of light, on the other hand, is 3 X 10⁸ m/s or roughly 186,000 miles per second, not mph! (It takes light about eight minutes to travel the 92,000,000 miles from the Sun to the Earth.) In the limit of small velocities, the results do in fact come out the same. So Galilean Relativity is the low speed limit of the more general case. (We do the calculations later.)

Postulates of Relativity:

     As we indicated earlier, relative motion between reference frames had been considered long ago. It helped people understand the consistency of ideas regarding how the world worked, especially the nature of motion. But there were indeed, many other studies ongoing throughout the world. Reconciling the outcome of them and integrating their results was necessary and led ultimately to the way we currently understand what relativity theory tells us. One of the interesting discoveries was the outcome of the Michelson-Morley experiment in 1887.

        According to Galilean relativity, velocities add. Michelson and Morley conducted an experiment that attempted to find an absolute reference frame in the universe by measuring the velocity of light in different directions. By adjusting for the differences, the movement of the Earth in the universe could be known.

        They mounted an interferometer on a stone slab floating in a container of liquid Mercury.They would then rotate the apparatus to measure the speed of light as the Earth moved through aether, the medium that supposedly filled the universe. Since the days of the Greeks, aether was thought to exist. Pythagoras discussed its existence and properties.  But instead of finding the speed of the "aether wind" the results showed there was no difference. In other words, the speed of light was the same regardless of the reference frame used. They were awarded the Nobel Prize in 1907 for this work which basically showed that aether did not exist.

        The real meaning of this outcome was not just that aether did not exist. The real significance that it implies is that no matter what reference frame an observer is in, he or she will always measure the same value for the speed of light. This suggests then that even if two reference frames are moving relative to each other, the speed of light observers in each frame measure is always the same. An extreme example might be to take two vehicles heading towards each other at high rates of speed, each a significant fraction of the speed of light itself. (We've never been able to reach such speeds but in fact the idea is still valid to consider.) If each vehicle shines a light forward, the speed of light measured by an observer in the oncoming vehicle is the same as if measured in the stationary lab onboard the vehicle. Something weird seems to be happening.

        Einstein correctly interpreted what this implies and stated two postulates that form the basis for the special theory of relativity. These are:

1. The speed of light is constant in all reference frames. This is not just the outcome of the Michelson-Morley experiment, but the Dutch astronomer De Sitter also showed that the velocity of propogation of light does not depend on the velocity of the body that emitted it.  Einstein remarked that this simple statement, readily accepted by a child in school "... has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties."
2. The laws of physics are valid throughout the universe. This indicates then, that results of experiment conducted in any inertial reference frame will give the same results. In other words, there is no absolute reference frame in the universe and that one need only describe outcomes of measurements in one interial reference frame relative to another. This idea is called the Principle of Relativity.

Simultaneous Events:

        Consider an event that happens. This event happens in both reference frames, but is perceived differently. We note things because of how we observe them, or their effects on the rest of the world. To observe an event, like an atom splitting, or seeing a clock strike 12, it takes time to get the information of that event to the observer in any reference frame. The time for the information to get to the observer in the stationary reference frame in which the event occurs depends on the speed of light. Information travels at the speed of light. Whether we think of it as a photon carrying the information, or light as a wave, we see the event at a finite time later The observer in the moving reference frame sees the event at a different time later.

        Suppose two reference frames are moving apart from each other at a constant speed (in this case the motion is along the x-axis.)

        This brings us to the definition of what we mean by simultaneous events. Basically we mean they take place at the same time. Now here is where it gets interesting. In particular we are interested in what is seen by two different observers in two separate refernce frames. Keep in mind that photons transmit events. If an atom decays, emitting a particle, the knowledge of this event is transmitted to the observer at the speed of light. Two events happen at the same time, if for an observer, midway between them, in some stationary  frame, observes both events at the same time. This is called simultaneity, or simultaneous events. Events that are simultaneous for one observer in a uniformly moving reference frame are not necessarily so for another observer. Time, the measurement of the event taking place is itself, relative.

        Consider a situation in which camera flash bulbs are set off simultaneously (as determined by an observer in the railroad car) at the two ends of the car. The observer in the middle of the railroad car sees the flashes at the same time and she asserts that these are simultaneous events. In Case 2, however, an observer staionary with regards to the moving railroad car, sees the flashes occuring at different times because of the motion of the RR Car. In other words, events that are simultaneous in one refernce frame are not simultaneous in another moving uniformly with respect to the original refernce frame.

        Suppose one observer (in S) moves at a speed v relative to the stationary reference frame (S'). The information of an event in the stationary reference frame must travel (at the speed of light) a distance ct'  (as seen by the stationary observer) while the moving observer is traversing   a distance vt . The information from the event to the moving observer must travel a distance ct . We can, directly from the Pythagorean Theorem, calculate a relationship between the two times t' and t (the times viewed in the moving and stationary reference frames respectively):

Solving, we get:

        In other words, the two observers see different times. The subtle point here is that as v approaches c in the limit, the value of v²/c²   approaches the value of 1 and 1 minus 1 then approaches zero! Dividing by zero is of course undefined mathematically, but as a quantity gets smaller and smaller in the denominator, the value of the result gets exceedingly large. In other words, as v gets very close to c, time dilates to the point that it becomes infinitely large.   t_o is the time measured in a stationary refernce frame and seen by an observer in that reference frame.   L_o is the length measured with a metre stick in that same stationary reference frame.

         Einstein did not even have to do the above calculation for time that shows it is dilated. But he did have to bring meaning to the reults. The set of transformations that give that result, and the following others are called the Lorentz transformations. They were already in place, but need a comprehensive theory to interpret and give meaning to them. They still bear Lorentz's name. We do not care to derive them from elementary principles. They can be done with simple mathematics (high school algebra).  The conclusions are important to us:

        These tell us several things about the world. Note from the form of these relationships that when v approaches c, the speed of light, either the time and mass become infinite and the length goes to zero. This suggests then, that the density becomes infinitely large (density is mass divided by volume.) Einstein said that nothing can ever exceed the speed of light. (It is for our information that theoretical particles, again from a "what if" kind of experiment, that go faster than the speed of light, are called tachyons.) As something approaches the speed of light, it gets infinitesmally small and gets infinitely desne and massive!

        Einstein pointed out that as an object goes fast, it becomes more massive and requires more energy to make it go faster yet. Thus, the speed of light can never be reached because it would require an infinite amount of energy to accelerate an infinite mass.

        We shall explore some examples, but let us keep in mind, that Einstein's great contribution is that the nature of the world depends upon the observer. Now, for very small speeds, even hundreds of thousands of miles an hour like satellites and our space probes (very small compared to the speed of light,  which is 186 thousand miles per second) the ratio of v/c is infinitesmally small and the results look the same as the Newtonian or Galilean transformations.
 

        Recall how velocities added classically. Two vehicles approaching at 30 mph each had a relative closure rate of 60 mph. But, relativistically, if two approached, each at .9 times the speed of light, the closure rate is less than the speed of light, not 1.8c! We do not concern ourselves with the mathematics of addition of velocities, but only the conclusions of the theory.

Time Dilation:

        Time is dilated - or moving clocks appear to run slow. Suppose that Mo is in a Rocket ship, moving at a constant (relatively high) speed relative to Jo. Let us put coordinate systems in each person's lap and make measurements. As a second (or some other convenient unit of time) passes by on Mo's clock, Jo sees a longer period of time! It looks to Jo as if Mo's clock must be running slow. Experiments verify this.
 
 

Mu-mesons, called muons, disintegrate in about 2 X 10^-6 seconds. If we had one in the laboratory, that traveled even at the speed of light,(which is faster than it can go,) it could travel:

         Yet, muons come to Earth in the form of cosmic rays, traveling distances greater than 10 km in the atmosphere. From their point of view, they only live about 2 millionths (10^-6) of a second. But we see that their clocks must run slow, for it looks to us as if they really are living longer because they are traveling greater distances than they could in our reference frame.

        Not only is the measure of time noticed to slow down in moving refernce frames, but this could also be interpreted as the slowing down of physical processes in general. Note that the amount that time is slowed down, or dilated is by a factor of

      For the example above, while the muon "sees" itself living only 10^-6 sec, we see it live

seconds. So if the speed of the particle is say .99C (.99 times the speed of light or getting pretty darn close to the speed of light) then the quantity  Ö ( 1 - v² / c² ) has a value  of  0.14. So 1 / 0.14 is about 7 so the time is dilated or slowed down by a factor of seven times!  So at that rate it can travel the greater distances that we observe it to do. No idea in Newtonian mechanics can explain this phenomena.

Length Contraction:

        Length is contracted. Consider measuring the length of an automobile driving down the street. We could take a picture, or use photocell- detectors, or try to lay our metre stick along side it as it passes us. At slow speeds we could get a good value. At high speeds this gets tough. At relativistic speeds, in fact, we measure short values. In fact, as Moe goes past us at high speeds, Moe measures the length of a metre stick as 1 metre. We see it forshortened.
 
 

        This effect is physically verified in particle accelerators where relativistic speeds are achieved.

        Think for a moment, if you will, what it might mean to travel at the speed of light. As time gets infinitely long and space becomes infinitely short, the photon is everywhere in all space at the same time!

   Effect on Mass:

     Mass gets heavy! As speed is increased, mass increases. This becomes a limiting factor. Recall simple rules of mechanics. A force causes an object to accelerate and the constant of proportionality is the mass. For a given mass, the acceleration is directly proportional to the force. As the mass increases, the force required to accelerate the object must be greater. As mass approaches infinity, an infinite force must be applied to make it go faster. It limits and prevents physical objects from going faster.
 
 

Suppose we apply the Polynomial expansion to this statement where the last term, v /c is a small value. Then we can write:
 
 

        This statement is perhaps one of the most extraordinary outcomes of Einstein's Special Theory of Relativity. It shows clearly that matter and energy are related. In fact, it shows, because of the relative observations, that the total energy of an object depends on its energy  content at rest, plus some kinetic energy of motion. From this point forward, it makes sense to talk, not about energy in the universe, or about mass in the universe, but really to talk about matter-energy itself. It shows the equivalence between matter and energy themselves. It was not until later that the significance of this result became useful, but it is the key to the future, the link that gave us nuclear energy and also nuclear destruction. Einstein himself doubted that the result would ever become useful to mankind.

one other result of the Lorentz Transformations is a description of the universe itself. In normal, Cartesian coordinate systems we can describe the length of line by the standard Pythagorean theorem:
 
 

If we apply the transformation of one system moving relative to another we get a different result, or a measurement of events in two systems moving relative to each other:
 
 

 Note how now we have not just the spatial coordinates involved as we normally might expect, but the additional effect of a fourth dimension, time. Thus space-time is a more appropriate description of the two reference frames, not just space. This is a consequence of the constancy of the speed of light.

Twin Paradox:

      Relativity does not require extremely advanced mathematics. It is accessible to a reasonably literate person capable of doing high school algebra. Consequently many people played with various situations that relativity could be applied to. One of the more famous cases is called the "Twin Paradox."

Discussed in class ... see class notes
 
 

We can summarize the consequences of the Special Theory of Relativity. These include:

1. Simultaneity

2. Space - Time

3. Equivalency of matter-energy

4. Time Dilation

5. Length foreshortening (contraction)

6. Mass Increasing

QUESTIONS
 

1. Describe the qualitative effects of relativity.

2. What is meant by the "special theory of relativity"? i.e. be able to distinguish it from General Relativity.

3. List the postulates and what do they imply?

4. Explain what is meant by the Twin Paradox.

5. Einstein said that objects could not exceed the speed of light. Why can't they?

6.

7.

8.

9.

10.

BIBLIOGRAPHY
 

1. Einstein, Albert, Relativity, Crown Publishers, New York, 15th Edition, 1961.
 

2. Epstein, Lewis C., Relativity Visualized, Insight Press, San Francisco, 1983.