PHYSICS 107 Lecture 10 Relativity: The Postulates Introduction Relativity represents yet a further step in the direction of abstraction and mathematization of the laws of motion. We are getting further and further away from the common-sense notions of Aristotle. Perhaps Einstein recognized this when he presented his original 1905 theory of relativity as following from two postulates. Math texts always proceed from basic axioms or postulates from which everything else flows by pure logic, and this is exactly how Einstein presented his work. This is actually a bit unusual in physical science. Postulates are specified by the author, whereas the premises in a scientific argument should come from nature (as of course they do in Einstein s theory). That s why even though the logic of premises, arguments and conclusion is present in scientific papers, one does not usually state postulates up front. In any case, Einstein has made it easy for us, since his original presentation is simple and clear. Here goes. Einstein s Postulate Number 1: All inertial frames are equivalent: If the laws of physics hold in one inertial frame then they must hold in all inertial frames. What is not relative, what is absolute if you like, is that the laws, in this case the principle of the conservation of momentum, are unchanged. The laws do not depend on the reference frame. The 1 st postulate is a sort of law of laws it makes a statement about any theory you construct. It seems that Einstein later regretted calling his theory the relativity theory. It was often described outside of science as "everything is relative". But that's only half of the principle of relativity, and not the most important half.
Ironically, even though this is the 1 st postulate of relativity, it also holds for Newton s laws, as we have seen! It's also important to stress that the principle as stated only applies to inertial frames. Inertial frames are those in which Newton's first law of motion (the law of inertia) is valid. If the frame is accelerating, then things are completely different. For example, if you're in a car and there is a ball on the floor when you start the car up and get going the ball will roll backwards. In other words an object at rest will not remain at rest. So it's easy to tell when you're in an accelerated reference frame: things will just start going by themselves. You have a similar experience if you take a tight turn in a car really quickly. Again something might be at rest before you start the turn and you're going at a constant velocity in a straight line but once the turn starts it will be thrown to the side. There is a kind of force, sometimes called a fictitious force, which is opposite in direction to the acceleration of the frame. We will mostly be concerned with inertial frames, but we will again have occasion to discuss non-inertial, accelerated, frames. So that's the first postulate of relativity: that the laws of physics hold in any inertial reference frame. Before we get to the second postulate we need to say something about another development in physics in the 19th century. In the 19th century it was worked out that light is an oscillating electromagnetic wave. The speed of light has been measured pretty accurately. It's about c = 3 10 8 m/s. (It s always called c for some reason. Its exact value is 2.99792458 10 8 m/s.) This is of course so fast that it is essentially instantaneous. When we turn on the lights in a room the walls are actually not illuminated immediately but the time delay is only about 10-8 seconds and the shortest time that we can perceive is only about 10-1 seconds. So for practical purposes the speed of light is infinite for us. And yet with today s very accurate clocks and measuring sticks the speed of light has been measured to very great accuracy. The Speed of Light All of this was worked out in the theory of electricity and magnetism that came to its essentially final form in the work of James Clerk Maxwell in the mid-19th century. But the theory had a strange problem: there didn't seem to be sensible transformation laws from one reference frame to another. It seemed as if there had
to be a preferred reference frame in which the light moved at speed c and then if one moved with respect to this preferred frame the light would be moving faster if you went against its direction and slower if you went along the direction, just like with the cars we have talked about. However an experiment, a famous one called the Michelson-Morley experiment, was done on earth to detect this effect. The speed of light was measured at different times of the year so you would have to be going with the preferred reference frame at one time of year and against it six months later. Therefore you should see the speed of light change. This didn't happen. Instead, the speed of light was the same no matter which direction the earth was moving. (Einstein referenced the Michelson-Morley experiment in his original paper but later said that it was not the main influence on his ideas. Instead it was the structure of the Maxwell equations that motivated him.) By the way, notice that I shifted back to speed here instead of velocity. The velocity v of light can be different in different frames because its direction can be different. But the magnitude c = v is always the same. So Einstein took this as the second postulate of relativity. Einstein s Postulate Number 2: The speed of light is the same in all reference frames. Does this make sense? The short answer is NO! We have a transformation law for velocities that contradicts Einstein's second postulate. Let's go back to our example of me on the side of the road and you in the car in the passing lane. But now, instead of us observing another car, we look at a flash of light going east. Maybe I'm taking a picture of something east of me with a flash. I observe the speed of that burst of light as being c = 3 10 8 m/s. According to our transformation law you should observe it is going slightly slower: v' = c - v rel. So the magnitude of v' is less than c. Our velocity transformation law is in contradiction to Einstein's postulate. And it s in contradiction to the Michelson Morley experiment. How can we resolve this problem? Einstein's answer was to say that our transformation law for velocities is based only on our experience of objects that are traveling much less than the speed of
light. In applying it to our flashbulb experiment we are pushing it beyond its range of validity. He proposed a new transformation law for velocities. This is v' = (v - v rel ) / (1 - v v rel /c 2 ). This transformation law has two great virtues. Virtue #1: It explicitly embodies the second postulate. If observer 1 sees something moving at the speed c, then observer 2 should also see it moving at speed c. So we set v = c and calculate v'. We have v' = (c - v rel ) / (1 - c v rel / c 2 ) = (c - v rel ) / (1 - v rel / c) = c (c - v rel ) / (c - v rel ) = c! So if the speed is c in one reference frame then it is c in all frames! This new transformation law resolves the contradiction between the 2 postulates. Virtue #2 It reduces to the old transformation law when v << c or v rel << c. We set v v rel /c 2 << 1 and so 1 - v v rel /c 2 1. This means that v' = (v - v rel ) / (1 - v v rel /c 2 ) v' = (v - v rel ) / 1 = v v rel. This is just our old transformation law, which is now seen to be still perfectly good at low speeds. (The textbook uses a different notation for the various v s in the problem. You should make sure you understand that the equation are the same with appropriate changes in the definitions.)
Historical Aside The theory of relativity is a nice example of how revolutions in science generally work. A new scientific theory must still explain all of those things that were explained successfully by the previous theory. Quite often this happens similarly to the way that Einstein's relativity replaced Newtonian mechanics. The Newtonian formula appears in Einstein's formula as a limit when all of the velocities of objects are much less than c. In this sense Einstein's theory is a generalization of Newtonian theory rather than a contradiction of it. We might say that it's rather less revolutionary than Newton's theory which really did overturn Aristotelian theory in almost every conceptual way. Nevertheless we're going to see that many things that were taken for granted before Einstein turn out to be wrong. It's not only the velocity transformation law that changes. Further Reading Einstein himself wrote a very nice short book on relativity for the general reader: Relativity The Special and General Theory,15 th ed., (Methuen, London, 1954). Another very nice treatment is Relativity for the Millions (Cardinal, NY, 1965), by Martin Gardner