# Special Relativity

Michael Fowler, UVa

## Galilean Relativity again

At this point in the course, we finally enter the twentieth century—Albert Einstein wrote his first paper on relativity in 1905.  To put his work in context, let us first review just what is meant by “relativity” in physics.  The first example, mentioned in a previous lecture, is what is called “Galilean relativity” and is nothing but Galileo’s perception that by observing the motion of objects, alive or dead, in a closed room there is no way to tell if the room is at rest or is in fact in a boat moving at a steady speed in a fixed direction.  (You can tell if the room is accelerating or turning around.)  Everything looks the same in a room in steady motion as it does in a room at rest.  After Newton formulated his Laws of Motion, describing how bodies move in response to forces and so on, physicists reformulated Galileo’s observation in a slightly more technical, but equivalent, way: they said the laws of physics are the same in a uniformly moving room as they are in a room at rest.  In other words, the same force produces the same acceleration, and an object experiencing no force moves at a steady speed in a straight line in either case.  Of course, talking in these terms implies that we have clocks and rulers available so that we can actually time the motion of a body over a measured distance, so the physicist envisions the room in question to have calibrations along all the walls, so the position of anything can be measured, and a good clock to time motion.  Such a suitably equipped room is called a “frame of reference”—the calibrations on the walls are seen as a frame which you can use to specify the precise position of an object at a given time.  (This is the same as a set of “coordinates”.)  Anyway, the bottom line is that no amount of measuring of motions of objects in the “frame of reference” will tell you whether this is a frame at rest or one moving at a steady velocity.

What exactly do we mean by a frame “at rest” anyway?  This seems obvious from our perspective as creatures who live on the surface of the earth—we mean, of course, at rest relative to fixed objects on the earth’s surface.  Actually, the earth’s rotation means this isn’t quite a fixed frame, and also the earth is moving in orbit at 18 miles per second.  From an astronaut’s point of view, then, a frame fixed relative to the sun might seem more reasonable.  But why stop there?  We believe the laws of physics are good throughout the universe.  Let us consider somewhere in space far from the sun, even far from our galaxy.  We would see galaxies in all directions, all moving in different ways.  Suppose we now set up a frame of reference and check that Newton’s laws still work.  In particular, we check that the First Law holds—that a body experiencing no force moves at a steady speed in a straight line.  This First law is often referred to as The Principle of Inertia, and a frame in which it holds is called an Inertial Frame.  Then we set up another frame of reference, moving at a steady velocity relative to the first one, and find that Newton’s laws are o.k. in this frame too.  The point to notice here is that it is not at all obvious which—if either—of these frames is “at rest”.  We can, however, assert that they are both inertial frames, after we’ve checked that in both of them, a body with no forces acting on it moves at a steady speed in a straight line (the speed could be zero).  In this situation, Michelson would have said that a frame “at rest” is one at rest relative to the aether.  However, his own experiment found motion through the aether to be undetectable, so how would we ever know we were in the right frame?

As we mentioned in the last lecture, in the middle of the nineteenth century there was a substantial advance in the understanding of electric and magnetic fields.  (In fact, this advance is in large part responsible for the improvement in living standards since that time.)  The new understanding was summarized in a set of equations called Maxwell’s equations describing how electric and magnetic fields interact and give rise to each other, just as, two centuries earlier, the new understanding of dynamics was summarized in the set of equations called Newton’s laws.  The important thing about Maxwell’s equations for our present purposes is that they predicted waves made up of electric and magnetic fields that moved at 3×108 meters per second, and it was immediately realized that this was no coincidence—light waves must be nothing but waving electric and magnetic fields.  (This is now fully established to be the case.)

It is worth emphasizing that Maxwell’s work predicted the speed of light from the results of experiments that were not thought at the time they were done to have anything to do with light—experiments on, for example, the strength of electric field produced by waving a magnet.  Maxwell was able to deduce a speed for waves like this using methods analogous to those by which earlier scientists had figured out the speed of sound from a knowledge of the density and the springiness of air.