Special Relativity: BasicsHigh-energy astrophysics involves not only light, which is intrinsically relativistic, butalso particles that are accelerated to nearly the speed of light. Newtonian mechanics thereforehas to be supplanted by special relativity. In this lecture and the next, we will go over someof the principles and applications of the special theory. In later lectures, we will considergeneral relativity, which generalizes these principles to accelerated frames and turns out tobe our best current theory of gravity. For basic physics such as this, by the way, my opinionis that any serious physicist should at some point read the Feynman Lectures on Physics.His clarity of thought was exceptional, and probably the best way to approach those volumesis to look at them after you have already had a course in a given subject; it allows you toappreciate his profound insights better.PhilosophyFirst, let’s start with a little philosophy. After the fact, it is easy to present physicalprinciples as if they are self-evident and derivable from pure mathematics. This is not thecase. We can marvel at the brilliance of Einstein and the other pioneers of relativity, andappreciate the philosophical way that they drew their conclusions, but to be scientific onemust at some point have contact with experiments. Therefore, ultimately, we have to pointto the universe as a whole (or at least, what we’ve probed observationally) to argue that thetheory is correct.A second philosophical point that many people mistakenly derive from relativity, prob-ably because of the name of the theory, is that the essential point is “everything is relative”.In fact, one of the postulates of relativity, and one of its deepest points, is that there aresome quantities that are invariant, meaning that all observers will measure the same valuefor those quantities. We’ll try to emphasize such invariants when we derive aspects of specialrelativity.Galilean RelativityWe should also not get the idea that Einstein was the first one to suggest a principle ofrelativity. In fact, Galileo used thought experiments quite similar to Einstein’s to show thatsomething coasting along at a constant velocity should experience all the same local effectsas something at rest. He asked his readers to consider experiments performed by someone ina ship’s cabin if the ship is moving at a constant speed. He notes that a ball tossed straightup will appear to come straight down; a tank of water will remain level; and in general theexperimenter will not be able to tell that the ship is moving. From our standpoint a morefamiliar and extreme example is traveling in a plane. We might be going 75% of the speedof sound relative to the ground, but we can still be served bad food without it ending up inour faces!Put more formally, all local experiments we do in an inertial frame will turn out thesame independent of our velocity relative to a given frame. However, note the restrictions tolocal experiments and inertial frames. If you somehow opened the window of your plane andstuck your head out, it would be the last thing you ever did; there is a quite clear differencein physical effects when you have contact with other frames! In addition, when the planeaccelerates (e.g., by hitting turbulence) it is sickeningly clear that you are not at rest. Inmore benign situations, such as experiments on a rotating Earth, the non-inertial nature ofthe frame leads one to introduce fictitious forces such as the Coriolis force.How, then, would we phrase Galilean relativity mathematically? A useful way to do thisis to consider two observers moving at a constant velocity v relative to each other. Let usset up Cartesian coordinate systems for both: for one frame the coordinates are (t, x, y, z)and for the other are (t0, x0, y0, z0). We will refer to these as, respectively, the unprimed andprimed frames. Here t means time, and we will make our lives easier by ensuring that the xaxis is parallel to the x0axis, and similarly for y and z.Suppose that, as seen in the unprimed system, the primed system is moving in the +xdirection with speed v. Note that we can always rotate our coordinate axes so that the xaxis lines up with this speed; if you prefer making your algebra messier you can always doit more generally, but we won’t bother. If we set up our initial conditions so that at timet = t0= 0 we have x0= 0 (i.e., the origins of the two systems are coincident), this impliesthat at time t, the origin of the primed system is at x = vt as measured in the unprimedsystem. Of course, in the primed system, the origin is always at x0= 0. In addition,the perpendicular directions y and z are equal to their primed counterparts, and t = t0.Therefore, the coordinate transformation for Galilean relativity becomesx0= x − vty0= yz0= zt0= t .(1)We also find that Newton’s laws of motion are invariant in form under these transfor-mations. This is as expected, and is a consequence of our inability to tell whether we aremoving steadily or not from purely local experiments. Among other things, this law tells ushow velocities should add. Consider, for example, something that moves with speed u in thex direction as seen in the unprimed frame. Therefore, dx/dt = u. In the primed frame wehaveu0= dx0/dt0= d(x − vt)/dt = dx/dt − v = u − v . (2)This is the simple, intuitive result. If a train goes by me at 100 km/hr and I throw a baseballparallel to the train at 100 km/hr, someone inside the train sees the ball not moving in thatdirection at all. If I throw antiparallel to the train at 100 km/hr, the person in the train seesa speed of 200 km/hr. Note, by the way, that if we want to transform from the primed frameto the unprimed frame, all we have to do is reverse the sign of v and switch the primed withunprimed variables. Very simple.The Problem with Maxwell’s EquationsIn the mid-1800s, however, a problem emerged. After many people had for severaldecades experimented with electricity and magnetism, James Clerk Maxwell came up witha compact set of equations that beautifully described all the phenomena. To this day,Maxwell’s achievement ranks among the very greatest in the history of physics. Surprisingly,though, Maxwell’s equations are not invariant under a Galilean transformation. For example,a blatant contradiction emerges when one tries to determine the speed of light in differentframes with this theory. According to this theory, the propagation speed was the samewhether