Introduction to Special Relativity

When Einstein was first working on developing his special theory of relativity, he was studying both Newtonian mechanics and Maxwell's equations: the two pillars of modern physics of his time. He noticed that these two theories contradicted one another in a very deep way. One of these pillars must fall. Newtonian mechanics implies the Galilean transformation equations—something which we are all already familiar with. These equations are consonant with our everyday intuitions: they tell us that velocities add. This makes sense: If I watch someone standing on top of semi throw a baseball at 5 m/s (in the same direction the truck is moving) and the truck wizzes by me at 30 m/s, then I'll see the baseball travel away from me at 35 m/s. But Maxwell's equations predict something very strange. These equations predict that a light beam will move away from you at the same speed whether your standing still or, as in another one of Einstein's thought experiments, "chasing" the light beam at 99% the speed of light. The reconciliation between Newtonian mechanics and Maxwell's equations is called the special theory of relativity. When we modify Newtonian mechanics to allow for the constancy of light speed for all observers, we get some very strange, bizarre, but also very profound consequences.

Time Dilation

Time dilation is one of the many bizarre consequences of the speed of light being the same for everybody. For any observers in inertial frames of reference moving away from one another at a relative velocity which is close to that of the speed of light, they will see each others clocks run more slowly. In this section, we'll explore one of Einstein's original thought experiment: a train with a light clock moving past an observer standing idly at the train station. We'll see that the constancy of light speed with respect to all observers implies that time must slow down when watching events unfold in one frame of reference from another frame of reference if those two frames of reference are moving relative to each other. We'll start off by showing this for light beams bouncing off of two mirrors, but at the end we'll realize that this same argument applies to photons emitted between two atoms. The latter results in all physical clocks, chemical or biological, running more slowly.