P. Nelson PHYS240/250 Spring 2005
Survey of Special Relativity
Galileo believed that the Earth moved around the sun. Many found this proposition absurd. If the Earth moves, why doesn’t it feel like we’re moving? Why aren’t we thrown off? Galileo patiently constructed arguments about how you can play ping-pong on a ship moving on a calm sea and never notice that the ship is moving. While he didn’t have it completely straight, his successors (Huygens and Newton) eventually elevated this idea to the status of a fundamental principle, which we have come to call The Principle of Relativity: R1 No experiment done within an isolated system can determine whether or how fast that system is moving. In fact if we put all our apparatus in a box the results of any experiment will be the same regardless of whether that box is at rest relative to the Sun or moving in any direction in a straight line at uniform speed. Around the turn of the century the relativity principle seemed once again to be in doubt: Maxwell’s equations didn’t seem to obey it. Some people wanted to abandon relativity; others wanted to modify Maxwell. What Einstein realized was very simple: There is a little freedom in how we interpret the Principle of Relativity. The obvious interpretation used by Newton isn’t the right one, though it’s very nearly right in everyday life. There’s another interpretation that lets us have both Maxwell and relativity. Moreover it has experimentally testable consequences, which have since given us great confidence that Einstein got it right. Einstein’s theory is not something you can deduce from pure thought. There is nothing logically inconsistent with Newton’s theory. Nature just doesn’t happen to work that way. To get at the truth we have to do experiments.
1. Prolog: Rotations Let’s go back to high school for a moment. You know how to take any figure in geometry and rotate it. A square remains a square, an isosceles triangle remains an isosceles triangle, etc.: the identity and character of a figure doesn’t change if we just rotate it. The situation is similar if we leave a 3-dimensional figure unchanged but view it from a different angle. Its perspective will change; maybe its width will seem to shrink while its depth seems to increase. But it’s the same object. Millions of years of evolution have given us brains that automatically compensate for changes in perspective, so that we know it’s the same object. Mathematically, if we set up Cartesian axes we can label every point in the plane by two numbers x . Then the same point viewed from a rotated point of view will be labeled by two y x different numbers y .Wecan find the new coordinates using trigonometry, and the fact that the new coordinate axes are rotated by some angle θ relative to the old ones. There’s a simple, precise formula expressing this: x cos θ sin θ x = (1.1) y − sin θ cos θ y