Recent DKos discussions of some science have revealed a lot of interest in the weird side of modern physics, but also showed that few had much of the needed background. The two great developments of 20th century physics- relativity and quantum mechanics- have each sparked a lot of philosophizing. Of the two, quantum mechanics is the spookier. So though that's where the real interest might lie, I thought it might be good to start an introduction to these topics with the less problematic one, relativity. This is not intended to provide any working knowledge, just some advertisement for fascinating topics and some immunization against bullshit.
One often hears people say that "Einstein showed that everything is relative". Einstein heard people say that a lot more than he liked, prompting him to try to rename the theory "invariants theory"- the theory of what aspects of the world are the same no matter how you look at them. The great point of his work was that such aspects exist, but they are not the ones that you intuitively believe in.
The idea that the world should have some features whose description changes depending on viewpoint and others that don't comes pretty naturally to us. Think of a rectangular cardboard soup box. Its volume doesn't depend on which angle you look at it from. Its width and depth do- they change depending on whether you look at it from the front picture side or the ingredients list side. So under choice of viewing angle, volume, surface area, maximum diagonal length etc. are invariants, but height, width, depth are not.
One great surprise in the physics developed by Galileo and Newton was that the basic laws of physics were the same if you replaced your description with one from the point of view of someone moving steadily with respect to you. That meant that velocity was not an invariant- there's no objective test to say who is moving and who isn't.
That conclusion (which we now call Galilean relativity) was shaken by the development of the laws of electromagnetism- Maxwell's equations. These laws predict, among other things, that light travels at a fixed speed in all directions. According to whom? If the laws work for you, and your friend looks at things from an airplane, should she find that light seems to travel slower toward the front of the plane and faster toward the rear? Your intuition that this should be true is a rigorous consequence of two principles: that the lengths of rods and the time intervals between events are the same according to you and your friend. So either those principles break down or relativity (the equivalence of these viewpoints) breaks down or the laws of electromagnetism break down.
That's an experimental question, and it has a clear answer. The laws of electromagnetism and the principle of relativity are fine. Our intuition about space and time is wrong. Time intervals and rod lengths are not invariant as we switch between different moving frames.
Does that mean that everything is relative? Not at all. Those old invariants are replaced with new, non-intuitive ones, e.g. the speed of light itself.
Starting from these basic assumptions (relativity and Maxwell's equations), one derives a whole new set of relations between quantities such as distances, times, and masses as seen in different frames. F=ma is replaced. Trying to keep the laws of conservation of energy and momentum in some form easily leads to E=mc^2. Relativistic physics is more constrained than non-relativistic physics, because it insists that all the laws (including ones that were quite unknown when it was formulated) must work in any of those "moving" inertial reference frames. This constraint has worked spectacularly well, to extraordinary precision, as a framework for describing the two nuclear forces, not just electromagnetism.
We can ask what things would look like from the point of view of somebody who was not only in steady motion with respect to a good frame, but who was also accelerating. We can logically piece together how their world looks, and we find it's bizarre. For example, identically constructed clocks will seem to run at different rates depending on whether they are being accelerated toward or from, and on how far they are from the observer. An observer on a merry-go-round would not think that its circumference was 2*pi*its radius. These are clearly not effects that belong in our nice laws of physics so we would like to simply exclude such reference frames from the acceptable club.
There's just one hitch. Galileo had noticed that in a gravitational field all objects accelerate in just the same way. As they look at each other, they won't notice any acceleration at all. What if there were no way to distinguish between free-fall acceleration in a gravitational field and no acceleration in no field? (That idea is called the principle of equivalence.) If that were true, identically constructed clocks would run at different rates in the basement and the attic. They do. So we have to accept all the crazy effects that show up in accelerating reference frames because we live in a world with gravity.
The gravity isn't uniform (Australians don't fall away from the Earth), and that makes the effects particularly odd. Clock rates depend on position, the distance around the earth is about an inch short of what Euclid would have calculated given the radius. The set of truths that we can know for certain a priori doesn't include what Kant and our intuitions say that it should. The set of truths is not, however, the empty set. You can calculate how much the satellite clocks used for a GPS system are affected by relativistic effects, and you can correct for it. In fact, your GPS must correct for it, because otherwise it would drift off about 10 km a day.
Relativity ends up being hard on the intuition, but not really of much philosophical import. There's a world, it has objective features, and we can observe them and make precise predictions for them.
That's probably too much for now. If people want more, we can explore some basic quantum mechanics next. You may feel some discomfort.