The Dog School of Mathematics Presents
Here was the situation at the turn of the century:
Einstein used these two situations and raised them to the status of postulates. He said, basically, assume it is true that
1) There is no preferred inertial frame. If you feel at rest
then you are at rest as much as anyone can be.
2) The speed of light in a vacuum is measured to give the
same value, c, by any observer.
Postulate 1) is simply Galileo's notion. As long as you feel at rest [your coffee doesn't slosh all over your shirt for no apparent reason] then you are in an inertial frame [the law of INERTIA works]. If I am in an inertial frame (my coffee doesn't slosh) and you are traveling at a uniform velocity WITH RESPECT TO ME then you, according to Galileo and Newton, are also in an inertial frame (i.e. your coffee doesn't slosh). Newton said that there is an absolute inertial frame that is "really" at rest and all other inertial frames are "really" moving WITH RESPECT TO IT. Since their coffee doesn't slosh on their shirts they think that they are "really" at rest but they aren't (sez Newton) but there is no way to find out how they are "really" moving WITH RESPECT TO the "actual" "really-at-rest" inertial frame.
Einstein agreed but eliminated the undetectable preferred inertial frame. If MY coffee doesn't slosh and you are moving at a uniform velocity with respect to me then YOUR coffee won't slosh either. Since my coffee doesn't slosh I can consider myself at rest. Since yours doesn't either you can consider yourself at rest and ME as moving. We are both equally correct since all inertial frames are equally entitled to call themselves "at rest." This is a bit hard for kids to grasp but adults seem to find it to be OK. In either case it is a postulate of special relativity.
Postulate 2) is the one that seems to make no sense on the face of it. It says that if I am in a rocket ship heading toward the sun at 1/2 c and you are in a rocket ship heading away from the sun at 1/2 c then we still both measure the speed of a sunbeam traveling through our rocket-based labs as traveling at c relative to our rocket. On first blush it would seem that if I measure the sunbeam as having velocity c then you should measure it as having velocity 0. This is certainly the way we experience it working for automobiles. When I drive down the road at 55 and the cars in the opposite lane are doing 55 then I see the cars going 110 in the opposite direction that I ("at rest") am pointed. The guy standing by the roadside sees the cars each doing 55 but in opposite directions and the two drivers each see the guy AND the road, trees etc. doing 55 WITH RESPECT TO them.
This is just the Galilean addition of velocities that we worked out before. Einstein sez that light-speed is somehow different. Light is measured by every observer as moving at c in a vacuum.
Einstein's approach was to assume the truth of the two postulates and then find what else had to be changed to make things consistent.
Scientific theories must have two qualities. First, they must be self-consistent; one part of the theory must not contradict another. The second quality they must have is a correspondence with experiment. They must "explain" experimental results both past and future. They must agree with the way nature really works.
Here in these lessons we will concentrate mainly on the self-consistency question. We'll try to get a feel for what SR actually says and see if it makes sense. Then we can worry about whether the universe really works that way.
DISCUSSING POSTULATE 1)
One result of the first postulate is that any "point" in space is really a "line" in space too. To see this consider the traveler on a train who lays a book down on the seat next to herself. She expects the book to be at the same "point" on the seat next to her when she awakens from a short nap. But to the observer who watches all this from the station platform the "point" where the book rests is "really" a line that the book sweeps out as the train rushes by. To the person on the ground the book was laid down in Pittsburgh and picked up, after the nap, in Philadelphia.
So it makes no sense to speak of a point in space without referring to some inertial reference frame. If all inertial frames are equivalent then the concept of a stationary point in space is a relative concept and thus when speaking of a "point" in space we must clarify which inertial frame we have in mind. In other inertial frames (fully equivalent to the first one) that "point" is actually a trajectory.
The concept of an event.
Now if a firecracker went off on the train both the train people and the station people could (ignoring the non-zero duration of the pop that might be evident on a slow-motion film) say that it happened at a certain point at a certain time. Something which is fixed in one frame in both place and time will be fixed in any other inertial frame in both place and time. Such a thing is called an EVENT. Since both space and time are needed to nail down when/where an event occurred we say that an event is a "point" in space-time.
If we come by later and ask where the event happened then there will be disagreement. The station people will say it happened "here" and will point to a spot on the track. The train people, however, will point to a spot on the train (which, according to the station people, is still moving and is a few miles down the track) and say "here."
So we have the two postulates and we're going to try to kludge anything else that we must to avoid violating the two postulates. That which results from this process will be SR. Then the question then will be "does it work in the real world?"
We'll leave that question for later.
Next: Space Trains