Theory of Special Relativity

The postulates of the special theory of relativity may be stated as follows:

1. There is no preferred inertial frame. That is, the laws of physics must take the same form in all frames moving with constant velocity relative to each other.

2. The speed of light is independent of the motion of the source and the observer.

or

Light propagates without an ether.

A. Simultaneity

A train is moving with velocity v relative to "Earth." Observers on the train will be indicated by primed letters on those on the Earth by unprimed letters. The following events occur.

Two light flashes occur. One flash occurs at point A′ in the primed frame which corresponds to point A in the unprimed frame. The second flash occurs at point B′ in the primed frame which corresponds to point B in the unprimed frame.

Event 1
		A	B

At some later time, the wave fronts of the two light flashes come together at point C′ in the primed frame which corresponds to point D in the unprimed frame. These events didn't have to occur this way.

	C’
A’		B’

Event 2
		A

This is just one possible sequence of events out of many that was observed to occur. It is also determined from measurement that A′C′= C′B′; point C′ is the midpoint of the train.

The question is, "Did the flashes at A′ and B′ occur at the same time and did the flashes at A and B occur at the same time?."

a) From postulate 1, we know that we can reason this out in any frame.

b) In the primed (train) frame, one concludes that flashes at A′ and B′ occurred at the same time since they met at a point equidistant from A′ and B′. In other words, since the speed of light does not depend on the motion of the source or the observer and since the light traveled equal distances to reach point C′, both flashes must have occurred simultaneously.

c) In the unprimed frame, a different story emerges. Point D is not midway between points A and B. Therefore, AD≠DB. In the unprimed frame light flashes occur at points A and B, and they meet at point D. Since the light from point A had to travel a larger distance to reach point D than the light from point B, the flash at point A must have occurred before the flash at point B. The events are not simultaneous in the unprimed frame.

Who's right? Were the events simultaneous or not? The answer, of course, is that the results in each inertial frame are equally valid so that we must conclude that

Spatially separated events which are simultaneous in one inertial frame will not be simultaneous in another inertial frame that is moving relative to the first frame.

B. Length

We define the length of an object as the distance between the endpoints of the object measured at the same time in a given inertial frame. There is only one inertial frame in which we don't have to worry about measuring the endpoints of the object at the same time. That is the rest frame of the object. In the object's rest frame, the endpoints are not moving so we can measure them at out leisure.

Now consider the following event.

Rumor has it that a train is to pass by the unprimed frame at about noon. The observers in the unprimed frame decide to measure the length of the train. Since they have observers everywhere with synchronized clocks, they decide that the observers who are at the endpoints of the train at noon will emit light flashes. Since the flashes will occur simultaneously in the unprimed frame, the length of the train is just the distance between the observers A and B in that frame.

The length of the train is A′B′= L₀ as measured in the train frame. The question is, "What length of the train do the unprimed observers measure?" Since it is all right to do the calculation in any frame, let us do it in the train frame.

According to train observers, lights A and B were not flashed at the same time (see Simultaneity A, above). They say that A flashed his light after B (reason this out for yourself as in case A). Thus the primed observer reasons as follows:

At some time t₀, B flashed his light at one end of the train. A time δt later, A flashes his light at the other end of the train, but the train has moved in that time.

Therefore, L = AB, which is what an unprimed observers call the length of the train, is less than L₀!

Since our calculations may be done in any frame, this must be the result of the measurement in the unprimed frame. Consequently,

The length of an object depends upon the inertial frame in which it's measured. Its length is longest in its rest frame.

The actual equation is

L = L₀√[1-(v/c)²]

where v = train speed and c = speed of light. Since (v/c)<<1 for objects seen in our everyday life, this effect is negligibly small for such objects.

C. Time Intervals

In some sense, the relativity of time intervals is complementary to that of space intervals. Consider the following "mystery." A clock is moving towards a wall in the unprimed frame. At t=0 (in both frames), it is a distance L from the wall as measured in the unprimed frame. It is traveling with speed v relative to the unprimed frame. Eventually it hits the wall and it stops. The question is, "Does it stop at a time equal to L/v, greater than L/v, or less that L/v?" We can do this calculation from the clock frame. In the clock frame the distance to the wall is less than L; therefore, it takes less time than L/v to reach the wall. The time will be less than L/v. The conclusion from the unprimed frame is that the clock must be running slow. Our conclusion is that

Time intervals depend on the inertial frame in which they are measured. The shortest time interval occurs in an object's rest frame. In other words, if a time interval τ₀ (proper time) passes in an object's rest frame, a longer time interval will pass in any inertial frame moving relative to the objects rest frame.

The equation is

τ=τ₀/√[1-(v/c)²].