SPECIAL RELATIVITY

At the beginning of the last century, physics underwent two major revolutions, the discoveries of relativity and quantum mechanics. Here, and for the next two or three weeks we will talk about Einstein's special theory of relativity. It, in a certain sense, is based on a simple idea. However, from its seemingly mundane underpinnings, profound and bizarre implications follow. We look at some of these issues in the following sessions.

SPACE AND TIME

Let us set the stage. How do we describe the location of events which occur in the Universe? Is it sufficient to give simply the where of the event? No, in order to define uniquely the event, we must also say when the event takes place. That is, when we define events in our Universe we must give both the location and time of the event. We give the space-time location of the event. In this sense, time then acts as just another coordinate in the Universe much as we define x,y,z positions for objects in space. This, however, immediately reminds us, however, that time is held in higher esteem than space because although time is apparently just another coordinate for Universal events, it has been accorded special properties--we return to the specialness of time later. Space and time form a structure, spacetime, on which (in which) events in our Universe unfold. This opens up the question, then, of whether space and time are things or simply mathematical conveniences (constructions).

Issac Newton and the Clockwork Universe

In Philosophiae Naturalis Principia Mathematica (1687), Newton gave his three laws of motion which laid out a prescription for how motions in our Universe marched forward in time, into the future, based only conditions given today. (Future events can be predicted from what we observe today; events are completely deterministic).

Newton, after due consideration, decided that the space and time of our Universe were fixed; space was the rigid framework within which events took place, and time acted as a smoothly (and uniformly) flowing quantity, the same for all observers in the Universe. Time was kept by some Universal clock which was synchronized for all obsevers and inexorably marched forward uniformly and at the same rate for all observers. The Newtonian theory and these notions for space and time work nicely for normal daily events (such as placing astronauts on the Moon) and are perfectly valid for most commonplace situations. (They do breakdown under certain conditions, high speeds, small scales, and/or low energies which we address later.)

Newtonian Velocity Addition

Vstationary = vtruck + vthrow

    In Newton's Universe things behave as your common sense tells you they must. To As an example of how things change in Einstein's world view, consider the following simple exercise.

    In the panel shown to the left, we consider the notion of velocity addition. There are two observers, one on the truck which moves with speed 15 m/s and another observer standing on the ground. The truck guy throws a ball to stationary guy (ground guy). The ball leaves truck guy's hand with a speed of 15 m/s. Sensibly enough, ground guy catches a ball traveling toward him with a speed of 15 m/s + 15 m/s = 30 m/s. This is our common sense simple addition of velocities rule. This is true in Newton's world where space is rigid and time flows smoothly and at the same rate for both observers.

This all seems sensible (it agrees with our notions of common sense). However, toward the mid to late 1800s, issues were bubbling up which weren't consistent with this notion of Newton. The most important of which was the theoretical suggestion that perhaps light traveled at the same speed in all frames of reference even if the frames of reference were moving (such as the truck) (James Clerk Maxwell and his theory of Electromagnetism). This set the stage for Einstein.

Einstein's Universe

Einstein made several bold assertions, a consequence of which was that Newton's simple velocity addition rule breaks down for objects traveling at high speeds, speeds close to the speed of light, c = 186,000 miles per second or 670,000,000 miles per hour! Things are okay when we move at speeds slower than this. Cars move at 70 mph and even space probes travel at only 25,000-40,000 mph. These speeds are much smaller than c and so Einstein's bold predictions don't strongly affect our eeryday acitivites.

A consequence of Einstein's assertions is illustrated in the panel to the right. You sit on a stationary platform and shine a flashlight to the right. The beam travels away from you with speed c. Einstein sitting in a bullet train passes by you which travels with speed v nearly the speed of c (pretty zippy). Mr. Newton would say that because Einstein is in a train moving at the same speed as the beam of light, that Einstein would see the beam of light traveling with speed

V = c - c ~ 0

In Mr. Einstein's Universe, the remarkable prediction is that Einstein would see the beam of light traveling away from him at the speed of light c. Whoa, what happened to the law of velocity addition?

Einsteinian Velocity Addition

Einstein made two assertions. One is straightforward enough, but the second on reflection is odd and leads to many strange and wonderful conclusions. Einstein asserted that:

The first assertion is sensible enough as we are already familiar with such things. For example, if you are riding in an airplane which travels at a speed of 500 mph, you notice that if you drop your pen it falls straight down (with respect to you). It also falls down at the rate predicted by gravity. This occurs despite the fact you are in the moving airplane. In your little world (your reference frame), all things behave as if you stationary in your home sitting on your couch on the Earth. (Uhhh, but are you really stationary in your home on the Earth; there are tweaks that should be made to some of my comments.)

The second assertion (in a sense, simply helps to guarantee that assertion one is correct), however, on its face it seems much more remarkable and leads to some bizarre consequences which we now explore.

Oh, before we get too far afield, let me point out that Einstein's postulates given above are simply assertions. The way science works is then that the implications of these assertions are figured out (that is, the consequences which follow if the above are true are predicted). These predictions (in many cases bizarre predictions) are then experimentally tested. So far in no instance has Special Relativity been shown to be incorrect. I point this out before we go further as many of the things I will get into will be outside the realm of your common sense and thus seem fantastical.

    The strange idea that the speed of a beam of light measured by different observers will always be c requires some gyrations on our part. What this result leads to is that time must flow differently for different observers. This strange result is time dilation. Let's try to explain how this works.
In the left panel, we see a series of pictures (from a topside view) of a train moving to the right. In the train, a person shines a beam at a mirror on the opposite wall. The beam reflects off the wall and returns to the observer. According to the observer in the train, the path followed was of length 2 x d and the trip took an amount of time Train Guy Time = 2 x d/c.

An observer on the ground watches the train go by and sees the same events; the beam traveling to the far wall, reflecting off the wall and returning to the observer. Because the train is moving, the stationary observer sees the beam of light follow a longer path 2 x D on its round-trip. Because the observer on the ground also sees the beam of light travel at c, he measures a length of time for the trip of Stationary Guy Time = 2 x D/c > Train Guy. Because we are talking about the same trip, this must mean that Train Guy's watch ran more slowly than did stationary guy's watch since less time elapsed on the trip. Moving guy's clock ran more slowly than did Stationary guy's clock. Let's compare the two times. We have

T = 2 x D / c = 2 (d2+[vT/2]2)1/2 / c

After a slight manipulation,

T = (2 x d/c) / (1-[v/c]2)1/2

Note that 2 x d /c is the time as measured by the moving observer so that

T = T' / (1-[v/c]2)1/2

Here, we let T' denote the time as measured by the moving observer. Note that if the train moves at the speed of light, then no time elapses according to the clock of the moving observer, T' must be 0! This effect is known as Time Dilation.

The remarkable thing is that all that is required is for there to be relative motion between the observers so that from moving guy's perspective, staionary guy's clock would run more slowly than his own because staionary guy appears to be moving with respect to him! This notion of relativity leads to the Twin Paradox.

Click on the attached for a nice time dilation video