Outer Limits of Reason (43 page)

These two simple postulates are all we need to derive the results of special relativity.

Length Contraction and Time Dilation

How can we unify these two postulates? How can two observers moving at different speeds agree about the speed of light? First let us meditate a little on measuring speeds. To calculate how fast a car is going, we set a particular distance and measure the time it takes the car to cover that distance. The speed is then the distance divided by the time. So 50 miles per hour means that the car can cover 50 miles in one hour. Imagine that we know that a car is traveling 50 miles per hour but for some odd reason we measure it traveling at only 30 miles per hour. How can that be? We must be making a mistake in our measuring. Since speed is distance divided by time, we must be making a mistake about the distance or the time. Our error must come from the fact that the measuring stick we use to calculate the distance is wrong, or the clock we use to calculate the time is off, or it could be a combination of both. This is the only way we can account for our error.

Now let us return to the speed of light. It is measured by the distance (space) it traveled in a certain interval (time). If there is an “error” in the way the light is observed, then there must be something wrong with the way space and time are measured. Imagine Captain Kirk firing a phaser gun while two space shuttles are observing the action. One space shuttle is stationary and the other is moving in the same direction as the light (as in
figure 7.24
).

Postulate 2 tells us that they both see the light traveling at 186,000 miles per second. Let us say that the stationary space shuttle has measured the “correct” distance and time to calculate the speed. What about the moving space shuttle? Since it is moving, one would expect its passengers to perceive the light going a little slower. But in fact they also see the light traveling at 186,000 miles per second. The only way this “error” can occur is if they measure the distance and time “incorrectly.” That is, their measuring rods must have shortened so that the distance they measured is “incorrect,” and their time clock must have slowed down so that the duration they measured is “incorrect.” In fact, this is exactly what happens! Their measuring rods get shorter in a phenomenon called “length contraction” and their clocks go slower in a phenomenon called “time dilation.” Since this is a natural process, it is wrong to call one view correct and the other incorrect. Both views are correct.

The first thing we must realize is that it is not only the measuring rods that shrink: everything in the moving space shuttle shrinks. In fact, the shuttle itself shrinks. Astronauts standing in the moving spaceship will be thinner. When they are lying down in the direction in which it is moving, they get shorter. Since everything goes through this length contraction, it is not noticeable to people in the moving shuttle. In contrast, it is noticeable to the person in the stationary shuttle.

Similarly, not only do the stopwatches used to measure speeds slow down, but all clocks and all processes slow down. These processes include the heart rate and chemical reactions in the bodies of astronauts. Their aging process slows down as well. Again, this will not be noticeable to any observer in the moving shuttle, but it will be perceptible to someone in the stationary shuttle.

It must be stressed that it is not the case that the moving space shuttle
appears
to shrink or
seems
like it is shrinking. Rather,
it shrinks
. This is also true of time dilation. These are not illusions. They are basic facts about the universe we live in: moving objects go through space contraction and time dilation.

These effects are noticeable to a stationary observer only when the rocket moves near the speed of light. We do not see things move even remotely close to the speed of light. Remember, even 1 percent of the speed of light is 1,860 miles per second. There are no vehicles that can even come close to that. In most cases no length contraction or time dilation will be perceptible. However, in a laboratory setting these changes can be measured.

Physicists have formulated equations that indicate exactly how much length contraction objects will experience and how slow time will progress relative to the stationary observer. These equations take into account the velocity of the observer. The faster the movement, the more space contraction and time dilation will occur. What is the limit of this process? What if people could go very, very fast? If they were actually able to go the speed of light, they would shrink to nothingness and time would totally stop for them. That is, they could not exist. This is yet another consequence of special relativity: there is a type of cosmic speed limit. Nothing can move as fast as, or faster than, the speed of light. This is, quite literally, a limitation described by science.

There is an urge to simply wave away all this talk of relativity and insist on absolute space and time. One wants to merely declare that the measurements done from a stationary position on Earth are the absolute measurements and every other measurement is relative. This would be an error. Although it seems like the Earth is not moving, it is, in fact, constantly moving in a wild pattern. Remember that the Earth is spinning on its axis at about 1,000 miles per hour. It is rotating around the sun at about 67,000 miles per hour. Furthermore, our solar system is moving around our galaxy at about a half a million miles per hour. Poke your finger into the air. Wait a second. Now poke your finger “in the same place.” Realize that the two places where you poked your finger are hundreds, if not thousands, of miles apart. A stationary observer on Earth is far from stationary. There are no absolute observers, no absolute measurements, and no absolute space and time. All is relative.

How does time slow down? How can clocks possibly slow down? All clocks work with some type of chemical or physical process. Whether it is a battery or a wind-up spring, clocks move forward with such processes. To see an example of a way that moving fast affects time, consider a clock that works using light. Imagine a clock that works with two mirrors and a light pulse bouncing between them, as in the top part of
figure 7.25
.

The clock works by going forward one second every 10,000 times the light bounces back and forth. Since the speed of light is constant in the universe, this would make a good clock. Now imagine the same clock speeding across the universe. The light will still be bouncing back and forth but now the light is bouncing diagonally. The diagonal path is always longer, so the clock will still work but the light will have to travel longer in order to perform its 10,000 laps. Therefore, to the stationary person, the moving clock will appear to be ticking at a slower rate.

There is actually experimental evidence for time dilation. In 1971 four atomic clocks were placed on planes that flew around the Earth. When the planes returned, the time on the clocks was compared to atomic clocks that were stationary on the ground. It was found that the frequent-flier clocks had actually lost time. Science fiction writers take this idea to the extreme with something called the
twins paradox
. Imagine an astronaut who leaves a twin sibling on Earth and then travels through space very close to the speed of light. If the astronaut goes fast enough, they would come back to Earth without aging much while the stationary twin is old and decrepit. In a sense, special relativity permits you to visit the future. Unfortunately, there does not seem to be a way to return.

Consider the implications of time dilation when velocity increases. Examine the path at the left in
figure 7.26
.

A person is moving northward and only slightly to the east. At some point they start moving eastward. The more eastward they move, the less northward they are moving. That is the way the two dimensions are brought together. Now consider the right-hand diagram in
figure 7.26
. If a person is moving a little, they do not really change their position in three-dimensional space. At some point they start moving fast and then they are changing their position in space but are moving less in time. The time dilation shows that the faster they move, the slower time flows (as measured by someone who is not moving). This is a way of seeing that the three dimensions of space and one dimension of time are intimately linked. Just as north and east are linked in the left-hand diagram, so too are time and space linked in the right-hand diagram. Einstein showed that space and time are not different entities that can be looked at individually. Rather, they are part of a four-dimensional object called
spacetime
. We usually think of motion as a movement in space while time passes. We now know that the proper way to think about motion is as a path in spacetime. Later, when we study general relativity, this spacetime will be very important.

Simultaneity

Another consequence of special relativity is that the notion of simultaneity, or that two events happen at the same time, is problematic. After all, if two people traveling at different speeds can argue about how much time elapsed, then they can definitely argue about whether two events occurred at the same time. Imagine that two astronauts observe an event and then board spaceships going at different speeds. Since they have traveled at different speeds, their perceptions of how much time has passed are different. Eventually they both land on another planet and are told that a second event happened before they landed. One astronaut may look at her watch and conclude that the two different events occurred at the same time. The other astronaut might look at his watch and conclude that the events happened at different times. Who is correct? There is no correct answer.
Time
,
duration
, and
simultaneity
are all relative terms.

Einstein described a beautiful little thought experiment that underscores this point. Imagine standing in a train station and observing a speeding train pass. When the train is right in front of you, two bolts of lightning appear to you at the exact same instant. The bolts hit the front and back of the train, as in
figure 7.27
. The distances from the two ends of the train to you are identical and so the light had to travel the same distance. You conclude that the lightning struck the two ends of the train at the same time. But now consider a traveler at the center of the train. They are traveling toward the front and away from the back lightning bolt. The light from the front bolt hits them first. Only later will the light from the back bolt hit them. They therefore come to the conclusion that the front bolt came first.
³⁷
Both you and the passenger agree on the laws of physics but come to two different conclusions about whether the bolts were simultaneous. Who is correct? We can only conclude that both are correct and that the very idea of whether two events happened at the same time depends on who is viewing them.