Relativity

We've already discussed how units of time differ in importance depending on who is observing what and when they're observing it. However, more than relative perception, time is, in very fact, relative to the observer. That is, the speed at which a clock ticks depends on who is watching it. The overarching theory being described is called special relativity and was introduced in 1905 by Albert Einstein (while he was working as a patent clerk, no less).

Imagine that you invent a machine that, when sitting on a pitcher's mound, could throw a baseball at exactly 100mph every time. If you put that machine on top of a car traveling at 60mph and then shot a baseball, the ball would be moving at 160mph as a police officer would calculate from a parked car on the side of the road. However, to the person in the car, the ball still looks like it is traveling at 100mph away from him. In other words, the velocity of the baseball depends on who is watching it move. Specifically, it depends on how fast the observer is moving.

We call this principle Galilean Relativity since it was first described by Galileo. There was something that Galileo missed, however, which has to do with the speed of light. It's not his fault, really. He tried to measure the speed of light by sending an assistant to a nearby hilltop. The idea was that he would uncover a lamp and that his assistant would uncover his own lamp when he saw the light from Galileo's. The distance between the two scientists divided by the time between the unveiling of the two lamps would yield the speed of light. However, since light can circumnavigate the earth seven times in a second, he was grossly outmatched. Galileo's conclusion was that light goes really really fast ("if not instantaneous, it is extraordinarily rapid"). Little did he know the fire he was playing with.

Special relativity is founded on the principle that light travels at 3.00×10^{8 m}⁄_s (we'll call it c from now on) for all observers. Unlike the baseball, if you are in a spaceship traveling at half that speed (^c⁄₂) and you shoot a beam of light at a "stationary ship", the ship would see the beam of light approaching at c (not at c + ^c⁄₂ as you might expect).

That may seem like an inconsequential—albeit strange—fact, but it ends up being extremely important. That means for someone traveling at 99% of the speed of light who shoots a beam at a "stationary" observer, both the moving and the stationary observer perceive the beam of light traveling at c. If you think about that fact in such an extreme case, some discrepancies seem to arise.

Consider the perspective of the moving ship. If the beam of light is really moving away from it at c, it will pass the stationary observer at some time in the future (we'll call that time t₁). In that amount of time, the beam of light will be some distance away from the moving ship (which we'll call x₁). Since light travels very quickly, you can imagine that the distance between the moving ship and the light beam at t₁ is pretty big (even if it only took a second, the light would be 186,000 miles away).

Now consider the perspective of the stationary observer. He sees the light traveling toward him at c and the ship traveling toward him at 99% of that speed. From his perspective, because the two velocities are so close, the distance between the moving ship and the light beam (in his perspective, x₂) will appear to be significantly less than x₁.

In the car/baseball example, this is like saying that when the machine shoots the ball, the person in the car sees it moving away from him at 100mph while he is traveling 60mph, but the police officer also thinks the baseball is traveling at 100mph (not 160mph). In his perspective the ball is only moving 40mph faster than the car. So when it hits the police car, the baseball has spent the same amount of time traveling away from the moving car at 100mph in one perspective as it has traveling away from the car at 40mph in the other perspective. The two distances, x₁ and x₂, aren't the same!

That sounds ridiculous for a car and a baseball, but that's exactly what happens with light. The problem is that only one thing happens. That is, the light beam only hits the stationary ship one time, and the moving ship would only pass the stationary ship once. But if the distance between the beam and the moving ship is different, it seems like the moving ship would pass the stationary ship twice (once in one perspective then later in the other). Obviously that's impossible. At the end of the day, the ship and the light can only pass the stationary observer once. So when does it happen, and where?

This is the problem that relativity solves. When talking about relativity to my students in their introductory physics lab, I discovered that they couldn't answer a simple yet important question: In relativity, what is relative? For instance, in Galilean relativity, we could say something like, "Relative to the police car, the baseball is traveling 160mph; velocity is relative." So, in special relativity, what is relative?

The answer is the genius of Einstein's theory and is two-fold. It's pretty clear that distance, or space, is relative since each ship perceives the separation between the moving ship and the beam of light to be different. If that's true, then for the light to pass the stationary ship exactly once without repeating itself, then each ship must perceive the passage of time differently. That is, for the stationary ship, the distance between the moving ship and the beam of light is small, so the light and the ship pass him in some small amount of time (for the sake of argument, let's say that the clock on the observer's ship ticks away one second between the time that the beam of light hits him and the time that the ship passes him). But in the perspective of the moving ship, the distance between itself and the beam is much larger, which means that it will take much longer than a second to get to the observer once the beam of light hits him.

Remember that two different things cannot happen. If the stationary observer saw exactly one second tick away on his clock, the moving observer absolutely must see only and exactly one second tick on the same clock. The only way that's possible is if the clock ticks slower for the moving observer so that the pendulum swings only once in what must be for him an amount of time greater than one second. In other words, time is relative also.

And that's the answer to the question I posed earlier. In special relativity, what is relative? Time and space. Depending on how fast you are moving and what you are looking at, the distance between two fixed places and the time it takes to get there at a constant velocity is different. For the moving ship in our example, he sees the stationary clock tick away only one second, but he would see an identical clock on his moving ship tick away several seconds. For either observer, the distance between the beam and the ship that we've called the "moving" ship (that's relative, too) is different.

We are so accustomed to thinking that space is constant. If I drive 10 miles to work today, then when I go back home by the same route, I will travel 10 miles. A second is a second is a second, right? Well, not exactly. Time and space are as changeable and as capricious as the velocity of the baseball. It's 160mph to the stationary police officer, 100 mph to the guy in the car, and an infinite number of other velocities to the various people on the road traveling at different speeds.

The thing that is constant is something called "spacetime", a combination (as the name suggests) of "space" and "time", which were considered to be separate, into one single continuum. We have used spacetime to better understand the universe since it, for the first time, explains the behavior of gravity and light with relative (ha!) clarity.