Why Does E=mc2_ - Brian Cox [20]
We are nearly done now. We know c, the speed of light, and let’s presume to know the speed of the train, υ. Then we can use this equation to figure out T. The crudest way to do it would be to guess a value of T and see if it solves the equation. More often than not the guess will be wrong and we’ll need to try another guess. After a while we might hone in on the right answer. Fortunately, we can avoid that tedious process because the equation can be “solved.” The answer is T2 = 1/ (c2 - υ2), which means, “first work out c2 - υ2 and then divide 1 by that number.” The forward slash is the symbol we will use to denote “divide by.” So 1/2 = 0.5 and a/b means “a divided by b,” etc. If you know a bit of maths, then you’ll probably feel a little bored by now. If not, then you might wonder how we arrived at T2 = 1/(c2 - υ2). Well, this isn’t a book on maths, and you’ll just have to trust that we got it right—you can always convince yourself that we got it right by putting some numbers in. Actually, we have the result for T2, which means “T multiplied by T.” We get T by taking the square root. Mathematically, the square root of a number is such that when multiplied by itself we regain the original number; for example, the square root of 9 is 3 and the square root of 7 is close to 2.646. There is a button on most calculators that computes the square root for you. It is usually denoted by the symbol “√” and one would normally write things like. As you can see, the square root is the opposite of squaring, 42 = 16 and
Returning to the task at hand, we can now write the time taken for one tick of the clock as determined by someone on the platform: It is the time for light to travel up to the top mirror and back down again—that is 2T. Taking the square root of our equation above for T2, and multiplying by 2, we find thatThis equation allows us to work out the time taken for one tick as measured by the person on the platform, knowing the speed of the train, the speed of light, and the distance between the two mirrors (1 meter). But the time for one tick according to someone sitting on the train next to the clock is simply equal to 2/c, because for them the light simply travels 2 meters at a speed c (distance = speed x time, so time = distance / speed). Taking the ratio of these two time intervals tells us by how much the clock on the train is running slow, as measured by someone on the platform; it is running slow by a factor of, which can also be written, with a little more mathematical rearranging, as. This is a very important quantity in relativity theory, and it is usually represented by the Greek letter γ, pronounced “gamma.” Notice that γ is always larger than 1 as long as the clock is flying along at less than the speed of light, because υ/c will be smaller than 1. When υ is very small compared to the speed of light (i.e., for most ordinary speeds, since in units more familiar to motorists the speed of light is 671 million miles per hour), γ is very close to 1 indeed. Only when υ becomes a significant fraction of the speed of light does γ start to deviate appreciably from 1.
Now we are done with the mathematics—we have succeeded in figuring out by exactly how much time slows down on the train as determined by someone on the platform. Let’s put some numbers in to get a feel for things. If the train is moving at 300 kilometers per hour, then you can check that υ2/c2 is a very tiny number: 0.000000000000077. To get the “time stretching” factor γ we needAs expected, it is a tiny effect: Traveling for 100 years on the train would only extend your lifetime by a matter of 0.0000000000039 years according to your friend on the platform, which is slightly above one-tenth of a millisecond. The effect would not be so tiny if the train could whiz along at 90 percent of the speed of light, however. The time-stretching factor would then be bigger than 2, which means that the moving