Why Does E=mc2_ - Brian Cox [19]
The upshot is that we must either cling to the comforting notion of absolute time and ditch Maxwell’s equations, or ditch absolute time in favor of Maxwell and Einstein. How should we check which is the correct thing to do? We must find an experiment in which we should, if Einstein is right, observe time actually slowing down for moving objects.
To design such an experiment, first we need to work out how fast something should move in order to reveal the proposed effect. It should be quite clear that moving at 70 mph down the highway in a car does not cause time to slow down very much, because we don’t come home after a trip to the store to find that our children have grown older than us while we were away. Silly as this seems, taking Einstein at face value means that this is exactly what does happen, and we would certainly notice the difference if only we could travel fast enough. So what constitutes fast enough? From the viewpoint of the person on the station platform, the light travels along the two sides of the triangle shown in the diagram. Einstein’s argument is that because this is a greater distance for the light to travel than if the clock were standing still, time will pass more slowly because the tick takes longer. All we have to do now is to calculate how much longer (for a given train speed) and we have the answer. We can do this with a little help from Pythagoras.
If you do not want to follow the maths you can skip over the next paragraph, but then you will have to take our word for it that the numbers all work out. That goes for any other maths we might bump into as the book progresses. It is always an option to skip past it and not worry—the mathematics helps provide a deeper appreciation of the physics but it isn’t absolutely necessary to follow the flow of the book. Our hope is that you will have a go with the maths even if you have no prior experience at all. We have tried to keep things accessible. Perhaps the best way to approach the maths is not to worry about it. The logic puzzles that appear in the daily newspapers are much harder to tackle than anything we will do in this book. That said, here comes one of the trickier bits of maths in the book, but the result is worth the effort.
Take a look at Figure 2 again and suppose that the time taken for half of one tick of the clock on the train as measured by the person standing on the platform is equal to T . It is the time taken for the light to travel from the bottom mirror to the top mirror. Our goal is to figure out what T actually is and double it to get the time for one tick of the clock according to the person on the platform. If we did know T, then we could figure out that the length of the longest side of the triangle (the hypotenuse) is equal to cT, i.e., the speed of light (c) multiplied by the time taken for light to travel from the bottom mirror to the top mirror (T). Remember, the distance something travels is obtained by multiplying its speed by the time of the journey. For example, the distance a car travels in one hour at 60 miles per hour is 60 x 1 = 60 miles. It is not hard to work out the result for a two-hour journey. All we are doing here is invoking the formula “distance = speed x time.” Knowing T, we could also figure out how far the clock moves in half of one tick. If the train is moving at a speed, υ, then the clock moves a distance υT each half-tick. Again we did nothing except use “distance = speed x time.” This distance is the length of the base of a right-angled triangle and because we know the length of the longest side, we can go ahead and figure out the distance between the two mirrors using Pythagoras’ theorem. But we know what that distance actually is already—it is 1 meter. So Pythagoras’ theorem tells us that (cT) 2 = 12 + (υT)2. Note the use of parentheses: In mathematics they are