Why Does E=mc2_ - Brian Cox [37]
Imagine that you are back on the proverbial train, sitting down in a carriage wearing a wristwatch. For you, it is convenient to measure distances relative to your own position and times using your wristwatch. Your train journey takes two hours from station to station. Since you never leave your seat throughout the journey, you have traveled a distance x = 0. This is the principle we established right at the start of the book. It is not possible to define who is moving and who is standing still, and therefore it is perfectly acceptable for you, seated on a train, to decide that you are not moving. In this case, only time passes. Since your journey takes two hours, then, from your perspective, you have traveled only in time. In spacetime, therefore, you have traveled distance s given by s = ct where t = 2 hours (because the distance in space as measured by you is x = 0). That is all straightforward. Now consider your journey from the standpoint of your friend, who is not on the train but who instead is sitting on the ground somewhere (it does not matter where he actually is, just that he is at rest relative to the earth while you are whizzing by on the train). Your friend would prefer to measure times using his own wristwatch and distances relative to himself. To simplify things a little bit, let us suppose your train journey is on a perfectly straight track. If you travel for 2 hours at a speed of υ = 100 miles per hour, then your friend notes that, at the end of the journey, you have traveled a distance X = υT. We are using capital letters when we talk about distances or times measured by your friend in order to distinguish them from the corresponding quantities measured by you (i.e., x = 0 and t = 2 hours). So, according to your friend, you have traveled a spacetime distance s given by s2 = (cT)2 - (υT) 2 .
Here is the crucial part of the whole argument: You must both agree on the spacetime distance of your journey. According to your measurements, you did not move (x = 0) and your journey took 2 hours (t = 2 hours), while your friend says that you have traveled a distance of υT (where υ = 100 miles per hour) and your journey takes a time T. Well, we are obliged to equate the corresponding distances in spacetime and so (ct) 2 = (cT)2 - (υT)2. This formula can be jiggled around to give us. So, although your wristwatch registers that your journey lasted for 2 hours, according to your friend your journey lasted a little longer. The enhancement factor is equal to, which is exactly what we got in the last chapter but only if we interpret c as the speed of light.
Are you beginning to feel the Ionian Enchantment? We have deduced the same formula that emerged from thinking about light clocks and triangles in the previous chapter. Then, we were motivated to think about light clocks because Maxwell’s brilliant synthesis of the experimental results of Faraday and others strongly suggested that the speed of light should be the same for all observers. This conclusion was supported by the experimental work of Michelson and Morley, and taken at face value by Einstein. In this chapter we arrived at exactly the same conclusion but with no reference to history or experiment. We didn’t even need to give light a special role. Instead, we introduced spacetime and, as a result, insisted that there should exist the notion of an invariant distance between events. On top of that we demanded that cause and effect be respected. We then constructed the simplest possible distance measure and remarkably arrived at the same answer as Einstein. This reasoning is perhaps one of the most beautiful examples of the unreasonable effectiveness of mathematics in the physical sciences. Thales would be so enchanted that he would already be reclining in a bath of asses’ milk having been scrubbed by eunuchs. For his concubines to enter his bathroom carrying wine and figs, all we have to do is establish that c must be the speed of light