Why Does E=mc2_ - Brian Cox [34]
We are faced with what the English biologist Thomas Henry Huxley famously described as “the great tragedy of science—the slaying of a beautiful hypothesis by an ugly fact.” Huxley, known as Darwin’s bulldog for his sterling defense of evolution, was once asked by William Wilberforce whether it was from his grandfather or grandmother that he claimed his descent from a monkey. Huxley is said to have replied that he would not be ashamed to have a monkey for his ancestor, but he would be ashamed to be connected with a man who used his great gifts to obscure the truth. The tragic truth in our case is that we must reject the simplest hypothesis if we are to preserve causality, and move on to something a little more complicated.
Our next and in fact only remaining hypothesis is that the distance between points in spacetime is to be calculated using s2 = (ct)2 - x2. In contrast to the plus-sign version, this is a world where Euclidean geometry does not apply, as in the case of geometry on the surface of the earth. Mathematicians have a name for a space in which the distance between two points is governed by this equation: It is called hyperbolic space. Physicists have a different name for it. They call it Minkowski spacetime. The reader might take this to be a clue that we are on the right track! Our top priority must be to establish whether Minkowski spacetime violates the demands of causality.
FIGURE 6
To answer this question we need once again to take a look at the lines in spacetime that lie a constant distance s from O. That is, we want to consider the analogue of the circles in Euclidean spacetime. The minus sign makes all the difference. Shown in Figure 6 are the same old events, O and A, along with the line of points that lie the same spacetime distance s from O. Crucially, these points no longer lie on a circle. Instead they lie on a curve known to mathematicians as a hyperbola. Mathematically speaking, all the points on the curve satisfy our distance equation—i. e., s2 = (ct)2 - x2. Notice that the curve tends toward the dotted straight lines that lie at 45 degrees to the axes. Now the situation as viewed by observers in rocket ships is completely different from the plus-sign version because event A always stays in the future of event O. We can slide A around but never into O’s past. In other words, everyone agrees that we wake up before we finish our breakfast. We can breathe a sigh of relief: Causality is not violated in Minkowski spacetime.
It’s worth repeating this because it is one of the most important points in the book. If we decide to define the distance in spacetime between the two events O and A using Pythagoras’ equation but with a minus sign, then no matter how anyone views the two events, A never crosses into O’s past; it just moves around on the hyperbola. This means that if event A is in O’s future according to one observer, then every other observer will also agree that A is in O’s future too. Because the hyperbola never ever crosses into O’s past, everyone agrees that eating breakfast comes after waking up.
We’ve just completed a subtle piece of reasoning. It certainly does not mean that we are correct in our original hypothesis that there should be an “invariant” distance in spacetime that is agreed upon by all observers. What it does mean, though, is that our hypothesis has survived an important test—it has survived the demands of the requirement of causality. We are not finished, however, because