Why Does E=mc2_ - Brian Cox [36]
Let us imagine the alien-laser-exploding-alarm-clock incident again, but now subject it to a cosmic speed limit. That is to say, we will not allow the laser beam to travel infinitely fast from the spaceship to our alarm clock. Covering ourselves in a thin mist of chalk dust for the last time, we call the laser-firing event B, as illustrated in Figure 7. If the spaceship fired the laser (event B) very shortly before the alarm clock-ringing event O, from a very great distance away, then there is no way the spaceship could possibly prevent me from waking up because the laser beam simply hasn’t got enough time to travel from the ship to my clock. This must be the case if the laser beam is constrained to travel at or below some kind of cosmic speed limit. If this is the situation, the events O and B are said to be causally disconnected.
As illustrated in the figure, we are supposing that B happens just before O such that it lies in the right-hand wedge region, which is the “dangerous” region for causality. Different observers will generally disagree on whether B happens before or after O, because their different points of view correspond to moving B around on the hyperbola, which crosses the space axis from the future to the past. This is unavoidable, but cause and effect can still be protected if there is absolutely no way that event B can influence event O. In other words, who cares whether B happened in O’s past or future, if it makes no difference to anything because B and O cannot influence each other? There are four distinct regions in Minkowski spacetime, separated from each other by the 45-degree lines. If we are to protect causality, then any event that occurs in either of the left-hand or right-hand wedges must never be able to send a signal that can possibly reach O.
To interpret the delineating lines, look again at our spacetime diagrams. The horizontal axis represents distance in space, and the vertical axis represents distance in time. The 45-degree lines therefore correspond to events that have a distance in space from O that is equal to the distance in time (ct). How fast must a signal travel from O if it is to influence an event lying exactly on the 45-degree line? Well, if the event is 1 second in O’s future, then the signal must travel a distance c x 1 second. If it’s 2 seconds in the future, then it must travel a distance c x 2 seconds. In other words, it must travel at the speed c. For a signal to travel between B and O, therefore, it must travel faster than the speed c. Conversely, for any events that lie between the 45-degree lines but in the upper and lower wedges, it is possible to communicate between them and the event at O using signals that travel at speeds slower than c.
We have finally managed to interpret the speed c: It is the cosmic speed limit. Nothing can travel faster than c because if it did it could be used to transmit information that could violate the principle of cause and effect. Notice also that if everyone is to agree on the distance in spacetime between any two events, then they must also agree that the cosmic speed limit is c, regardless of how they are moving around in spacetime. The speed c therefore has an additional interesting property: No matter how two different observers are moving, they must always measure c to be the same. The speed c is beginning to look a lot like another special speed we have encountered in this book: the speed of light, but we haven’t proved the connection yet.
Our original conjecture is still very much alive. We have managed to build a theory of space and time that looks capable of reproducing the physics we met in the last chapter. Certainly, the existence of a universal speed limit offers promise, especially if we can interpret it as the speed of light. We also have a spacetime in which space and time are no longer absolutes. They have been sacrificed in favor of absolute spacetime. To convince ourselves that we have constructed