Why Does E=mc2_ - Brian Cox [35]
FIGURE 7
Remember, we needed c in order to have any chance of defining the notion of distance in spacetime, because we had to measure space and time in the same currency, but so far we have no idea what it actually represents. Is it the speed of anything interesting? The key to the answer lies in an intriguing property of the Minkowski spacetime we have just constructed. Those lines at 45 degrees are important. In Figure 7 we’ve drawn several other curves, each of constant spacetime distance from O. The important point is that there are in fact four types of curve that we can draw. One lies wholly in the future of event O, one lies always in the past, and two others lie to the left and right. They look a little bit worrying because they cross the horizontal line in just the same way that our circles crossed it in the case of the plus-sign version of Pythagoras. In the plus-sign case, this led us to reject the hypothesis because it meant that causality was violated. Are we in the same boat with the minus-sign version? Are we sunk? Well, no, there is a way out. Figure 7 shows an event B sitting in the troubling region. It lies in O’s past according to the figure. But the hyperbola of constant distance from O for this event crosses the space axis, with the implication that it is possible for some observers to consider event B as occurring in O’s future, while for others it is in O’s past. Don’t forget: Every observer must agree on the spacetime distance between events even if they do not agree on the distances in space and time separately. It looks like a breakdown of causality, but fortunately that is very definitely not the case.
How are we to restore causality to our theory of spacetime? To answer this question, we need to think a little more carefully about what we mean by causality. This next piece will involve rocket ships and lasers, so if the abstract reasoning of the previous sections has left you drained, then you can relax for a while. Let’s think about event O again: waking up in bed in the morning. To be a little more precise, the event could correspond to my alarm clock going off. Shortly beforehand, on a planet in the Alpha Centauri system, the nearest star system to Earth at a distance of just over 4 light-years, a spaceship lifts off and heads toward Earth. Must everyone agree that the spaceship started its journey before I woke up? From the point of view of causality the issue depends critically upon whether information can travel infinitely fast or not. If information can travel infinitely fast, then the alien spaceship might conceivably be able to fire a laser beam that travels in an instant to the earth and destroys my alarm clock. The result is that I oversleep and miss breakfast. Missing breakfast might be the least worrying issue given this particular scenario, but we are doing a thought experiment, so let us ignore the emotional consequences of having our alarm clock vaporized by an alien laser and continue. The firing of the spaceship’s laser caused me to miss breakfast, and therefore the ordering cannot be swapped without violating our doctrine of the protection of causality. This is easy to see because if some observer were able to conclude that the spaceship took off after I woke up, then we would have a contradiction because I cannot oversleep if I have already woken up. We are forced to conclude that if information can travel at arbitrarily high speeds, then it can never be permissible to switch the time ordering of any two events without violating the law of cause and effect. But there is a loophole in our reasoning that permits the time ordering of certain pairs of events to be flipped,