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things look from the viewpoint of the astronaut twin. For her, the earthbound twin is the one doing the traveling while she snakes along her own time axis. It looks like the paradox is back again; since the astronaut twin is at rest relative to her spaceship, it seems that she should speed maximally through time and hence age the most. But there is a very subtle point here. The distance equation does not apply if we set out to use the astronaut twin’s clocks and rulers to measure distances and times. More precisely, it fails when the astronaut twin undergoes the acceleration that turns the spaceship around. Why does it fail? The arguments we presented when we figured it out seemed pretty watertight. But if one uses an accelerating system of clocks and rulers to make measurements, as the astronaut twin must, then the assumption that spacetime is unchanging and the same everywhere that we used to write down the distance equation is wrong. Over the time of the acceleration, the astronaut twin will be pushed back into her seat, in much the same way that you are pushed back into your seat when you press the accelerator pedal on a car. For a start, that immediately picks out a special direction in space: the direction of the acceleration. The existence of that force must be accounted for in the distance equation, and that is where the loophole resides. It is a little too complicated for us to go into the mathematical details, but the upshot is that when the spaceship fires its rockets to turn around, the earthbound twin ages rapidly relative to the astronaut twin and that more than makes up for the fact that she ages more slowly during the nonaccelerating phases of the expedition. There is no paradox.

We can’t resist quoting some numbers, because the effect can be startling. Space travel is most comfortable for those onboard the spaceship if the rockets are firing in order to sustain an acceleration equal to “one g.” That means that the space travelers feel their own weight inside the rocket. So let’s imagine a journey of 10 years at that acceleration, followed by 10 more years decelerating at the same rate, at which point we turn the spaceship around and head back to Earth, accelerating for 10 more years and decelerating for a further 10 before finally arriving back. In total the travelers onboard the spaceship will have been journeying for a total of 40 years. The question is how many years have passed on Earth? We’ll just quote the result because the mathematics is (only a little) beyond the level of this book. The result is that a breathtaking 59,000 years will have passed on Earth!

This has been a remarkable journey, and we hope the reader has followed us into the world of spacetime. We are now ready to head directly to E = mc2. Armed with spacetime and our invariant definition of distance, we ask a simple but very important question: Are there other invariant quantities that also describe the properties of real objects in the real world? Of course, distances aren’t the only things that are important. Objects have mass, they can be hard or soft, hot or cold, solid, liquid, or gas. Since all objects live in spacetime, is it possible to describe everything about the world in an invariant way? We will discover in the next chapter that it is, and the consequences are profound, for this is the road that leads directly to E = mc2.

5

Why Does E=mc2?

In the last chapter we showed that merging space and time together into spacetime is a very good idea. Central to our whole investigation was the notion that distances in spacetime are invariant, which means that there is consensus throughout the universe as to the lengths of paths through spacetime. We might even regard it as a defining characteristic of spacetime. We were able to rediscover Einstein’s theory but only if we interpreted the cosmic speed limit c as the speed of light. We haven’t proved that c has anything to do with the speed of light yet, but we’ll dig much more deeply into the meaning of c in this chapter. In a sense, however, we have already begun