Now let’s change point of view and consider everything from Grey’s perspective. In Figure 3.6 we’ve changed the grid such that it now represents measurements made using clocks and rulers at rest with respect to Grey. Relativity is so-called because of this relative aspect of motion – who is at rest and who is moving is just a point of view. Now Grey doesn’t move, which means his worldline snakes along a grid line. As before, Grey and Black clap their hands once every three hours according to their individual watches, but now it’s Black that claps only seven times a day. Grey concludes that Black is living her life in slow motion; their roles have been entirely reversed.
Figure 3.6. The trajectories of two observers moving over spacetime. Grey is moving at constant speed from left to right as determined by Black. The grid now represents a set of clocks and rulers at rest relative to Grey.
At this point you are well within your rights to exclaim loudly that this is nonsense. How can Grey age more slowly than Black according to Black’s clocks while Black ages more slowly than Grey according to Grey’s clocks? This sounds impossible, but surprisingly there is no contradiction. The ‘problem’ arises because, following in the footsteps of Newton,††† we are fixating on the concepts of universal time and space. Instead, we need to rewire our brains and focus on the worldlines – the paths traced out over spacetime by the immortals – and the grids of rulers and clocks they erect to describe the world. Black’s grid, shown in Figure 3.4 is different to Grey’s grid, shown in Figure 3.6. The grid lines that run roughly horizontal across the Penrose diagrams represent all of space ‘now’ for each immortal. The grid lines that run vertically represent all of time. But the grids are not the same. Grey’s space is a mixture of Black’s space and time, and vice versa. It’s hard to accept that the delineation between space and time is subjective, because our personal experience is that they are fundamentally different things that cannot be mixed. But this is not true. The separation between them is personal; it depends on our point of view.
The Twin Paradox
All very well, you may say, but what happens if the immortals decide to meet up again in the future? Then we really would appear to have a paradox, because we’ll be able to tell who has aged more. That’s a direct observation of reality, and we can’t then have it both ways. Indeed, we can’t. This apparent paradox is sometimes known as the Twin Paradox.
To see why the Twin Paradox isn’t a paradox, let’s introduce a third immortal called Pink. We’ve added her worldline onto the Penrose diagram in Figure 3.7. We now have a triplet paradox. Our three immortals meet up for a fleeting moment at Day Zero. Grey zooms past Black at half the speed of light, exactly as before, and Pink uses her spaceship to fly along a path that allows her to meet up with Grey and Black again in the future. Let’s look at Pink’s worldline to work out what she’s doing. After Day Zero, Pink accelerates away from Black, moving slowly at first, which is why their worldlines almost overlap for two handclaps. She then begins to speed up and catch up with Grey. When Pink and Grey meet (at the end of Day One), we can ask a question to which we must receive a definitive answer: who is older, Pink or Grey? Counting the dots along their respective worldlines informs us that Pink clapped her hands six times, while Grey clapped his hands seven times: Grey is older.
Figure 3.7. The Twin Paradox.
Though it is very counter-intuitive, the idea that Grey ages more than Pink is simple to appreciate if you recall from the previous chapter that the length of a timelike worldline is the time measured by a watch carried along that worldline. With that single idea, it’s easy to see that Pink and Grey age at different rates because Pink’s worldline is different from Grey’s worldline between their meetings. What’s not obvious without calculating is whose worldline is longer. You can see that by counting handclaps on the diagram; we did the calculation for you using the spacetime interval equation from Chapter 2.
Let’s continue following Pink’s journey. After around a day, she’s travelling very close to the speed of light, which you can see from the Penrose diagram because her worldline is almost at an angle of 45 degrees. She then swings her spaceship around and fires her rockets to decelerate, eventually reversing direction. She meets Grey again, and the two immortals can compare their ages once again. Counting the handclaps, Pink has aged 17 x 3 = 51 hours and Grey 20 x 3 = 60 hours since they first met at Day Zero. Finally, Pink returns back to Black. 28 x 3 = 84 hours have passed on her watch, and 120 have passed on Black’s watch (you can see this without counting dots because this rendezvous occurs after five days using Black’s grid, which is shown on the figure). This is nothing less than time travel into the future. Black has aged more than Pink when they meet up. Fascinatingly, there is no limit on how far into the future one can travel with access to a fast enough spaceship. The Andromeda galaxy is 2.5 million light years away from Earth. If Pink had access to a spaceship that could travel at 99.9999999999 per cent of the speed of light, it would take her 18 years to make the round trip to Andromeda. She would, however, return to Earth 5 million years in the future.
There is a general principle at work here known as the Principle of Maximal Ageing. Black and Grey will age more than anyone else who sets off from Day Zero and takes any route over spacetime before returning to meet up with them again. The thing that is special about Black and Grey is that they never turn a rocket motor on to accelerate or decelerate. We call Black and Grey’s routes between events ‘straight lines’ over spacetime because they don’t accelerate.‡‡‡
Horizons
For our final foray into flat spacetime, we’ll explore acceleration, and in doing so follow in Einstein’s footsteps on the road to general relativity. In Figure 3.8, the purple dotted line corresponds to an immortal who starts out in the distant past travelling from right to left at close to the speed of light. We’ve named this immortal ‘Rindler’, after the physicist Wolfgang Rindler, who first introduced the term ‘Event Horizon’. Rindler steadily decelerates until he reaches his closest approach to Black at Day Zero. From the diagram, you should be able to see he is momentarily stationary relative to Black at a distance of just over half a light day. His constant acceleration then takes him away again, out towards infinity, moving all the time ever closer to the speed of light. For the first half of the journey, the rockets are slowing him down and for the second half they are speeding him back up again relative to Black. Rindler is always accelerating at the same rate, as measured by accelerometers onboard his spaceship, although he won’t need instruments to tell him that he’s accelerating. He will feel the constant acceleration as a force pushing him into his seat. He won’t be ‘weightless’ inside his spacecraft. Hold that thought, because it’s going to be very important.
Figure 3.8. An immortal ‘Rindler’ observer undergoing constant acceleration.
This trajectory of a constantly accelerating observer is known as a Rindler trajectory. Notice that although Rindler accelerates forever, his worldline never quite makes it to 45 degrees on the Penrose diagram. That is because, no matter how long he accelerates for, he cannot travel faster than the speed of light. The most striking thing about Rindler’s trajectory is that he always remains inside the shaded right-hand region of the Penrose diagram we’ve labelled ‘1’. He can see anything that happens in this region at some point during his journey. By ‘see’ we do mean ‘see’ in the sense that light can travel from any event inside region 1 and reach his eyes. To confirm this, pick any point in region 1 and check that 45-degree light beams emitted from that point will intersect Rindler’s trajectory. Likewise, Rindler could have sent a signal to any event in this region of spacetime at some point during his journey.§§§