All the horizontal ‘now’ lines, representing infinite slices of space, begin and end at the left and rightmost vertices of the diamond. The spacetime distance between any two events on one of these slices is the distance as measured on a ruler between the events. The two remaining vertices of the diamond are thus identified with events that are an infinite distance in space from O and are known as spacelike infinity.
The Penrose diagram has brought infinity to a finite place. This ability to picture infinite space and eternity on a single diagram will be tremendously useful when we come to think about black holes. But first, let’s have some fun with the Penrose diagram of flat spacetime in order to explore some of the famous consequences of Einstein’s Special Theory of Relativity.
The immortals
In the last chapter we were transported back to Christmas 1974 and family arguments around the television set, valves blazing. We also saw that events that happen simultaneously from one point of view do not happen simultaneously from a different point of view. More generally, observers moving relative to each other will disagree on the distance in space and the time difference between events, but they will always agree on the interval between them. Armed with our Penrose diagram, we can develop a more detailed picture of what’s happening.
Let’s consider two observers, Black and Grey, moving at constant speed relative to each other. Figure 3.4 shows a Penrose diagram with the worldlines of the two observers. They are immortals who have chosen to spend their infinite lives carrying out a visual demonstration for our benefit. Being immortal, their (timelike) worldlines begin at the bottom vertex of the diamond (past timelike infinity) and end at the top vertex (future timelike infinity). The immortals carry identical watches and have agreed to clap their hands every three hours. The dots along their worldlines mark the events in spacetime corresponding to their claps. The repetitive but illustrative cause embraced by the immortals for pedagogical benefit is more noble than that of Douglas Adams’s character ‘Wowbagger the Infinitely Prolonged’ who decided to relieve the boredom of immortality by insulting everybody in the Universe in alphabetical order. He called Arthur Dent a ‘complete kneebiter’. Wowbagger’s worldline would begin inside the diamond, not at past timelike infinity, because he became immortal at a finite time in the past, allegedly in an accident involving a rubber band, a particle accelerator and a liquid lunch. His worldline would still end at future timelike infinity, leaving him plenty of time to complete his task.
Figure 3.4. The trajectories of two observers moving over spacetime. Grey is moving at constant speed from left to right as determined by Black. The grid measures distances and times using a set of clocks and rulers at rest relative to Black.
The grid on the Penrose diagram is the same as that in Figure 3.3. It corresponds to a system of clocks and rulers at rest with respect to Black. Grey is moving at a constant speed from left to right relative to Black. Let’s start by making sure we can appreciate that fact using the diagram. Black and Grey are at the same point in spacetime in the middle of the Penrose diagram, which means they fleetingly meet up there. Let’s refer to the time when that happens as Day Zero. From her point of view, Black does not move through space, which means she travels along one of the vertically oriented lines in the grid. Since she started out in the middle of the diagram, she travels along the vertical grid line. If she’d started out from a point somewhere to the left or right then she would follow one of the curving vertical lines instead, but in both cases, she would not be moving relative to the grid. The curved appearance of a straight line is familiar to anyone who has been bored enough on a long-haul flight to stare at the map on the screen in the seat. Figure 3.5 shows the ‘Great Circle’ route from Buenos Aires to Beijing on a Mercator projection. This is a straight line on the curved surface of the Earth – the shortest distance between Buenos Aires and Beijing – but it looks curved because the map is a distorted projection of the surface of a sphere onto a flat sheet of paper.
Grey does move relative to the grid. Two days after passing Black we see he’s travelled one light day away from her (according to Black’s clocks and rulers, i.e. Black’s grid). After a further two days of travel, Grey is two light days from Black, and so on. We can conclude that Grey is travelling at half the speed of light relative to Black.¶¶ It is worth checking that you understand the diagram well enough to see this before you read on.
Don’t be confused by the fact that it looks like Black and Grey meet up in the distant past and the far future. They don’t, because there is an infinite amount of space being squashed down at the top and bottom of the diamond (you can see all the grid lines bunch up there). The immortals only meet once, on Day Zero.
Figure 3.5. The Great Circle route from Buenos Aires to Beijing on a Mercator projection map. This is a straight line – the shortest distance between the two points on the Earth’s surface.
The next challenge is to use the diagram to see that Grey’s watch runs slow compared to Black’s. Look at Black’s worldline. She claps her hands every three hours by her watch, which means she lays down eight dots every day along her worldline. Now look at Grey’s worldline. He does the same, but according to Black’s grid (i.e. Black’s watch), he claps his hands only seven times per day. Crucially, this isn’t some sort of optical illusion caused by the way we’ve drawn the diagram. Everything Grey does is slowed down as measured by Black, which means that Grey’s whole life is running in slow motion from Black’s point of view.***
BOX 3.1. The relativistic Doppler effect
At the risk of confusing matters, but for the sake of deepening understanding, notice that Black draws her conclusions by recording events using a grid, which we might think of as corresponding to a network of clocks and rulers at rest relative to her. Very importantly, she does not draw conclusions based on what she sees with her eyes. In fact, Black sees Grey living in fast-forward (the opposite of slow motion) before they meet on Day Zero, and in even slower motion when Grey has passed her. It’s possible to work this out from Figure 3.4 and it is well worth the effort if you fancy a challenge.
Here’s how it works out. Since light moves on 45-degree lines and since we see things using light, it follows that on day minus one (the day before Day Zero), Black sees Grey when he is at day minus two. During the next 24 hours, in the period leading up to their brief encounter at Day Zero, Black lays down the usual eight dots while Grey lays down 14 dots, which means Black sees Grey clap faster. After Grey passes by, things flip around and Black sees Grey clap slower; she sees Grey clap just under five times per day. We’ll leave that for you to work out by counting dots and thinking about 45-degree light beams.
The point is that, in relativity theory, it is very important to say exactly how time differences are being determined. Seeing things (with instruments like eyes) can be very different from measuring the passage of time using a network of clocks and rulers. The effect we just discussed is known as the relativistic Doppler effect and it is sensitive to the location of the light detector (i.e. where the eyes are). That’s why Grey went from fast-forward to ultra-slow-motion as he passed Black. There is a more familiar and similar effect for sound (also called the Doppler effect) in which we hear a change in the pitch of a siren when an ambulance drives past. The lesson is that we need to be careful about using the word ‘see’ in relativity.