Black Holes: The Key to Understanding the Universe

3

Bringing Infinity to a Finite Place

Physicists often describe general relativity in aesthetic terms; it is the theory to which the word ‘beautiful’ is most often attached. ‘Beautiful’ implies an elegance and economy not easily visible in the mathematics, which is notoriously arcane. There is a well-known anecdote about Arthur Eddington who, when it was put to him that he was one of only three people in the world who understood Einstein’s theory, paused for a moment and replied, ‘I’m trying to think who the third person is.’ Rather the adjective applies to the elegance and economy of the ideas that underlie the theory and to the beautiful idea that gravity is geometry. John Archibald Wheeler expressed this central dogma of general relativity in a single sentence: ‘Spacetime tells matter how to move; matter tells spacetime how to curve.’ The hard part of general relativity is to calculate how spacetime is curved, and for anything other than very simple arrangements of matter and energy, exact solutions to Einstein’s equations are not easy to find. Black holes are one of the few cases in Nature for which we can precisely calculate the spacetime geometry, and once we have the geometry, we can represent it pictorially. The challenge is to find the most useful way of drawing the spacetime around a black hole on a flat sheet of paper. Flat paper is two-dimensional and spacetime is four-dimensional, which makes it difficult to draw (to say the least). If the spacetime is curved, that introduces an additional headache and distortion is inevitable. The trick is to draw the minimum number of dimensions necessary, which as we saw in the previous chapter is often a single dimension of space plus the time dimension, and to choose the distortion such that the features we are interested in are rendered in a way that enhances our understanding.

Figure 3.1. Hand with Reflecting Sphere by M. C. Escher, 1935. (M.C. Escher’s ‘Hand with Reflecting Sphere’ © 2022 The M.C. Escher Company-The Netherlands. All rights reserved. www.mcescher.com)

There is an image with which we are all familiar that introduces distortion in a well-chosen way to represent a curved surface on a flat piece of paper – a map of the surface of the spherical Earth. Many ways of representing the surface have been devised but the one of particular interest is shown in Figure 3.2. The Mercator projection, introduced in 1569 by Gerard de Kremer,* is designed specifically for navigation. Sailors care about compass bearings, and presumably other stuff that’s not relevant to this book, so the Mercator projection is defined such that angles on the map at any point are equal to compass bearings on the Earth’s surface at that point. This means that a navigator can draw a straight line on the map between two places and the angle between the line and the vertical will be the bearing from North that the ship must sail to travel between those places. The price to pay is that distances on the map are distorted, and the distortion increases with latitude. Greenland appears to be the same size as Africa on the Mercator projection, when in fact in area it is over 14 times smaller. The distortion becomes infinite at the poles, which cannot be represented on the map. The Mercator projection is an example of a ‘conformal’ projection, which means that angles and shapes are preserved at the expense of distances and areas.

Figure 3.2. Mercator projection of the Earth’s surface between 85° South and 85° North.

The spacetime diagrams we drew in the last chapter extend infinitely in every direction: time carries on forever up and down, and space extends forever left and right. That’s not necessarily a problem unless we are interested in depicting the physics of forever, but that’s precisely what we would like to do if we want to visualise the spacetime in the vicinity of a black hole. If we are to develop an intuitive picture of a black hole, therefore, we’d like a way of bringing infinity to a finite place on the page. Roger Penrose found an elegant way of doing so.†

Penrose diagrams

Figure 3.3 shows the Penrose diagram for ‘flat’ spacetime. By flat spacetime, we mean a universe without gravity, which is the spacetime of special relativity we discussed in the last chapter. Flat spacetime is often referred to as Minkowski spacetime, after Hermann Minkowski who first introduced the idea of spacetime: ‘The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.’‡ As we’ve discussed, matter and energy distort spacetime, and so in the presence of a planet, star or black hole, the geometry will change. But to warm up, we will focus first on simple, common or garden flat spacetime; a universe with nothing in it to distort the fabric.

Figure 3.3. The Penrose diagram for flat spacetime in the simplified case of one space dimension.

The Penrose diagram for flat spacetime is a thing of beauty: all of space and all of time has been squeezed into a finite-sized diamond-shaped region. Every event in the history and future of an infinite, eternal universe is located somewhere on this diagram. The distortion is extreme, as it must be (we have captured infinity on a single page) but the distortion has been carefully chosen.§ Just as for the Mercator projection, the Penrose diagram is a conformal projection; angles are preserved at the expense of distances on the page. This means that light rays always travel along 45-degree lines, and all the light cones are oriented vertically, just as they were in the spacetime diagrams of the last chapter. At any point, therefore, we can think of time as pointing vertically upwards. Since the light cones define the notion of past and future and tell us whether a particular event can influence another event, they are of utmost importance. If we are interested in how events in spacetime are related to each other in the vicinity of a black hole, a simple and intuitive rule for light cones is what we’re after. The other beautiful feature of the Penrose diagram is that, as advertised, it brings infinity to a finite place on the page; not just one place, but five.

The diamond in Figure 3.3, representing an infinite, flat universe, is centred on a particular event, which we’ve labelled ‘O’. There is nothing special about this central point. We could choose any event in spacetime around which to centre the diagram, but of course we’ll usually choose an event that we are interested in. It would be an eccentric choice to depict things we want to study in the most distorted regions out towards the corners of the diamond, but if we really want to then there’s nothing stopping us. Remember that we are capturing all of space and all of time in this diagram, so every event that ever happened and ever will happen is depicted somewhere. ‘All of space’ is limited to one dimension – left and right – just to make things easier to draw. We could have drawn another spatial dimension, but it wouldn’t add much to our understanding and would complicate the diagram. We’ve shown what adding another spatial dimension looks like in Box 3.2. The freedom to choose where to centre the diagram is also available for the Mercator projection. If we were interested in navigating around the polar regions, for example, we could choose Singapore as the ‘pole’. The distortion would then be extreme around Singapore, but the map would be good for Greenland.

Since we can choose any event in spacetime around which to centre the diagram, let’s go with an important one: your birth. For the purposes of this discussion, and only this discussion, the entire Universe – or at least this representation of it – revolves around you. The diagonal wavy lines that pass through O mark out the future and past light cones of O. We’ve labelled these two regions inside the light cones as ‘O’s causal future’ and ‘O’s causal past’. Any event inside the future light cone can be reached from O and any event inside the past light cone can communicate with O. Since O is your birth, your worldline must snake around inside the future light cone.

The light cones can be thought of as the paths over spacetime of two pulses of light that started their journey in the distant past and happen to pass each other at event O before heading off into the far future. One light beam heads in from the left and another from the right. Remember that there is only a single dimension of space represented on this diagram. We’ve sketched this on a ‘space-only’ diagram at the bottom left of Figure 3.3. Two light beams travel towards and then past O from opposite directions, crossing at O. The flashes, marked on both diagrams, tell us the location of the light beams one day before they reach O and one day after they pass O.

We said that the light beams began their journey in the distant past and headed off into the far future. As we’ve drawn them, each beam originates from one of the bottom diagonal edges of the diamond and ends on the opposite top diagonal edge. The top edges are where all light beams end up if they fly through the Universe forever. You can see that any light beam emitted from any event on the diagram will end up there, because all light beams travel at an angle of 45 degrees. For this reason, the top edges are known as future lightlike infinity. Likewise, any light beam that was emitted in the infinite past will have begun its journey on one of the bottom edges. These edges are known as past lightlike infinity.¶

Now let’s focus on the grid lines on the figure. In the vicinity of O the grid looks like a sheet of graph paper, but closer to the edges the grid is increasingly distorted. This is the ultimate fish-eye lens effect: an infinite amount of space and time are squeezed into the diamond by shrinking spacetime by different amounts from place to place. The further away from the centre we go, the more shrinking is applied. On the Mercator projection of the Earth’s surface, the stretching increases as we move away from the equator and an infinite stretch is applied at the poles, which is why we can’t draw the polar regions. On the Penrose diagram, the shrinking increases as we head away from the centre of the diamond and an infinite shrink is applied at infinity, which is why we can represent an entire infinite universe within a diamond.

The distorted grid is a way to measure the coordinates of events in spacetime, just as the grid of latitude and longitude allows us to measure the coordinates of places on the Earth’s surface. Looking at Figure 3.2, you can see how the distortion of the lines of latitude on the Mercator projection becomes more pronounced towards the poles, which affects the visual representation of the distance between points on the Earth’s surface on the map. It is important to appreciate that the grid itself and the corresponding choice of coordinates is completely arbitrary. On Earth the choice is partly driven by the Earth’s spin and the location of the geographical poles, but it’s also historical. There is nothing about the geometry of a sphere that forces us to measure longitude relative to the meridian that passes through Greenwich Observatory in London.

Likewise, in spacetime any grid can be used although, just as for the Earth, some grids will be more useful than others. For example, non-rotating black holes are spherical and so when we are describing spacetime in the vicinity of a black hole we’ll choose coordinate grids suited to a spherical geometry. It’s worth emphasising, though, that the grids we choose don’t even need to correspond to anybody’s idea of space or time. It’s just a grid, laid over spacetime such that we can label events. All that matters is that when we calculate the interval along a particular path using the grid, that distance will be an invariant quantity (which means it is independent of the grid choice). Analogously on Earth, the distance between London and New York doesn’t depend on how we choose to define latitude and longitude.

That said, the grid on the Penrose diagram in Figure 3.3 does correspond to somebody’s idea of space and time: yours. Let’s focus on the moment of your birth, which we’ve identified as event O in the centre of the diamond. The horizontal dashed line passing though O represents all of space – ‘now’ – according to you. Every event on the ‘now’ line happened at the same time from your point of view at the moment you were born.

We now need to be clearer on what a ‘point of view’ is. Imagine the ‘now’ line is populated by a series of clocks which are all synchronised with the clock present at your birth. They are spaced at uniform intervals along the line; we might imagine them being connected by little rulers. Each clock is at rest relative to the clock at O. As time ticks, this line of clocks will march towards the top of the Penrose diagram into the future. The worldlines of these clocks are represented by the curving vertical lines on the diagram. The clock at O – the clock present at your birth and at rest relative to you when you were born – goes straight up the vertical line into the future. If you don’t move, then the vertical line will also be your worldline. The other clocks also travel along what we will refer to as straight worldlines, even though they are curved on the Penrose diagram.

Let’s say ‘tomorrow’ is exactly 24 hours after your birth. The whole line of clocks will have advanced up the diagram into the future and will now lie along the dashed line labelled ‘O’s tomorrow’.** Likewise, if ‘yesterday’ is exactly 24 hours before you were born, then all the clocks would be located along the line labelled ‘O’s yesterday’. All the curving horizontal lines on the diagram are therefore slices of space that are different moments in time from your point of view.†† Only once we have been this careful can we say precisely what we mean by ‘O’s tomorrow’ (the day after your birth). As this book unfolds, you will come to appreciate the significance of this apparent pedantry. If you stay still for your whole life relative to this ensemble of clocks, the horizontal lines are all your tomorrows, stacked up one after the other. Your tomorrows will end when your worldline ends at the event of your death, but the tomorrows on the Penrose diagram extend into the infinite future. Tomorrow, and tomorrow, and tomorrow, Creeps in this petty pace from day to day, To the last syllable of recorded time.‡‡

We will now identify the remaining three of the five infinities on the diagram. Assuming that our imaginary clocks have always existed and always will exist, the worldline of every clock begins at the bottom vertex of the diamond and ends at the top vertex. If you recall from the previous chapter, anything other than light must move along a timelike worldline and could therefore carry a clock along with it.§§ Any (immortal) object that follows a timelike worldline will therefore begin at the bottom vertex of the diamond and end at the top vertex. The bottom vertex is known as past timelike infinity and the top vertex is known as future timelike infinity: ‘the last syllable of recorded time’.

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.

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.§§§

Can you see that Rindler cannot receive signals from regions 2 and 3? This is because there is no way for anything travelling less than or equal to the speed of light to get from those regions into region 1. We say that regions 2 and 3 are beyond Rindler’s horizon. In fact, region 3 is particularly isolated because it is also impossible for anyone in region 3 to receive signals from Rindler. These two regions are completely causally disconnected. Region 4 is different; Rindler could receive signals from this region, but he couldn’t send signals into it. Rindler’s situation is very different to that of Black and Grey, for whom the entirety of spacetime is causally accessible. Rindler lives in a smaller Universe than Black and Grey. By virtue of his acceleration, he has cut himself off from some regions of spacetime. The 45-degree boundary lines to his region are generically referred to as ‘horizons’ because information cannot flow both ways across them.

Previously we encountered horizons in the context of gravity and black holes. Now we see that they also appear for accelerating observers. Is there a conceptual connection between acceleration and gravity? Indeed there is, and when Einstein first realised it, he called it the happiest thought of his life.

The happiest thought

We’ve all seen pictures of astronauts aboard the International Space Station. They float. If an astronaut lets go of a screwdriver, it floats next to them. Even globules of water float around undisturbed as mesmerising, gently oscillating bubbles of liquid. Why? The astronauts, the screwdriver, the water and indeed the International Space Station itself have not escaped Earth’s gravitational pull. They are only 400 kilometres or so above the surface: just forty times the altitude of a commercial aircraft. If you were to jump out of an aircraft, you would be unwise to assume you’ve escaped gravity and not deploy your parachute. The space station is falling towards the Earth in just the same way as you would if you jumped out of an aircraft, but it’s travelling fast enough relative to the surface of the Earth – around 8 kilometres per second – to continually miss the ground. It can continue to orbit in this way with very little intervention from rockets because there is very little air resistance at an altitude of 400 kilometres. We say the space station is in freefall around the Earth; forever falling towards the ground but never reaching it. The crucial point is that freefall is locally indistinguishable from floating freely in deep space, far away from any stars or planets; that is to say, if the astronauts had no windows and could not look outside to see the Earth below, they would be unable to do any experiment or make any observation to inform them that they are in the gravitational field of a planet. This is the reason every object floats undisturbed in the space station; there is no force to disturb them, and this is the idea that Einstein famously described as the ‘glücklichste Gedanke meines Lebens’, the happiest thought of my life. It immediately suggests that there is something interesting about the force of gravity, because gravity can be removed by falling. Likewise, its effects can be simulated by accelerating. Acceleration is locally indistinguishable from gravity, and vice versa.¶¶¶ That very important idea is known as the Equivalence Principle.

Figure 3.9. Rindler’s spacecraft.

Imagine that Rindler accelerates at 1g.**** Inside his spacecraft, Rindler’s experience would be precisely the same as if he were sitting comfortably in an armchair or wandering around his cabin on the surface of the Earth. If he had no windows, there is no experiment or observation he could perform to tell him otherwise. If he reduced the power of his rocket and dropped his acceleration down to around 0.3g, he might imagine he was sitting on the surface of Mars. No other force in Nature behaves like this. It’s not possible to remove the force between electrically charged objects by accelerating or moving around. And yet this is possible for gravity. This is the clue that led Einstein to formulate his theory of gravity purely in terms of the geometry of spacetime. Gravity as geometry. Let’s explore that remarkable idea.

BOX 3.2. Extending the Penrose diagram to two space dimensions

We have been working in a world of one space dimension where our observers can only move along a line. Much of relativity theory can be understood without needing to invoke the other two space dimensions we move around in, just as we can appreciate much of Newton’s mechanics by considering things moving along a straight line. But we should talk a little about those other two space dimensions. The left-hand picture of Figure 3.10 shows the Penrose diagram of flat spacetime in 2+1 dimensions (this is the standard notation for 2 space dimensions and 1 time dimension). It looks like two cones, glued base-to-base. At any given moment in time, we can draw a ‘now’ surface, rather than the ‘now’ lines we’ve been thinking about in our 1+1 dimensional Penrose diagrams throughout this chapter. The ‘now’ surface at the junction of the two cones (Time Zero) is a flat disk, and as we move forwards in time the Now surfaces become distorted into domes just as our lines were distorted into curves.

Figure 3.10. Extending the Penrose diagram to two space dimensions. The diagram on the left is obtained by rotating the one on the right about the vertical dashed line.

The right-hand diagram in Figure 3.10 is more closely related to the 1+1 dimensional diagrams we have been drawing throughout this chapter. Our diamonds are obtained by reflecting the triangle about the vertical dashed line. The complete 2+1 diagram is obtained by sweeping the triangle around the vertical dashed line. The left-hand diagram is a complete representation of every spacetime point in 2+1 dimensions. The right-hand diagram loses some of the information because entire circles are drawn as single dots.

It’s only in the 2+1 diagram that light cones actually look like cones. In the right-hand diagram the light cones at A and B look like ‘crosses’. Note also that the dome on the left-hand diagram appears as a line on the right-hand diagram, and that the circle on the ‘now’ sheet and the circle on the dome are actually of the same radius. The dome-one just looks smaller because we are squashing space down as time advances to make it fit into the diagram.

We live out our lives in 3+1 dimensions. We can’t draw that of course, but we can imagine. The points in the right-hand diagram would now correspond to entire spheres rather than circles.

One final word: we have taken the opportunity to introduce the notation for the five different types of infinity we mentioned throughout the chapter. The vertices of the two cones labelled i+ and i– are future and past timelike infinity – the ultimate origin and destination of anything travelling less than the speed of light. The cone’s surfaces labelled ℑ+ and ℑ– are future and past lightlike infinity, accessible only to light beams or anything else that can travel at the speed of light. The circle at the junction of the two cones marked i0 marks out spacelike infinity – the infinitely distant regions of space at any moment in time.

* De Kremer renamed himself Gerardus Mercator Rupelmundanus, Mercator being a Latin translation of Kremer, which means merchant.

† Sir Roger introduced his diagrammatic methods in the early 1960s and they were later used to great effect by Australian theorist Brandon Carter. Today, conformal spacetime diagrams are often referred to as Carter–Penrose diagrams.

‡ Hermann Minkowski, in his 1908 address ‘Raum und Zeit’ to the Society of German Natural Scientists and Physicians.

§ Distortion in the sense used here has nothing to do with the distortion of spacetime due to matter. We are talking about the need to distort the representation to fit it on a sheet of paper.

¶ Future lightlike infinity is often labelled ℑ+ (pronounced ‘scri plus’) and past lightlike infinity is labelled ℑ– (‘scri minus’).

** In three dimensions, the clocks populate all of space, not just a line.

†† This picture can be extended to three dimensions of space. The clocks would form a three-dimensional lattice spanning all of space, and one could imagine a latticework of rulers connecting them together. In the terminology of relativity, such a latticework of clocks and rulers is known as an inertial reference frame.

‡‡ Shakespeare’s Macbeth.

§§ To be precise, we should say anything that is not massless.

¶¶ Actually, Grey moves at 48.4 per cent the speed of light relative to Black, which you can just about see if you look really carefully at the figure. We chose 48.4 per cent for reasons that become clear in the next footnote.

*** If you know a bit of special relativity (which you will if you have read our earlier book Why does E = mc2?) you might like to note that the factor 8/7 = 1.14 is equal to 1/ √(1 – v2) for v = 0.484.

††† Admittedly difficult when he himself is standing on the shoulders of giants.

‡‡‡ Incidentally, you might like to note that the straight-line path between events in spacetime is longer than any non-straight line. This is because the geometry of spacetime is not the geometry of Euclid. If it were, the interval would be given by Pythagoras’ theorem: (Δτ)2 = (Δt)2 + (Δx)2. But the interval contains a minus sign: (Δτ)2 = (Δt)2 – (Δx)2, and that makes all the difference. The geometry of flat spacetime is what mathematicians call hyperbolic geometry.

§§§ Another way to see this is to note that, because Rindler starts at the bottom of region 1, all of region 1 lies within his future light cone.

¶¶¶ We say freefall is locally indistinguishable because the Earth’s gravitational field is not uniform, and this is detectable over large-enough distances. For example, objects fall towards the centre of the Earth, and this means that two objects which begin falling parallel to each other at some height will get closer together as they head towards the ground. This is called a tidal effect. We’ll meet these effects, which in the context of black holes lead to spaghettification, later in the book.

**** 1g is an acceleration equal to that of an object falling in the vicinity of the Earth’s surface, i.e. 9.8 metres per second squared.