Black Holes: The Key to Understanding the Universe

2

Unifying Space and Time

‘The word “distance” by itself does not belong in a book on general relativity. The word “time” by itself does not belong in a book on general relativity.’

Edwin F. Taylor, John Archibald Wheeler and Edmund Bertschinger12

Black holes are perfect for learning about physics because understanding them requires pretty much all of it. Don Page begins his exhaustive ‘inexhaustive review of Hawking radiation’ with the sentence: ‘Black holes are perhaps the most perfectly thermal objects in the universe, and yet their thermal properties are not fully understood.’13 Thermodynamics is one of the cornerstones of physics, dealing with familiar concepts such as temperature and energy, and a possibly less-familiar concept, entropy. Thus, we will need to learn some thermodynamics. Stephen Hawking’s seminal paper ‘Particle Creation by Black Holes’ begins: ‘In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies …’14 Thus, we will need to learn some quantum mechanics. And, of course, there is Einstein’s General Theory of Relativity, wherein, as Misner, Thorne and Wheeler write in their great (in quality and in mass) textbook Gravitation, ‘… the reader is transported to the land of black holes, and encounters colonies of static limits, ergospheres, and horizons – behind whose veils are hidden gaping, ferocious singularities’.15 This is the land we will explore first.

We learn at school that gravity is a rather mundane thing – the force between everyday objects; you can’t jump too high from the surface of the Earth because there is a force that pulls you back down to the ground. In 1687, Isaac Newton formalised this idea and published it in The Principia Mathematica. Newton’s theory works well in most situations, allowing us to calculate the trajectories of spacecraft to the Moon and beyond, and at first sight has nothing to say about space and time at all. Newton did, however, assume two properties of space and time in formulating the theory. He assumed that time is universal: if everyone in the Universe carries a perfect clock and all the clocks were synchronised sometime in the past, they will all read the same time in the future. Newton put it more poetically: ‘Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external …’ He also assumed that space is absolute: a great arena within which we live out our lives. ‘Absolute space, in its own nature, without regard to anything external, remains always similar and immovable … Absolute motion is the translation of a body from one absolute place into another.’ These assumptions sound like common sense – so much so that it’s a testament to Newton’s genius that he even noticed he’d made them. His true genius is revealed when we discover that his care was prescient because both assumptions are wrong. The Universe is not constructed this way, and as the foundations of the theory crumble, so must the theory itself. Einstein’s General Theory of Relativity is the replacement, describing a Universe in which distances in space and the rate at which time ticks depend on an observer’s proximity to stars and planets and black holes or even on their route to the shops and back.

It is an experimental fact that the passage of time varies from place to place and depends on how fast things move relative to each other. In a wonderfully simple experiment, carried out in 1971, Joseph C. Hafele and Richard E. Keating bought round-the-world airline tickets for themselves and four high-precision atomic clocks. In their own carefully chosen words: ‘In science, relevant experimental facts supersede theoretical arguments. In an attempt to throw some empirical light on the question of whether macroscopic clocks record time in accordance with the conventional interpretation of Einstein’s relativity theory, we flew four caesium beam atomic clocks around the world on commercial jet flights, first eastward, then westward. Then we compared the time they recorded during each trip with the corresponding time recorded by the reference atomic time scale at the US Naval Observatory. As was expected from theoretical predictions, the flying clocks lost time (aged slower) during the eastward trip and gained time (aged faster) during the westward trip.’16 The eastward clocks lost 59 nanoseconds and the westward clocks gained 273 nanoseconds.* These are tiny time differences over such a long journey, but they are not zero and, most importantly, the experimental observations agree with the mathematical calculations performed using Einstein’s theory. The Hafele–Keating paper finishes in a similarly concise fashion: ‘In any event, there seems to be little basis for further arguments about whether clocks will indicate the same time after a round trip, for we find that they do not.’ And there we have it – a remarkable and highly unexpected feature of our Universe that relativity theory is designed to describe: time is not what it seems.

Space is not what it seems either: in a further affront to common sense, the distance between two points in space is not something everyone will agree upon. Hold your fingers apart in front of you. Who would dare say that the distance between your fingertips depends on the point of view? Einstein would. This is also a well-verified experimental fact. The Large Hadron Collider at CERN is the world’s most powerful particle accelerator. The giant machine’s job is to make protons travel around its underground tunnel at 99.999999 per cent the speed of light, before smashing them together. The purpose is to explore the structure of matter and the forces of Nature that animate our world. The LHC is 27 kilometres in circumference from the point of view of someone standing on the ground in Geneva, marvelling at this great engineering achievement. From the point of view of the protons orbiting around the ring, the circumference is 4 metres.

Einstein didn’t know about atomic clocks or airliners or the Large Hadron Collider in 1905, and no experiments had been performed that challenged Newton’s pictures of absolute space and universal time. Why, then, did Einstein decide to invent a new picture? The answer is that he realised there is a fundamental clash between Newton’s seventeenth-century theory of gravitation and James Clerk Maxwell’s nineteenth-century theory of electricity and magnetism.

The clash concerns the way the speed of light appears in Maxwell’s theory. The theory, which is based on experimental observations carried out by Michael Faraday, André-Marie Ampère and others throughout the nineteenth century, states that light is an electromagnetic wave that travels through the vacuum of empty space at a fixed speed: 299,792,458 metres per second. According to the theory, the speed of a beam of light is always this precise number, no matter how the person that measures it moves relative to the source of the light. That’s a very strange prediction, and not the way most other things in Nature behave.

The fastest ball ever bowled in international cricket, at the time of writing, was by Shoaib Akhtar for Pakistan against England in Cape Town in 2003. Nick Knight, opening for England, played a textbook defensive stroke to square leg, rounding off a maiden over for Akhtar. The ball travelled down the wicket at 100.2 mph.† If Akhtar had instead bowled the ball from a Grumman F14 Tomcat travelling at 600 mph directly towards Knight, then the ball would have reached the batsman at 600 + 100.2 = 700.2 mph, and he may not have guided it to square leg. This is not true for light. If, rather than the cricket ball, a laser beam had been sent towards Knight from the F14 Tomcat, the light would still have reached him at the speed of light (not the speed of light + 600 mph).

There are two possible resolutions to this bizarre feature of Maxwell’s equations. The obvious one would be to modify Maxwell’s equations so that this is not the case, and light behaves like a cricket ball. Ultimately this is an experimental question; a question about what actually happens in Nature. Innumerable observations of disparate physical phenomena for well over a hundred years tell us that Maxwell’s equations are correct as they stand and therefore light always travels at the same speed.

The other, less obvious, resolution is to change the way that observers travelling at different speeds relative to each other account for distances and time differences such that everybody always measures the speed of light to be the same. Einstein chose this route, thus rejecting Newton’s notions of absolute space and time, and this choice led him to relativity.

Einstein’s theory of relativity

Einstein’s theory is a model, which is to say it is a mathematical framework that allows us to make predictions about how objects that exist in the natural world behave. The model is inherently geometrical, which lends itself to intuitive visual pictures which require very few equations – a good thing for a book such as this. We believe that the best approach to explaining relativity is to describe this geometrical picture, rather than attempt to present its evolution historically. Our justification is that the model works, and that is the only justification necessary. Einstein could have simply plucked his theory out of thin air without any reference to Maxwell’s theory or experiments, and it would be equally valid because it is a good model in the sense that its predictions have passed every experimental test to date.

If Einstein could have plucked one single idea out of thin air that would have led him directly to his theory, including the explanation of what happened in Hafele and Keating’s experiment and the most famous equation in all of physics, E = mc2, it would be a concept known as ‘the spacetime interval’. The idea is beautifully simple.

Let’s return to Pakistan against England in Cape Town and Shoaib Akhtar’s record-breaking delivery to Nick Knight. We are going to simplify things for now by switching gravity off – we’ll switch it back on at the end of this chapter. This means that when the ball leaves Akhtar’s hand it will travel to Knight in a perfect straight line at a constant speed – 100.2 mph relative to the ground.‡ Let’s further imagine that the cricket ball has a clock inside. At the moment the ball leaves Akhtar’s hand, the ball emits a flash of light and records the time on its internal clock. At the instant the ball reaches Knight’s bat, the ball emits another flash of light and records the time of arrival on its internal clock. We’ll call the time interval between the flashes as measured on the cricket ball clock Δτ – pronounced delta tau.

In the commentary box, Jonathan Agnew (Aggers), for the BBC, notes the arrival of the two flashes of light from the ball and calculates the time interval between the emission of the flashes from his point of view: ΔtAggers.§ He also measures the distance between the place where the ball leaves Akhtar’s hand and the place where the ball hits Knight’s bat: ΔxAggers.

In his Grumman F14 Tomcat flying over the wicket in a straight line between the stumps at 600 mph Tom, the pilot, also notes the two flashes of light and calculates the time interval between the emission of the flashes from his point of view: ΔtTom. Like Aggers, he also measures the distance between the place where the ball leaves Akhtar’s hand and the place where the ball hits Knight’s bat: ΔxTom.

The Hafele and Keating result tells us that the time differences between the emission of the flashes as measured by Aggers, Tom and the cricket ball will all be different. Likewise, the distance the ball travelled from bowler to batsman will also be different. For those who have never encountered Einstein’s ideas before, these differences should come as a tremendous shock. They are counter-intuitive because they mean that distances and time intervals are not something everyone can agree upon. However, here is a remarkable and important result. If Aggers calculates the quantity (ΔtAggers)2 – (ΔxAggers)2 and Tom calculates the quantity (ΔtTom)2 – (ΔxTom)2 then they will both get the same result, and the result will be equal to the square of the time interval measured using the cricket ball clock, (Δτ)2:

(Δτ)2 = (ΔtAggers)2 – (ΔxAggers)2 = (ΔtTom)2 – (ΔxTom)2

(Δτ)2 is known as the spacetime interval between the two events: event 1 is the ball leaving the bowler’s hand and event 2 is the ball striking the bat. You may well ask: ‘What does it mean to subtract a distance in space squared from a time difference squared?’ The answer is that we must specify the distance between two events as the time it takes for light to travel between those events, which means we should compute the distance in light seconds. The spacetime interval (or ‘interval’ for short) is important because it is a quantity on which everyone agrees, no matter what their point of view. In physics we call such a quantity an invariant. Since Nature doesn’t care about our point of view,¶ we should only seek to describe Nature in terms of invariant quantities. When we discover an invariant, it is a big deal because we learn a little more about the essential structure of the Universe.

In their book Exploring Black Holes, Taylor, Wheeler and Bertschinger describe the equation for the interval as ‘one of the greatest equations in physics, perhaps in all of science’. Kip Thorne and Roger Blandford, in Modern Classical Physics write that the interval is ‘among the most fundamental aspects of physical law’. The word ‘fundamental’ is important. You might reasonably ask: ‘Why is the interval like this?’ ‘Why does everyone agree on this particular combination of time and space?’ The answer, as Thorne and Blandford imply by their use of the word ‘fundamental’, is that this is the way the Universe is constructed. We know of no deeper explanation for the form of the interval.

A further question you may be asking is: ‘How should I think about the interval, this most fundamental aspect of physical law?’ This is a good question. Physicists usually endeavour to develop a mental picture of what’s happening in their equations: physical intuition brings equations to life. Fortunately, the interval does have a simple physical interpretation. It is related to what we will refer to as ‘the distance between two events’. This is not the usual distance between them in space, but the distance in spacetime. Let’s explore that idea.

Events and worldlines in spacetime

The concept of an event is fundamental to relativity. An event is something that happens somewhere and somewhen. You clicking your fingers is a good approximation of an event: it happens very quickly and in a well-defined location. The emission of a flash of light from our cricket ball is an event. Strictly speaking an event is an idealised concept, something that happens so fast and in such a small area that it corresponds to a single point in space and time. The theory of relativity is concerned with the relationship between events; how far apart they are in spacetime and whether they influence each other or not. This is a very intuitive way to think about the world, so much so that it’s how we speak in everyday life; ‘I’ll meet you tomorrow evening at eight o’clock at the pub.’ ‘I was born on 3 March 1968 in Oldham.’ Things that have happened to us and things that will happen to us are all events in space and time, and they happen somewhere and somewhen. A slight shift in wording, and we have the basis of the theory of relativity: things that have happened to us and things that will happen to us are all events in spacetime. What is spacetime? It’s the collection of all events. Everything that has ever been and will ever be in the Universe.

Here is a picture of spacetime. Picture the events of your life. Your first day at school. Christmas with your grandparents. That night down the pub. A chronicle of moments from joy to despair and everything in between. Events are the atoms of experience. From our human perspective, events come with labels; we speak of them in terms of the place and time they happen. Imagine carefully laying out the events of your life one by one to form a line snaking over spacetime; an unbroken path charting your journey through the world. This is called your worldline.

Figure 2.1. The events of a life in space and time. The line through the events is known as a worldline. The cones at each event are known as light cones. They are the paths of a flash of light, emitted at the event. Because nothing can travel faster than light, only future events inside these cones can be influenced by the original event.

Figure 2.1 depicts a worldline winding its way over spacetime. It is known as a spacetime diagram. Imagine this is your worldline, your life laid out before you. Change the dates, add your own events and memories, construct the map of your experiences. Spacetime is an evocative thing. The collection of all events in your life, past, present and future. Your memories are of events in spacetime. The moments that make up your life – Christmases long ago, summer afternoons with school-friends, first kisses and last goodbyes – have not been lost forever. Those moments are still out there, somewhere in spacetime. Your future – everything that has yet to happen to you – every event including your death at the end of your worldline – is waiting for you to arrive, somewhere in spacetime. If we lay out all events in this way, we have created a map of spacetime, and the distance between the events is given by the interval. How wonderful it would be to have freedom of movement over this map, the ability to revisit every moment. We can move anywhere over a map of space, so why should we not have the same freedom over a map of spacetime? The reason is to be found in the interval.

Let’s recap. From a particular point of view, the distance in space between two events is measured to be Δx and the difference in time between the events is measured to be Δt. From a different point of view, Δx and Δt will be different, which is very counter-intuitive. But crucially, the interval (Δτ)2 will not depend on point of view:

(Δτ)2 = (Δt)2 – (Δx)2

We can use the idea of the interval to introduce the concept of the length of a worldline. To be specific, think of the worldline in Figure 2.1 as it goes from being born in 1968 to the enigmatic future event marked X. How long is this portion of the worldline? If event X occurs at precisely the same location as the birth, then the equation above informs us that the interval between the two events (birth and X) is just given by the time interval, i.e. Δτ = Δt because Δx = 0. This is the interval between the two events, but it is not the length of the worldline. Rather it is the length of the worldline that goes straight up the time axis (the vertical line on Figure 2.1). Just like the distance of a journey between Oldham and Wigan depends on the route taken, so it is with distances in spacetime. They depend on the spacetime path taken by the worldline. The way to compute the length of the snaking worldline in Figure 2.1 is to imagine chopping it into lots of tiny segments. Each segment being approximately a straight line.** Then we can compute Δτ for each segment, using the formula above, and add up all the Δτ’s to get the total length.

We can also make the important observation that there are three different sorts of interval: (Δτ)2 can be positive, negative or zero. We might say that there are three different sorts of ‘distance’ in spacetime, in contrast to one sort of distance in space.

If the time difference between the events is larger than the distance in space between them, the interval is positive. Such pairs of events are referred to as ‘timelike separated’. All the events on your worldline are timelike separated from each other. There is a simple physical interpretation for the interval in this case. If you had a perfect stopwatch that you started at the moment you were born, and carried it with you for your whole life, the watch would measure the length of your worldline, from your birth to the present moment. The length of your worldline is therefore your age. This is the meaning of the interval for timelike separated events. It is the time measured on a watch carried along a worldline between events.††

If the distance in space between the events is larger than the time difference, the interval is negative. We say these events are ‘spacelike’ separated. We can now no longer interpret the interval in terms of a watch moving between the events. A physical interpretation does exist, however. For the case where the two events occur at the same time, we can interpret the interval as recording the distance between these events as measured on a ruler. It turns out that for spacelike separated events, it is always possible to find an observer (i.e. a point of view) from whose perspective the events happen at the same time. This means it’s not possible for someone or something to be physically present at both events since that would require being in two places at the same time. That’s just another way of saying that we could not arrange to carry a watch between the events.

There are therefore two fundamentally different regions of spacetime surrounding any event: a region containing those events that could conceivably be on the worldline of a watch passing through that event, and a region containing events that could not. We’ll see the significance of this division in a moment.

The third possibility is that the time difference between a pair of events is exactly equal to the distance in space between them. This is the case if a worldline between the two events is the path taken by a beam of light. To see this, recall that we measure time in seconds and distance in light seconds. Light travels 1 light second in 1 second, 2 light seconds in 2 seconds, and so on. So, for any pair of events that lie on the path of a beam of light, (Δt)2 = (Δx)2 and the interval is zero. These events are known as ‘lightlike’ separated. If we draw the paths of light rays out over spacetime from an event, they form what is known as the future light cone of the event. In Figure 2.1, the light cones are depicted as small cones at each event. The light cones spread out at an angle of 45 degrees from each event. Inside the future light cone, all events are timelike separated from the original event while outside the future light cone, all events are spacelike separated from the original event. Since we were present at every event in our own lives, our worldline snakes along inside the light cones.‡‡

It is important to understand the meaning of the light cones and what they tell us about the relationships between events in spacetime. They will be central to understanding black holes and the paradoxes they create. Let’s zoom in on a particular event on our worldline to get more of a feel for the light cone and the relationships between neighbouring events in spacetime.

Christmas in spacetime

Let’s imagine we’ve zoomed in on the region of spacetime in the vicinity of ‘Christmas 1974’ on our worldline. Your family are sitting around the TV arguing about whether to watch Bruce Forsyth and The Generation Game on BBC1 or Laurence Olivier in Henry V on BBC2. Presciently concerned about an incipient culture war, Granny springs to her feet and knocks a glass of Harvey’s Bristol Cream onto the electric fire.§§ This causes the main fuse to blow, rendering the debate meaningless.

Figure 2.2 shows the spacetime region around ‘Christmas 1974’, drawn from the point of view of someone sitting in your house. Event A is ‘Granny making contact with the glass of sherry’ and event D is ‘the fuse blows’. From this perspective, events A and D happen at very nearly the same place in space but at different times; D is in the future of A. The diagonal lines heading upwards and outwards from A trace out the future light cone of A. We’ve also drawn diagonal lines heading out into the past from A. These are known as the past light cone of A. All events in the shaded region inside the future light cone are timelike separated from A, which means that anyone who is present at A could also be present at any event inside the future light cone. All events inside the past light cone are also timelike separated from A. This means that anyone who is present at any event inside the past light cone could also be present at A. The expression for the interval between A and D is particularly simple: (Δτ)2 = (Δt)2, where Δt is the time difference between A and D as measured by a watch in your house.¶¶

Figure 2.2. An event ‘A’ in spacetime and its neighbouring region. The diagonal lines are the lines traced out by beams of light that pass through A. They form the future and past light cones of event A.

We’ve also marked two other events on the diagram, labelled B and C. From the house perspective, these happen at the same time as event A, but in a different place. Let’s say they are an alarm clock going off at the other end of the street and a car starting its engine in the neighbouring town. The interval between A and B is (Δτ)2 = – (ΔxAB)2, and the interval between A and C is (Δτ)2 = – (ΔxAC)2. The interval is negative, which means that events B and C are spacelike separated from event A; ΔxAB and ΔxAC are distances that could be measured on a ruler.

Here is the key point. Event A caused event D (Granny knocked over a glass and that caused the fuse to blow). However, event A could not have caused events B and C. For that to happen, some influence would have to travel instantaneously from A to B and C because these things all happened at the same time. This delineating of causal relationships is why light cones are so important. Events inside each other’s light cones can have a causal relationship because it is possible that some signal or influence could have travelled between them. Events outside each other’s light cones cannot have a causal relationship. The interval therefore contains within it the notion of cause and effect. Certain events can cause others, and the light cones at each event tell us where in spacetime the dividing lines lie.

Let’s now look at the same events in spacetime from two different perspectives. Figure 2.3 is a spacetime diagram constructed using measurements of distance and time made by an observer moving at constant speed past your house towards the car in the neighbouring town. As we have already discussed, such an observer will measure different times and different distances between events, but the intervals between events must remain the same because the interval is invariant. Nature doesn’t care about your point of view, and the interval is a fundamental property of Nature. For this to be the case, something quite surprising happens. Events B and C happen after event A according to this observer.

Figure 2.4 shows a spacetime diagram constructed by an observer moving in the opposite direction at constant speed past your house. This observer says that events B and C happen before event A.

Figure 2.3. Events A, B, C and D as described in the text, as seen by an observer moving past event A at constant speed travelling from left to right on the diagram.

At first sight, it seems that the spacetime picture has led to disaster. How can we countenance a theory that allows for the reversal in the time ordering of events? What if those events were your birth and death? Would someone be able to see you die before you were born?

Figure 2.4. Events A, B, C and D as described in the text, as seen by an observer moving past event A at a constant speed travelling from right to left on the diagram.

The resolution to this apparent paradox can be seen by looking at the light cones. The light cones are in precisely the same place on all three diagrams, as they must be because all observers agree on the speed of light. Notice that although events B, C and D all move around on the spacetime diagram with respect to event A as we switch between different points of view, event D always remains inside the future light cone of A and events B and C always remain outside both the future and past light cones of A. To see that this must be the case, remember that the interval between two events is invariant: if the interval is timelike from one perspective, it’s timelike from all perspectives. This means that events that can influence each other have their time-ordering preserved from all perspectives. Events that can’t influence each other do not have their time-ordering preserved, but that doesn’t matter because it does not mess with cause and effect. There is no contradiction if someone sees a house alarm going off or a car starting in the next town before or after Granny knocks over the glass, because these events could never have influenced each other – they are spacelike separated. There would of course be a contradiction if the lights fused before Granny knocked over the glass which caused them to fuse. But that can’t happen for events A and D because D is always in A’s future light cone, regardless of the point of view.

The future light cone of an event therefore tells us which regions of spacetime are accessible from that event and which regions are forbidden. Likewise, the past light cone of an event tells us which events in spacetime could have possibly had any influence on that event. If you look back at the worldline in Figure 2.1, you’ll see that travelling to revisit moments in your past, the people and memories left behind, is impossible because we can never move from inside to outside the light cone at any event in our lives. To do so, we would have to travel faster than light. But the interval is invariant, so we can’t do that. In a sense, our memories are out there, somewhere in spacetime, but we can never revisit them.

The picture of spacetime we’ve described above is that contained in Einstein’s Special Theory of Relativity, first published in 1905. It describes a Universe without gravity, which is why we took the unconventional step of switching gravity off when discussing England versus Pakistan in Cape Town. Incorporating gravity into the spacetime picture is the concern of Einstein’s General Theory of Relativity, published in 1915.

From special relativity to general relativity

The central idea of general relativity is that spacetime has a geometry that can be distorted. As we’ll see, this corresponds to changing the rule for the interval between events. Matter and energy distort spacetime in their vicinity, and Albert Einstein worked out the equations that allow us to calculate how it is distorted. This is shown schematically in Figure 2.5. Objects like the International Space Station moving close to the Earth will be travelling through a region of distorted spacetime, and if we were Newtonians, we would interpret its motion as being due to a force deflecting it from a straight line and into orbit. But there is no force in Einstein’s picture: gravity is to be understood purely as geometry.

Figure 2.5. Schematic picture of the distortion of spacetime in the vicinity of the Earth.

An entirely reasonable first response to that last paragraph is to ask what on earth are we talking about. What does it mean to talk about distorting space and time? How should we picture distorted spacetime? Thus far, we have been drawing diagrams like Figure 2.1, with time pointing upwards and one or two directions in space represented horizontally. But our world isn’t like that. We live in three spatial dimensions: forwards and backwards, left and right, up and down. Adding a fourth dimension – time – is very hard to picture.

To help us to comprehend the idea of spacetime, let’s take a step back and imagine a two-dimensional world called flatland, populated by flat creatures.*** The flatlanders can wander around, forwards and backwards, left and right, but never up or down. Their flat eyes can only see flat things on the flat surface and their flat brains can only comprehend flat things. Imagine the reception that renowned flat physicist Flat Albert would receive if he dared to say that space is really three-dimensional. ‘There is another dimension, another direction we cannot point in,’ he claims, and with his mathematics he would have no difficulty describing this three-dimensional world.

Suppose that Flat Albert is correct, and the flatlanders actually do live on the surface of a large messy table in an office, as shown in Figure 2.6. The third dimension is real; it’s the direction upwards from the surface of the table, but the flatlanders can’t see it. Their explorations haven’t yet revealed that space comes to an end at the edge of the table, but they have discovered that there are impenetrable regions where they cannot go. They have to walk around the coffee cup and the lamp and books, and they are left to wonder at why the forbidden regions are sometimes circular, sometimes rectangular and occasionally some other less regular shape. Moreover, the land is covered in regions of light and dark that shift and change in shape and size.

Figure 2.6. Flatland.

How did Flat Albert deduce that there is an extra dimension in the world, based solely on observations made on the flat tabletop? ‘It’s all to do with those changing light and dark regions,’ he says. ‘I know what they are. They are shadows.’

Albert used mathematics to figure out that the shadows are two-dimensional projections of objects that live in three dimensions (coffee cups and books) and to deduce the three-dimensional shapes of the objects that cast them. It helps that the shadows change shape occasionally, which Albert correctly interprets as the higher dimensional source of brightness changing. From our three-dimensional viewpoint, we immediately see that this is due to someone moving the table lamp.

Perhaps you can see the analogy. The interval – the thing that doesn’t vary with point of view – lives in the four dimensions of spacetime. Distances in space and differences in time are mere shadows; they vary as we adopt different points of view in the three-dimensional world of our everyday experience. We can’t picture something that lives in four dimensions, just as Flat Albert couldn’t picture a coffee cup or a lamp or a book. But that didn’t stop him lifting his gaze from the two-dimensional world of his experience to marvel at the true reality of three-dimensional space and the unchanging objects that live on the tabletop.

By pushing the flatland analogy a little further, we can also get a feel for how general relativity fits into this picture. The flatlanders may be inclined to assume that their tabletop is flat. If this were the case, parallel lines across flatland would never meet and the interior angles of triangles would add up to 180 degrees. We call such a flat geometry ‘Euclidean’.

If the table is slightly warped, however, the flatlanders will discover small deviations from Euclid. Using precise measuring devices, they will be shocked to find triangles whose angles do not add up to exactly 180 degrees and parallel lines that converge or diverge from each other. It is in precisely this sense that we speak of space being curved or warped in Einstein’s theory of gravity and it is what we are aiming to illustrate in Figure 2.5.

Flatland helps us to picture how space could have more dimensions than we directly experience, and it helps us to picture how space can be warped. By dropping down a dimension we can see the bigger picture of a two-dimensional space (the tabletop) embedded in a three-dimensional space (the room). We can’t step outside of ourselves to see the bigger picture of four-dimensional spacetime because our imaginations are limited to picturing things in three dimensions or less. In that sense we are very much like flatlanders, fated to view the world in too few dimensions.

It’s not easy to become comfortable with the idea of higher-dimensional spaces, but if it offers some comfort, professional physicists are no better at picturing four-dimensional spacetime than you are. When it comes to spacetime, we are all Flat Albert, peering at shadows. Fortunately, it is not necessary to try to visualise spacetime in all its four-dimensional grandeur. Often, we can drop a couple of dimensions in our mental picture and not lose anything important. We’ve already seen this in the spacetime diagrams we have used to explore the basics of special relativity, where (apart from Figure 2.1) we depicted only a single space dimension and the time dimension. Our understanding wouldn’t have been enhanced by trying to draw two-dimensional space, although it might have made our diagrams look prettier. If we’d attempted to draw all three space dimensions plus the time dimension, we’d have run into problems.

We will rarely need to keep track of more than one space dimension in our study of black holes, because our focus will be on the distance from the black hole. We will be most interested in how the warping of the geometry influences the way cause and effect play out and that means keeping track of the light cones. Physicists have had a century to come up with a nice visual scheme for doing this, and the most widely used is named after Sir Roger Penrose. In the following two chapters, we’ll introduce ‘Penrose diagrams’. Armed with these beautiful maps of spacetime, we’ll be ready to navigate beyond the horizon.

* A nanosecond is one billionth of a second.

† We will use imperial units when discussing cricket.

‡ In more technical language, we are assuming that the cricket ground is an inertial reference frame. We might imagine it detached from Earth and floating freely between the stars. We are also neglecting air resistance.

§ He’ll need to make a correction for the time it takes for the light to travel from the ball to his eyes in order to work out when the flash was actually emitted.

¶ This comes as a shock to a certain type of individual.

** This means that the person travelling along the worldline is not accelerating during the segment. Any curving path, through space or through spacetime, can be thought of as being made up of lots of tiny straight-line paths.

†† To see this, notice that the watch never moves relative to itself, so Δτ = Δt because Δx = 0.

‡‡ Light cones are cones in spacetime if space is two-dimensional because a flash of light will spread out in a circle of increasing radius. In three dimensions, a flash of light spreads out in a spherical shell making a kind of hyper-cone in spacetime. That is not possible to visualise, so we’ll stick to two dimensions of space to make drawing diagrams easier.

§§ Both bars are on, and there is Vim under the sink.

¶¶ Notice that, for pretty much all events we deal with in our everyday lives, (Δτ)2 = (Δt)2 is approximately true. That’s because the distances in space we are usually interested in range over a few metres or kilometres or even a few thousands of kilometres, and all of these are tiny when measured in light seconds. In everyday life Δx is very much smaller than 1 light second, and this is the reason why it feels to us as if time is universal.

*** We are inspired by Edwin Abbott’s 1884 novel Flatland: A Romance of Many Dimensions. And possibly Mr Oizo’s ‘Flat Beat’ featuring Flat Eric.