Black Holes: The Key to Understanding the Universe

4

Warping Spacetime

Less than five months before his death in 1916, while serving in the German army calculating the trajectories of artillery shells on the eastern front, the eminent astrophysicist Karl Schwarzschild discovered the first exact solution to the equations of Einstein’s General Theory of Relativity. Schwarzschild’s achievement was no less remarkable for the fact that he derived the solution and sent it to Einstein just a few weeks after the theory had been published. Einstein was impressed, writing in return, ‘I have read your paper with the utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way.’ Schwarzschild had found the equation describing, to very high accuracy, the geometry of spacetime around a star. Recall John Wheeler’s maxim: ‘Spacetime tells matter how to move; matter tells spacetime how to curve.’ Schwarzschild’s solution describes the curve of spacetime, and it’s then a reasonably straightforward task to work out how things move over it. Today, Schwarzschild’s solution is one of the first things taught in an undergraduate course on general relativity and, in most circumstances, it corresponds to a tiny improvement over the simpler Newtonian predictions for planetary orbits. But not in all circumstances, because the Schwarzschild solution, unbeknown to him and to Einstein in 1916, also describes black holes.

What does Schwarzschild’s solution to Einstein’s equations look like? We caught a glimpse of the answer when we thought about the warped tabletop of flatland. Flat Albert and his flat mathematician friends discovered that the geometry of Euclid no longer applies because the table’s surface is warped. The angles of triangles do not quite add up to 180 degrees and distances between points will not be described by the familiar form of Pythagoras’ theorem. If Flat Albert wanted to calculate the distance between two places in warped flatland, he’d have to find a way of representing the warping mathematically.

To warm up, let’s forget spacetime for the moment and return to Earth. The Earth’s surface is curved, and this means we can’t simply use Pythagoras’ theorem to calculate the distance between widely spaced cities such as Buenos Aires and Beijing. We can easily appreciate this because we are three-dimensional beings, and we know what a sphere looks like. For example, if we were to procure a 19,267-kilometre-long ruler and place it in downtown Buenos Aires, its tip would not land in Beijing. The reason is that the ruler is flat and the Earth isn’t. The ruler would stick out into the third dimension – its tip would end up off the surface, out in space. We could, however, imagine purchasing 20 million one-metre rulers and laying them end to end along the great circle route between the two cities. The rulers could be made to follow the curved contour of the Earth’s surface (in reality we’d have to tunnel through any mountains we encountered to keep our chain of rulers at sea level, so we are imagining a perfectly smooth spherical Earth). Give or take a few metres, we could measure the distance over the curved surface of the Earth this way. If we wanted to do better, we could make each ruler one centimetre long or even smaller. Smaller rulers have the advantage that they better track the curving surface of the Earth.

The idea that we can build up a curved surface using lots of little flat pieces is nicely illustrated by the Montreal Biosphere, designed by Buckminster Fuller for the 1967 World’s Fair. When viewed from afar, the biosphere is a perfect-looking sphere, but close-up we can see it’s made up of lots of small flat triangles, ‘sewn together’ but slightly tilted with respect to each other. Any shape could be constructed in this way; the geometry is determined by how the flat pieces are assembled.

Figure 4.1. The Montreal Biosphere, constructed for ‘Expo 67’. (meunierd/Shutterstock)

In general relativity, curved spacetime can be built up in the same way. Lots of little pieces of flat spacetime can be sewn together to make curved spacetime, and Schwarzschild’s solution to Einstein’s equations describes how they are sewn together in the vicinity of a star. The distance in spacetime between events on each little flat piece could be determined by an arrangement of clocks and rulers, i.e. the interval (Δτ)2 = (Δt)2 – (Δx)2. We can get a very accurate description by choosing the spacetime patches to be sufficiently small that the flat space formula for the interval is a good approximation over each patch. This is just like saying the distance between two points on one of the little triangles on Buckminster Fuller’s biosphere can be determined using Pythagoras’ theorem, despite the fact that the distance between two points separated by many little triangles requires a more difficult calculation because the surface is curved.

We’ve sketched this patchwork view of spacetime in Figure 4.2. If we were five-dimensional beings with an innate sense of hyperbolic geometry, we would be able to visualise how the little pieces sew together to make a ‘surface’ curved into a fifth dimension. Good luck with that, but the basic idea is quite simple. We are to think of curved spacetime as being tiled by lots of little flat pieces of spacetime, each slightly tilted with respect to their neighbours and adorned with their own grids of clocks and rulers. The challenge in general relativity is to specify how the pieces are sewn together. If we know that, we can calculate the interval between widely separated events on the curved surface by adding up all the intervals on each patch, just as we laid down the little rulers to measure the distance between Buenos Aires and Beijing.

Figure 4.2. Building up spacetime by sewing together a patchwork of tiny regions each of which is flat.

The idea that curved spacetime is well approximated by flat spacetime over sufficiently small distances and intervals of time, just as the Earth is flat over sufficiently small distances, is precisely what Einstein had in mind when he had his happiest thought:

‘At that moment there came to me the happiest thought of my life … for an observer falling freely from the roof of a house no gravitational field exists during his fall – at least not in his immediate vicinity. That is, if the observer releases any objects, they remain in a state of rest or uniform motion relative to him, respectively, independent of their unique chemical or physical nature. Therefore, the observer is entitled to interpret his state as that of rest.’

This quote is wonderful because it illuminates Einstein’s thinking. He didn’t think mathematically, at least initially. He thought in simple pictures and asked simple questions. What does the fact that gravity can be removed by falling tell me? If gravity can’t be detected in a freely falling observer’s immediate vicinity, spacetime must be flat in their immediate vicinity. Don’t get confused about what it feels like to actually fall off a roof by the way – we’re considering an idealised fall in a vacuum and ignoring air resistance. Simplify the problem down to its essence. All cows are spherical to a theoretical physicist, which is why they are clear thinkers but shit farmers. Gravity is a strange force because it can be removed by falling. Einstein’s genius was to see the connection between this idea and a geometric picture of gravity as curved spacetime. Gravity appears not because there is a fundamental force of attraction between things, as we learn at school, but because small patches of spacetime are tilted relative to their neighbours in the vicinity of massive objects.

If there is no force of gravity, why does a person fall off a roof and hit the ground, or the Moon orbit the Earth? The answer is that the person and the Moon are both following straight lines over curved spacetime. We can be more specific if we recall the Twin Paradox in the previous chapter. There, we encountered the Principle of Maximum Ageing. An astronaut that does not accelerate takes a path over spacetime between two events that maximises the time they measure on their wristwatch between those events. In general relativity, the Principle of Maximum Ageing is placed centre stage as a fundamental law of Nature that determines a freely falling object’s worldline over curved spacetime. As Einstein says in his quote, an observer in freefall ‘is entitled to interpret his state as that of rest’. This implies that the path a freely falling object takes over curved spacetime must be the path that maximises the time on a wristwatch carried by the object. On each little flat patch, this path will be a straight line across the patch, but in curved spacetime the patches sew together to make a curve. The result is entirely analogous to the case of the little flat rulers on Earth. The straight lines must match up with each other end-to-end, but the resulting path is curved. The result in spacetime is what we see as an orbit – the paths of the planets around the Sun. Or, for that matter, the fall as someone slips unfortunately off a roof. In a way, the path the unfortunate faller takes on their way to the ground is entirely logical – they are maximising the time they have left.

Schwarzschild’s solution for the curvature of spacetime, when paired with the Principle of Maximum Ageing, is all we need to calculate the worldlines of anything falling in the vicinity of a planet, star or black hole.

The opportunity to acquire a deeper understanding of general relativity and Schwarzschild’s solution lies within our grasp, and it would be a shame not to go all the way when we’ve come this far. So, over the next few pages there is a little more mathematics than in the rest of the book. There is nothing much more complicated than Pythagoras’ theorem, but if you really don’t like mathematics then don’t worry; normal diagrammatic service will be resumed shortly.

The metric: calculating distances on curved surfaces

By 1908, Einstein had the basic idea of gravity as curved spacetime, but it took him a further seven years to achieve its mathematical realisation in the form of general relativity. His challenge was to find a way of calculating the distance between two events if spacetime is curved. When asked why it took so long, here is what he said: ‘The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning.’17

To understand what Einstein meant, we’ll leave spacetime for a moment and return to two-dimensional Euclidean geometry. Let’s choose two points A and B and draw a straight line between them, as shown in the left-hand picture in Figure 4.3. The line will have a length that we could measure with a ruler. Call that length Δz. The line of length Δz is also the hypotenuse of a right-angled triangle with sides of length Δx and Δy. Pythagoras’ theorem relates these three lengths:

(Δz)2 = (Δx)2 + (Δy)2

In this equation, all the quantities are distances that can be measured by rulers. They also happen to be the difference in coordinates using the grid that you can see in the background. Specifically, A is at x = 3 and y = 2, which we write (3,2), and B is at (9,7), so Δx = 9 – 3 = 6 and Δy = 7 – 2 = 5. Using Pythagoras gives us Δz = √61.

Now look at the right-hand picture in Figure 4.3. It is the same pair of points, A and B, but now with a different grid. The new grid is perfectly good for labelling points: A is at (5,3) and B is at (7,9). But the differences in these coordinates cannot be used in Pythagoras’ theorem. This is a nice illustration of the arbitrariness of grids. Any grid will do for labelling points, but some grids are more useful than others. Here, the square grid makes it easier to calculate the length from A to B. On a flat surface, we can always choose a rectangular grid to make life easier, but on a curved surface, it is impossible to pick a single grid such that the Pythagorean equation works for the distance between every pair of points. The distinction between the coordinate grid as a mesh for labelling points and as a device to help calculate distances between points is what Einstein was referring to when he said it is ‘not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning’. We need to follow his words of warning and not get too attached to the grids we lay down over spacetime.

Figure 4.3. Left: The distance between A and B is related to the coordinates of A and B via Pythagoras’ theorem. Right: The same two points, A and B, can be located using a wavy grid but the distance between them is not related to the coordinates via Pythagoras’ theorem.

What is true is that there is always a way to compute distances using any grid. It’s just that the formula is not the one due to Pythagoras. If X and Y are the coordinates labelling the wavy grid on the right of Figure 4.3 then, for any two points that are sufficiently close together the distance between them can always be written:

(dz)2 = a(dX)2 + b(dY)2 + c(dX)(dY)

where a, b and c are numbers that vary from place to place on the grid. We changed the notation and wrote dz instead of Δz. The two quantities have the same meaning – the difference between two coordinates – but we are going to reserve using the d’s for the special case in which these distances are small. This formula is true for any grid and a similar formula can be written down in more than two dimensions. For any curved surface, the set of numbers like a, b and c provide the rule for how to compute distances. Collectively, that set of numbers (like a, b and c) is called the ‘metric’ for the surface. Once we know the metric for our chosen coordinate grid, we can compute distances. A big part of general relativity is deriving the metric for a particular situation. This is what Schwarzschild did for the spacetime around a (non-rotating) star.

The Schwarzschild solution

We can now step back into spacetime and return to general relativity. Schwarzschild’s solution tells us the metric in the vicinity of any spherically symmetric distribution of matter like a star or black hole. Using the spacetime grid that Schwarzschild used (more on that in a moment), the corresponding interval (distance) between two nearby events outside of a star or black hole is:

As in Chapter 1, the Schwarzschild radius is given by the formula:

where G is Newton’s gravitational constant, M is the mass of the star and c is the speed of light. Pretty much everything that we want to know about non-spinning black holes in general relativity is contained within this one line of mathematics. We see that Schwarzschild chose a grid labelled with a time coordinate, t, and a distance coordinate R. Ignore the dΩ2 term for now, it won’t be important and we’ll explain why in a moment.

Because the spacetime is curved, it is not possible to find a single coordinate grid such that the flat (Minkowski) formula for the interval holds everywhere. That’s the reason for the factors in front of dt2 and dR2. Conceptually, these factors are no different from those we had to introduce in the simpler two-dimensional case above: they encode the information about the curvature. The formula is telling us that the warping of spacetime at a particular location depends on how close that location is to the star and how massive the star is. The t and R coordinate grid chosen by Schwarzschild doesn’t have to correspond directly to anything that can be measured using clocks or rulers. However, the R and t coordinates do have a physical interpretation, which will allow us to develop an intuitive picture of Schwarzschild’s spacetime.

We’ve sketched what we might term a ‘space diagram’ of Schwarzschild’s spacetime in Figure 4.4. The star sits at what we’ll call the ‘centre of attraction’. We’ve drawn two shells surrounding the star. Each shell is at a fixed Schwarzschild coordinate R from the centre of attraction (at R = 0). The R coordinate is defined in terms of the surface area of these shells. In flat space, the surface area of a sphere A = 4πR2, and R is the distance to the centre of the sphere as measured by a ruler. In the distorted space around a star (or black hole) this is no longer true (for a black hole it’s not even possible to lay a ruler down starting from the centre of attraction – the singularity). We can, however, always measure the area of spherical shells like those on the diagram and R is the radius a shell would have had if spacetime were flat. That’s how Schwarzschild chose this coordinate.

No matter how the spacetime is distorted, the distortion must be the same at every point on these spherical shells. This is because Schwarzschild assumed perfect spherical symmetry when he derived his equation. Think of a perfect sphere; every point on the surface is the same as every other point. The dΩ2 piece in the equation deals with the distance between events on a particular shell. It is precisely the same as the piece we find in the metric used to calculate distances on the surface of the (spherical) Earth. We’ve discussed this in a bit more detail in Box 4.1. We would need this piece if we wanted to calculate the details of orbits around a star or black hole, but in what follows we’ll always consider things moving only inwards or outwards. This will simplify matters while still capturing the important physics.

The Schwarzschild time coordinate also has a simple definition: t corresponds to the time as measured on a clock far away from the centre of attraction, where spacetime is almost flat. As we move inwards towards the star, spacetime becomes more curved and that’s why we need the factors in front of dR2 and dt2. These factors are more important as we move inwards and are close to 1 far away. This makes sense because it means that far away from the star the interval is the same as it is in flat space.

Figure 4.4. Schwarzschild ‘space diagram’. The star sits at the centre. Two imaginary spherical shells surround the star.

The fact that Schwarzschild’s coordinates have a simple interpretation far from the star allows us to understand what the curvature of spacetime means for the passage of time and the measurement of lengths closer in. Let’s imagine we have access to a small laboratory that we can position anywhere in spacetime. Our laboratory has no rockets attached and is therefore in freefall. Inside the laboratory there is a watch to measure the passage of time and a ruler to measure distance. Our lab is small in both space and time, which means we can assume spacetime is flat inside the lab. Let’s now locate the laboratory in the vicinity of the outer shell of Figure 4.4 and observe the watch ticking. If the ticks are so short that our laboratory stays at roughly the same R coordinate for the duration of the tick, Schwarzschild’s equation tells us that the spacetime interval between ticks is:

We’ve used an ‘approximately equals’ sign to emphasise that we’re making an approximation. In this case, dR ≈ 0 between the ticks so we can ignore the dR piece of Schwarzschild’s equation.

This equation is the origin of our claim in Chapter 1 that time passes more slowly for an astronaut when they are close to a star or black hole. dt2 is the time interval (squared) we would measure between the ticks of our laboratory watch as measured by a clock at rest far away from the star, where spacetime is not curved. The (1 – RS/R) factor corrects for the fact that this does not correspond to time in our laboratory, as measured by the laboratory watch. The curvature has distorted time and so we need more ticks of the distant clock for one tick of the laboratory watch. Time has slowed down at the location of the laboratory relative to far away. On the inner shell in Figure 4.4, R is smaller still. If we place our laboratory there, the number in front of dt2 will be even smaller, and therefore watches on the inner shell will run even slower.

What about space warping? Imagine sitting in the laboratory at the outer shell and measuring the distance to a nearby lower shell using a ruler. If the lower shell is close by, the measured distance on the ruler between the two nearby shells is given by the second term in Schwarzschild’s equation:

Here dR would be the distance between the shells if space were flat. Since the factor (1 – RS/R) is now in the denominator, the distance between the nearby shells as measured by an observer on one of the shells is larger than it would have been in flat space. This means that space is being stretched and time is being slowed down as we get closer to a star.

To get a feel for the size of these effects, we can put the numbers in for the case of the Sun. The Sun’s Schwarzschild radius is approximately 3 kilometres and its radius is approximately 700,000 kilometres. This gives a distorting factor of 1.000002 at the surface of the Sun. This means that, for two Sun-sized shells whose radii differ by 1 kilometre in flat space, the measured distance between them would be 2 millimetres longer than 1 kilometre. Likewise, an observer far away from the Sun would see a watch at the Sun’s surface run slow by 2 microseconds every second, which is around a minute per year.

The Schwarzschild black hole: just remove the star

Schwarzschild’s solution was originally used to study the region outside of a star or planet (the region inside the star is filled with matter and his solution is not valid there). The remarkable thing is that the same solution can also be used to describe a black hole. All we need to do is to ignore the star. Schwarzschild’s solution then describes an infinite, eternal Universe in which the spacetime becomes more and more distorted as we head inwards towards the singularity at R = 0: a perfect eternal black hole.

Figure 4.5. Schwarzschild ‘space diagram’ with the star removed. There is no matter anywhere.

We’ve sketched Schwarzschild’s space without a star in Figure 4.5. The two imaginary shells we considered before are still there, but the star has disappeared, leaving only empty Schwarzschild spacetime. We’ve also drawn a shell at the Schwarzschild radius, which previously lay inside the star. Looking back at our equations, something very strange happens on the shell at the Schwarzschild radius: the (1 – RS/R) factors are equal to zero. Even more dramatically, as we continue further inwards, these factors become negative. What does this mean? From the perspective of someone in freefall across the shell at the Schwarzschild radius the Equivalence Principle informs us that nothing untoward happens. And yet, from a distant perspective, the shell is a place where clocks stop and space has an infinite stretch.

To understand what is happening, it helps to draw some pictures. Before plunging in with Penrose diagrams, we can learn something from a spacetime diagram. As with the diagrams of flat spacetime we’ve already met, there are many ways to construct these diagrams (corresponding to different choices of grid). We will use Schwarzschild’s coordinate grid since we have just seen that something interesting happens at the Schwarzschild radius. Figure 4.6 shows the light cones at each point in Schwarzschild spacetime. This should be contrasted with the corresponding diagram in flat spacetime. If spacetime is flat, the light cones are all aligned and point vertically upwards, but that is not the case in Schwarzschild spacetime. Far from the Schwarzschild radius, the light cones do look like those in flat spacetime, but as we approach the Schwarzschild radius the cones get narrower and narrower. At the Schwarzschild radius the light cones are infinitely narrow, which means an outgoing beam of light can only travel in the time direction and can never climb away from the hole.* Now we can appreciate that the Schwarzschild radius is also the event horizon: an outgoing beam of light emitted at the Schwarzschild radius stands still.

Inside the horizon, the light cones have flipped round. This is because the (1 – RS/R) factor has become negative, which means the factor in front of dt2 gets a minus sign and dR2 gets a plus sign. It is as if space and time have switched roles, but in fact what has switched roles is our interpretation of the Schwarzschild t and R coordinates. Because light cones open out around the Schwarzschild R direction, this is the direction of ‘time’ for anything inside the horizon, and the Schwarzschild t direction is now ‘space’. Since Schwarzschild coordinates correspond to measurements made using clocks and rulers for someone far from the black hole, this means that what is time for someone inside the black hole is space for someone far away, and vice versa. As we’ve been at pains to emphasise, the coordinates we use don’t have to correspond to anyone’s idea of space and time: to quote Einstein again ‘they do not have to have an immediate metrical meaning’. The Schwarzschild R and t coordinates do happen to have a nice interpretation far away from a black hole, but inside the horizon their roles flip. The startling consequence is that an object inside the horizon moves inexorably towards the centre of attraction at R = 0, just as surely as you move inexorably towards tomorrow.

Figure 4.6. Schwarzschild spacetime. The t and R coordinates are those used by Schwarzschild. Notice how the light cones tip over at values of R smaller than the Schwarzschild radius.

We haven’t said much about the centre of attraction yet. Inside a black hole, this is the singularity, the ‘place’ where Einstein’s theory and Schwarzschild’s solution break down. The quotation marks are appropriate because the singularity isn’t really a place in space. It is a moment in time: the end of time that lies in the future for all who dare to cross the horizon. Figure 4.6 illustrates very nicely that the singularity lies in everything’s future inside the horizon, because all the light cones point towards it. It’s also evident from Figure 4.6 that the singularity is not a point in space, which is what we are tempted to think when we look at Figure 4.5. The time and space role reversal means that it is an infinite surface at a moment in time. Let’s explore this remarkable claim in more detail by plunging in and drawing the Penrose diagram for a Schwarzschild black hole.

The Penrose diagram for the eternal Schwarzschild black hole

The Penrose diagram for the eternal Schwarzschild black hole is shown in Figure 4.7. It is built from two portions: the diamond shape to the right corresponds to the universe outside the black hole. The triangle at the top corresponds to the interior of the black hole and the dividing line between the two is the event horizon. It is a 45-degree line because the horizon is lightlike, which means that light can ‘get stuck’ on it. The singularity is the horizontal line at the top edge of the triangle. It is horizontal because it corresponds to the inexorable future of anything that falls inside the horizon. To see all of this, recall that light cones are always oriented vertically upwards on a Penrose diagram, and worldlines always head into future light cones.

Figure 4.7. The Penrose diagram corresponding to the eternal Schwarzschild black hole. The grid corresponds to lines of fixed Schwarzschild coordinates (any similarity to the grid drawn on the Penrose diagrams in the last chapter is accidental).

We have drawn a grid on the diagram, just as we did for flat spacetime. This grid is Schwarzschild’s grid. In the diamond region, the roughly horizontal lines are lines of constant t and the roughly vertical lines are lines of constant R. The event horizon lies at R = 1.† In the interior of the black hole, we can see the role reversal of space and time because the lines of constant Schwarzschild t now run vertically and the lines of constant Schwarzschild R are horizontal. Unlike Figure 4.6, the Penrose diagram is constructed such that the future light cones always point vertically upwards, which means that time is always up and space is always horizontal at every point.

A nice feature of this diagram is that we can use it to describe a black hole of any mass we want. The supermassive black hole in M87, for example, has a Schwarzschild radius of around 19 billion kilometres, corresponding to a mass of 6 billion Suns. An object at R = 2 would then be hovering 19 billion kilometres above the event horizon. If instead we want to describe a black hole with the mass of our Sun, an object at R = 2 would be hovering a mere 3 kilometres above the event horizon. The same thing works for Schwarzschild time too, with one unit of t corresponding to around 18 hours for M87* or (dividing by 6 billion) 10 microseconds for a black hole of one solar mass.

To get more of a feel for the Schwarzschild spacetime, we can classify the edges of the Penrose diagram, just as we did for flat spacetime. The two 45-degree edges on the right of the diamond correspond to past and future lightlike infinity. Only things travelling at the speed of light can come from or reach these places. The right-hand apex of the diamond, where these two edges meet, corresponds to spacelike infinity. The bottom and top apexes of the diamond are past and future timelike infinity. This is very similar to the Penrose diagram of flat spacetime. The new feature is the horizontal line at the top of the diagram, labelled ‘the singularity’. We can gain a good deal of insight by enlisting the services of two more intrepid astronauts who are exploring the black hole in the centre of M87. Their worldlines are illustrated in Figure 4.8, which shows that both the astronauts begin their voyage of exploration at R = 1.1; it’s as if we’ve plonked them gently into the spacetime. Blue is a very relaxed astronaut and chooses to do nothing at all. He has rocket engines, but he doesn’t bother to switch them on and freefalls across the horizon and into the black hole. Red is more sensible. She immediately flicks the switch on her rocket engines and accelerates away from the black hole. Her acceleration is sufficient to escape the black hole’s gravitational pull, and she later flicks the switch on her engine and coasts happily away to future timelike infinity.

Figure 4.8. The journeys of Blue and Red in the vicinity of a Schwarzschild black hole. The dots lie on their worldlines and are 1 hour apart in the case that the black hole is the one in the centre of M87.

Just as in Chapter 3, we have marked the astronauts’ journeys with dots, and the spacing of the dots corresponds to one hour as measured on their watches. There are an infinite number of dots along Red’s worldline, because she is immortal and lives into the infinite future. Things are very different for Blue, however. Undergoing no acceleration, he initially feels as if he is floating. In accord with the Equivalence Principle, there is no experiment he can do inside his spaceship to tell him he is in the vicinity of a black hole, but there is a shock in store in his future. After crossing the horizon, there are only 20 dots on Blue’s worldline. At some point during the twentieth hour, something bad happens. The relaxed immortal’s worldline ends. He meets the singularity. As you can see from the Penrose diagram, the singularity is unavoidable.

All the immortals in flat spacetime live forever, no matter how they move. In Schwarzschild spacetime, every worldline that enters the upper triangle must end on the singularity. Nobody is immortal once they travel beyond the horizon of a black hole. The interior of a black hole is a fascinating place; a Dantean wonderland where all hope would appear to be abandoned as space and time flip roles and the end of time awaits. But there is much more to say, so let’s cross the horizon and explore.

BOX 4.1. The surface of the Earth

A good way to see the difference between coordinate distances and ruler distances on a curved surface is to think about the surface of the Earth. The coordinates we often choose to label points on the Earth’s surface are latitude and longitude, and they are not simply related to the lengths of rulers. The latitude of London is approximately 51 degrees North. The City of Calgary in Canada also sits at around 51 degrees North, and its longitude is 114 degrees west of London. If we ask a pilot to fly between the cities at constant latitude, the distance the aircraft travels will be approximately 5,000 miles, corresponding to a journey of 114 degrees in our chosen coordinate system. If we made the same 114-degree journey from Longyearbyen, the most northerly city on Earth sitting inside the Arctic Circle at 78 degrees North, we’d only travel around 1,600 miles. We therefore need a metric that translates coordinate differences into distances at different places on the surface.

Think of the metric as a little machine that takes coordinate differences between two points and spits out the real-world ruler distance over the curved surface between those points. The metric encodes two things: it understands how to deal with our coordinate choice, which is completely arbitrary, and it encodes the geometry of the surface – in this case a sphere – which is a real thing. This is how the curvature and distortion of a surface is dealt with mathematically.

* You might be inclined to think ingoing beams are also stuck forever on the horizon. This is the case from the perspective of someone far from the hole (for whom Schwarzschild time is their clock time). But it does not mean things cannot fall into the black hole from their own perspective. We shall have more to say on this in Chapter 5.

† This is because we’ve chosen to label R in units of the Schwarzschild radius. With this choice, an object accelerating to hold at a fixed distance of twice the Schwarzschild radius from the black hole will be represented by a worldline that runs along the R = 2 grid line.