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.
5
Into the Black Hole
In the film Interstellar, Matthew McConaughey dives into a black hole called Gargantua and emerges inside a multi-dimensional reconstruction of his daughter’s bookshelves. That’s not what happens in Nature.* But what is the fate of an astronaut who decides to embark on a voyage beyond the horizon into the interior of a black hole? We are now equipped to answer that question for black holes that do not spin, according to general relativity. In Chapter 7, we’ll add some spin and explore the interior of what are known as Kerr black holes. This will allow us to embark on even more fantastical voyages into a wonderland of wormholes and other universes. But first things first.
For our purposes, we are going to recruit three more astronauts to join Red and Blue from the previous chapter in their exploration of the supermassive black hole in M87. Their journeys over spacetime are shown in Figure 5.1. We have marked out their positions as time advances using coloured dots.