Black Holes: The Key to Understanding the Universe

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.

Figure 5.1. The Penrose diagram corresponding to the eternal Schwarzschild black hole. Five astronauts start out at rest at R = 1.1. Four of them head towards the black hole. Blue freefalls, Green freefalls with Blue but switches her rockets on after crossing the horizon and attempts to accelerate away from the singularity. Magenta also travels with Blue and Green until she reaches the horizon, after which she accelerates towards the singularity. Red accelerates away from the black hole from the moment her journey begins and succeeds in escaping to infinity. Orange also accelerates away from the black hole, but not enough to escape.

We followed Blue in the previous chapter. He sets out from rest at R = 1.1, freefalls into the black hole and ends up at the singularity. His worldline is marked in units of one hour by his watch and nothing unusual happens from his perspective at the event horizon – he sails through it oblivious. There are 20 dots on his worldline inside the horizon, which corresponds to almost a day inside the black hole before the end of time.†

Green begins her journey alongside Blue and freefalls towards the horizon with him, but on crossing the horizon she panics, shouts ‘Burma’‡ and turns on her rocket engine in a vain attempt to escape. There are only 16 dots along her worldline once she crosses the horizon, which means that accelerating away has resulted in the end of time arriving sooner.§

Magenta takes what might seem, at first sight, to be a more fatalistic view. She decides to fall together with Green and Blue until the horizon, and then she gently accelerates towards the end of time at 480g (which is five times less acceleration than Green). She presses play on Joy Division’s ‘Unknown Pleasures’ and flicks the switch. Perhaps irritatingly for her, this lengthens her stay inside the horizon and she lives longer than Green – her worldline has 17 dots. The spacetime geometry inside the black hole is certainly counter-intuitive.

The maximum time anyone can spend inside the horizon of a black hole corresponds to someone who starts out from rest on the horizon and does absolutely nothing but fall freely to the singularity. Apathy pays. This corresponds to just over a day (28 dots) inside the horizon of the supermassive black hole in M87.

On passing through the horizon, Green and Magenta accelerate away from the apathetic Blue, who’s probably relaxing to Miles Davis. Green accelerates away from the singularity and Magenta accelerates towards it. Blue sees Green recede into the distance above him in the direction of the horizon, and Magenta heads away in the direction of the singularity. They both get smaller and smaller from Blue’s point of view as they disappear into the distance in opposite directions. So far so normal. It certainly looks like Magenta is heading towards her doom at the singularity and Green is doing her best to stay close to the horizon. None of this seems different to the way things would be in any other region of spacetime.

In Figure 5.1, we’ve drawn two light beams that Magenta shines out. The first is emitted just under nine hours after she passed through the horizon and the other at just under 14 hours. Drawing light beams like this is what we should do if we want to investigate what each of our astronauts actually sees. Remember that the beauty of Penrose diagrams is that the light cones all point vertically upwards and open out at 45 degrees. Notice that the earlier light beam intersects Blue’s trajectory. That means Blue sees Magenta emit the beam of light (he receives the light at the point on his trajectory where the beam intersects it). Now for the fun bit. Look at the second beam Magenta shines out. It never intersects Blue’s trajectory, and therefore Blue never sees Magenta turn on her torch to emit that second beam. In other words, Blue never sees Magenta’s final moments, even though she is accelerating away from him. The light from Magenta’s final four dots simply doesn’t have time to enter Blue’s eyes before he reaches the end of time. He would see her way down below him as the last light reached him before he winked out of existence.

If Blue turns around, he’ll see Green way above him attempting to accelerate away from the singularity. Again, he’ll reach the end of time before he sees Green end her days. What’s interesting is that every astronaut has the same experience. Nobody ever sees anyone else hit the singularity. The reason is the horizontal nature of the singularity on the Penrose diagram. It is a moment in time, and we can never see events that are simultaneous with a moment in time. We always see things slightly in the past, because it takes light time to travel to our eyes. This means that nobody falling into the black hole sees anyone else reach the singularity before they themselves reach it – they quite literally never see it coming. If you are struggling to see this, imagine drawing 45-degree light beams all over the diagram. They’ll tell you what each astronaut can and can’t see.

The singularity doesn’t arrive entirely unannounced to the unsuspecting astronauts though because as they approach the singularity they get stretched out by tidal forces: they get spaghettified. Standing on the Earth, the pull of gravity is slightly greater at your feet than at your head, but not so much that you notice that you are being stretched. The gravitational pull of the Moon on the Earth has a similar stretching effect, causing the more noticeable twice daily tides. We can see how tidal forces arise using the Penrose diagram in Figure 5.2.

Figure 5.2. Spaghettification.

The dotted lines correspond to the worldlines of two balls falling into the hole. One ball starts out at R = 2 and the other a little closer in at R = 1.8. Once again, the dots correspond to regular ticks of a clock (imagine a clock glued to each ball). The dots are close together to make it easier for us to see the tidal effects (though harder to count the dots). The 45-degree lines correspond to a light beam bouncing back and forth between the balls. We can use the bouncing light as a ruler to measure the distance between the balls, just as you might use a laser tape measure at home if you enjoy putting up shelves. The numbers are the number of ticks between successive bounces as measured on the watch attached to the lower ball. These numbers correspond to the roundtrip travel time of the light pulses. The key thing to notice is that the time between bounces increases as the two balls fall towards the horizon. This means that the balls are moving apart as they fall towards the black hole.

Imagine now that you are falling into the black hole feet first. Your head and feet will try to move apart but, since they are connected by your body, you’ll instead feel like you are being stretched. For the black hole in M87, the tidal effects at the horizon would be unnoticeable, but you’d begin to feel uncomfortable inside at around R = 5 million kilometres. At some point around 3 million kilometres your head would come off. You’d have been spaghettified. Closer to the singularity, your constituent atoms would be ripped apart. Even more dramatically, for a typical stellar mass black hole you’d be spaghettified before you even reached the horizon.

Let’s now return to Figure 5.1 and ask what things look like to an observer who remains outside the black hole. Red sets out alongside Blue, Green and Magenta but wisely decides to switch on her engines in good time and accelerate away from the black hole. In good time is a ‘relative’ term here – she pulls 864g until Schwarzschild t = 1.5, at which point she decides enough is enough and switches off her rocket engine. Red’s 864g acceleration should probably be accompanied by Kenny Loggins’s ‘Danger Zone’. You can see the moment when Red switches off her rockets because that’s when her worldline takes an abrupt left turn and heads for the apex of the diamond. Red has escaped the black hole and, being immortal, her worldline will continue to future timelike infinity at the top apex of the Penrose diagram. Remember that nothing peculiar happens to anyone, at any point before they start to experience tidal forces and Red manages to avoid those. She does feel the acceleration of her rocket for the first portion of her journey, but once that’s done, she floats around happily inside her spaceship as she coasts into infinity.

What does Red see as she observes her colleagues diving into the black hole? Before Blue, Green and Magenta reach the horizon we have drawn some more 45-degree lines. We can use these to confirm that Red sees the others move in slow motion and that this gets increasingly pronounced as they approach the horizon. To trace the journey of the light as it travels from Blue to Red’s eyes, follow the 45-degree lines from Blue to Red’s worldline. Can you see that two dots on Blue’s in-falling worldline correspond to very many dots on Red’s worldline? This means that Red experiences many hours for every hour that Blue experiences. Dramatically, Red never even sees anybody pass through the horizon because light emitted very close to the horizon heads off at 45 degrees and only reaches her in the far future. This means she continues to receive light from the other astronauts forever. She sees in-falling objects move ever slower as they approach the horizon, until they eventually freeze there. In principle, she can see everything that ever fell into the black hole.

There is a second important consequence of the fact that Red sees the in-fallers in ever-increasing slow motion as they approach the horizon. As we’ve emphasised, the slowing down of the astronauts and watches is not specific to astronauts and watches. Everything slows down, from the rate of ageing of the cells in the astronauts’ bodies to the inner workings of atoms. Time is distorted, and that means every physical process is distorted too. This includes light. Light is a wave and has a frequency, just like sound or waves on the surface of water. Water waves are perhaps the best way to picture a wave if you aren’t familiar with the terminology. If you throw a stone into a still pond, ripples radiate out from the stone. Standing still in the pond, you’ll feel a series of peaks and troughs as the wave passes by. The distance between two peaks is known as the wavelength, and the number of peaks that pass by per second is known as the frequency. For visible light, we perceive the frequency as colour. High-frequency visible light is blue and low-frequency visible light is red. Beyond the visible at the low-frequency end of the spectrum are infrared light, microwaves and radio waves. Beyond the high end lie ultraviolet light, X-rays and gamma rays.

A distant observer such as Red sees the in-fallers by the light they emit, and the frequency of that light reduces as time slows down. The images of the in-falling astronauts therefore become redder as they approach the horizon, and ultimately fade away as the frequency drops out of the visible range and into the microwave and radio bands beyond. This effect is known as redshift. Red sees her in-falling colleagues freeze and fade away as they approach the horizon.

We will end our investigation of the Schwarzschild black hole by commenting on an apparent paradox that’s confused many people in the past. We now understand that nobody outside the black hole ever sees anyone fall through the horizon. But we have also said that astronauts do fall through and can see each other inside the horizon. How does that work? Won’t it be the case that astronauts falling towards the horizon should never see their colleagues ahead of them fall through? Worse still, if they go in feet first, won’t their feet appear to freeze below them? Will they fall through their own feet? The answer is that nobody sees anyone else fall through the horizon until they themselves are inside. Even more bizarrely, nobody even sees their feet cross the horizon until their eyes have crossed it. Let’s look at the Penrose diagram to work out how this can be the case and why, in fact, it is not in the least bit bizarre.

In Figure 5.1 there is an astronaut we haven’t met before who we’ve called Orange. She is listening to Monty Python. She starts out with our other astronauts and attempts to avoid falling in but, being a little silly, she accelerates away from the black hole too slowly and ultimately crosses the horizon. As she approaches the horizon, she sees Blue, Green and Magenta approaching the horizon in slow motion. But, because the horizon is also a 45-degree line, she doesn’t see anyone cross the horizon until the moment her eyes cross it. This also applies to her own feet. She doesn’t see them cross the horizon until the moment her eyes cross it. This sounds weird, as if everything is piling up on the horizon and that Orange falls through some kind of ghostly mirage of everything that ever fell into the hole.

But there is nothing unusual here. Orange no more falls through her own feet than you fall through your own face when you walk towards a mirror. Let us explore that sentence in more detail.

Figure 5.3 shows Orange at two moments in her journey across the horizon; the moment when her feet cross and the moment when her eyes cross. The flash indicates light emitted from her foot at the moment it reaches the horizon. This light remains stuck on the horizon while she falls. From Orange’s perspective, the horizon and the light whizz past her eyes. She sees her feet, but only when her eyes reach the horizon. But this is what always happens when you look down at your feet: the light from them travels up to your head and you see them after the light has been emitted.

Figure 5.3. Orange not falling through her own feet.

What about the light from everyone else who crossed the horizon? All that light is simply waiting around on the horizon until Orange’s eyes pass by and collect it. Again, there is nothing unusual about that. Presumably you aren’t puzzled by the fact that you can see distant cows and nearby cows standing in a field at the same time.

To make these unfamiliar ideas clearer we’ll introduce another way to think about spacetime around a black hole: the river model. The river model was so-called by Andrew Hamilton and Jason Lisle and it has an impeccable pedigree.18 It was formulated by Allvar Gullstrand in 1921, who had previously won the 1911 Nobel Prize in Physiology or Medicine for his work on the optics of the eye. French mathematician Paul Painlevé discovered the model independently in 1922, in between his two stints as Prime Minister of France. (Given the intellectual abilities of some recent holders of high office in the United Kingdom and elsewhere, the preceding sentence assumes an almost comedic quality.) In 1933, Georges Lemaître showed that the river model correctly describes a Schwarzschild black hole but with a different choice of grid.

Figure 5.4. The river model of a black hole. (Wendy lucid2711/Shutterstock, annotated by Martin Brown)

In the river model, we are entitled to think of a Schwarzschild black hole by analogy with water flowing into a sink hole, as illustrated in Figure 5.4. The water represents space, which flows into the hole at ever-increasing speed. Light, and indeed everything else, moves over the flowing river of space in accord with the laws of special relativity. We might imagine our astronauts swimming around in the flowing river of space. Far from the black hole, the flow is sedate and they can easily swim away upstream. As they approach the black hole, the flow gets faster and they find it increasingly difficult to escape. At the horizon, the flow reaches the maximum speed that anything can swim (the speed of light), and since nothing can travel faster than light, this is the point of no return. Inside the horizon, the river flows faster than the speed of light and gets ever faster as it approaches the singularity. Anything that strays beyond the horizon will be caught up in the superluminal flow and inexorably swept to its doom. On the horizon, something swimming radially outwards at the speed of light will go precisely nowhere. It will remain frozen forever on the horizon.

This picture makes Orange’s experience on crossing the horizon very clear. She is stationary in the river of space, but the river is flowing across the horizon, sweeping her inwards. Particles of light (photons) emitted from her feet will head outwards at the speed of light as normal, but because the river is flowing inwards at the speed of light, the photons from her feet that will encounter her eyes remain frozen at the horizon. Orange’s head is swept across the horizon in the river at the speed of light, where she meets these photons and sees her feet. Thus, the photons from her feet enter her eyes after they’ve been emitted, travelling at precisely the speed of light. If you look down at your feet now, this is also exactly what happens. As she crosses the horizon, Orange experiences the world precisely as you are doing now.

We can invoke the river model to visualise other phenomena we have previously described using the Penrose diagram. Tidal effects arise because the river flows faster closer to the hole, which will cause two astronauts swimming at the same speed but at different radial distances to drift apart as they fall inwards. On our Penrose diagram, we charted only a single spatial dimension – the radial direction. The river model is two-dimensional, and that allows us to see some additional effects. Because the flow is converging inwards to the singularity, objects close to the hole will be squeezed in the tangential direction as well as being stretched in the radial direction. Two astronauts will get closer together tangentially as they head inwards but be pulled apart radially, experiencing a kind of double spaghettification. Heading towards a black hole, feet first, you get thinner and longer.

We can also picture why a distant observer far from the hole never sees anything cross the horizon. Think of photons as fish. Suppose someone in a canoe heading towards the horizon drops fish overboard once every second by their watch. The fish swim away upstream to a boat far away in comparatively still water. At first, the fish can swim upstream easily and arrive at the boat close to one second apart. But as the canoe drifts closer to the horizon, the fish struggle to swim away against the quickening flow and so the arrival time of the fish at the boat increases. This is the redshift effect we discussed above. On the horizon, the fish dropped from the canoe enter a river flowing at the maximum speed they can swim. They never escape upstream to reach the boat, and so the observer on the boat never sees the canoe cross the horizon.

In this chapter, we have explored the topsy-turvy world in and around a Schwarzschild black hole, and we have learnt what it feels like to jump into a black hole or to watch someone as they jump into one. Now, it is time to explore further, and to introduce a feature of general relativity beloved of science fiction writers and which may ultimately prove to be a key idea if we are to understand what space really is. Wormholes.

BOX 5.1. Small. Far away. Why does an astronaut see their colleagues as far away when they cross the horizon if the light from them is frozen on the horizon?

Let’s imagine that Orange is falling through the horizon feet first. In the river model, we would picture her as floating in the river with her feet pointing downstream. At the moment her feet reach the horizon, a pair of light-speed fish set off from her feet. Let’s call these light-speed fish ‘photons’ because that’s what they represent. These particular photons are travelling at just the right angle to enter her eyes. There will be photons heading out in all directions from her feet of course, just as there are photons reflecting off your feet now in all directions, but only those that are heading in the correct direction will enter your eyes.

Figure 5.5. Photon-fish from Blue’s feet enter Orange’s eyes at a smaller angle than the photon-fish from Orange’s own feet. Both enter the eyes at the same time, but Orange sees her own feet to be normal-sized and Blue to be distant and small.

The photons we’re considering are emitted at just the correct angle such that they will have moved inwards to meet Orange’s eyes as those eyes pass by. It helps to think about the photons as fish swimming against the river, which is flowing vertically downwards in Figure 5.5. If they swim vertically at the same speed as the river, they will miss Orange’s eyes. But if their path is tilted slightly inwards, they’ll head inwards.¶ That is what they must do to reach Orange’s eyes.

At the moment Orange’s eyes reach the horizon they also encounter the photons coming from Blue’s feet that were trapped on the horizon when he fell through. These photons have been there for longer than the photons from Orange’s feet and they’ve therefore had more time to make their way to meet Orange’s eyes. This means that the photons from Blue’s feet that happen to be at the right position to enter Orange’s eyes as she passes by must have been emitted at a steeper angle, closer to the vertical, than the photons from her own feet. This means that Orange will see Blue to be smaller and, therefore, far away, because the size we perceive something to be is determined by the angular spread of the light arriving on our retina. For example, if we look out into a field, the distant cows look smaller than the ones nearby because they subtend a smaller angle.

Figure 5.6. Small. Far away. (© Hat Trick Productions)

* Our PhD student Ross Jenkinson has a different take on it: ‘My interpretation was that the 5D beings picked him up in a 5D box and saved him from the black hole, carrying him through an unseen dimension, which they represented as him being able to travel through time as if it were a dimension in space. Analogous to picking up a flatlander in some tupperware as they fall through a 3D black hole.’ That’s also not what happens in Nature.

† If the black hole were one solar mass, Blue would have only 14 microseconds after crossing the horizon before reaching the singularity. If you want to explore the interior of a black hole, you should choose a big one otherwise the adventure will be over very quickly.

‡ This is an obscure Monty Python reference. Every popular science book should contain one.

§ The acceleration depicted here for Green is a bone crushing 2,400g. If this were a solar mass black hole, Green would be experiencing 15 trillion g, which would be even more uncomfortable. It is just as well that our astronauts are immortal. To experience 1g, Green would have to dive into a black hole 2,400 times the mass of M87*. The most massive black hole known at the time of writing is ten times the mass of M87*.

¶ This means that the photons reaching Orange’s eyes were actually emitted from her feet ever-so-slightly before they crossed the horizon.