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.