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.
6
White Holes and Wormholes
Penrose diagrams bring infinity to a finite place on the page, and in Chapter 3 we explored how the different types of infinity are depicted at the edges and points of the diamond-shaped representation of flat spacetime. To refresh your memory, we’ve drawn the diagram for flat spacetime again, on the left of Figure 6.1. The upper and lower vertices of the diamond represent the distant past and the far future for anything or anyone who travels along timelike worldlines. We called these past and future timelike infinity. The worldlines of immortals begin and end there. Eternal light beams begin their journeys on one of the bottom edges and end on the opposite top edge. These are past and future lightlike infinity. All infinite ‘now’ slices of space stretch from the left-hand vertex to the right-hand vertex of the diamond. These are spacelike infinity. Every vertex and edge of the Penrose diagram represents infinity in one form or another.