Black Holes: The Key to Understanding the Universe

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.

Figure 6.1. The Penrose diagrams for Minkowski spacetime (left) and the eternal Schwarzschild black hole (right). Rindler is accelerating through the spacetime on the left and Dot is hovering outside the black hole (at R = 1.01) on the right.

Now look at the Penrose diagram for the eternal Schwarzschild black hole shown on the right of Figure 6.1. Just like the diagram for flat spacetime, this represents an infinite spacetime. We might expect that every vertex and edge of the diagram should lie either at infinity or the singularity. But this is not the case. What does the entire left-hand edge of the Penrose diagram represent? It doesn’t lie at infinity but at an R coordinate of 1. This is the horizon of the black hole. Up to now, we’ve been focusing on the part of the horizon that runs along the top left edge of the diamond-shaped region, beyond which lies the interior of the black hole, because this is the gateway for our brave astronauts. But what about that left-hand edge? We did not worry about it before because nothing travelling in the diamond can cross it, but since it does not lie at infinity, could there be something lurking beyond?

Figure 6.2. The Penrose diagram of the ‘maximally extended’ Schwarzschild spacetime. The grid lines correspond to so-called Kruskal–Szekeres coordinates as described in Box 6.1.

On the flat spacetime diagram of Figure 6.1, we’ve drawn the worldline of Rindler, the ever-accelerating astronaut we met in Chapter 3. He found himself hemmed in by horizons and lived out his existence in a smaller region of spacetime than his fellow immortals by virtue of his acceleration. Now look at the astronaut whose worldline is depicted on the Schwarzschild spacetime diagram. Let’s call her Dot. She is also accelerating constantly, but in the curved spacetime in the vicinity of the black hole, this means that she hovers at a constant distance just outside the horizon at R = 1.01. Nevertheless, her experience inside her accelerating spacecraft is very similar to Rindler’s. She can send signals across the event horizon of the black hole but cannot receive signals from inside. Likewise, Rindler can send signals into region 2 but cannot receive signals from it. For Rindler, we also recognise the presence of a horizon isolating him from region 4. He cannot travel to region 4, but he can receive signals from there. What is the meaning of the lower left-hand edge of Dot’s diamond? Is the same true for her? Can she receive signals from across the lower horizon, and if so where are they coming from? There is something strange about this line. It is the edge of the Schwarzschild Penrose diagram, but it does not lie at infinity. Why can’t there be something on the other side? In 1935, Albert Einstein and Nathan Rosen were the first to realise that there can be something on the other side.* In their words: ‘The four-dimensional space is described mathematically by two … sheets … which are joined by a hyperplane … We call such a connection between the two sheets a bridge.’19 Today, this ‘Einstein–Rosen Bridge’ is also known as a wormhole.

It turns out that we’ve only been drawing half of the Schwarzschild solution to Einstein’s equations. It is part of a larger space known as the maximally extended Schwarzschild spacetime, which is a bit of a mouthful. The eternal Schwarzschild black hole is to maximally extended Schwarzschild spacetime as Rindler’s quadrant is to Minkowski space; a piece of a larger whole. We’ve drawn the maximally extended Schwarzschild spacetime in Figure 6.2.

The most striking things about Figure 6.2 are the entirely new regions of spacetime that have appeared: regions 3 and 4. Given that region 1 was the entire infinite universe outside of the black hole and region 2 was the region inside containing the end of time, you may be forgiven for wondering what regions 3 and 4 could possibly be. Let’s explore.

Being a Penrose diagram, time runs upwards and all light rays travel at 45 degrees. We can therefore draw light cones at any point on the diagram and immediately see how the different regions are connected to each other. Things can travel from region 4 into regions 1, 2 and 3, but the reverse is not possible. Region 3 is inaccessible from region 1 and vice versa. This means that the 45-degree lines that cross in the middle of the diagram are horizons. An astronaut from region 1 could jump into region 2, the interior of the black hole, and another astronaut could jump in from region 3. They could meet up to have a chat inside the black hole before their rendezvous with the singularity at the end of time (the horizontal line at the top). We see that region 3 is a whole other infinite universe and it is completely separated from region 1, but linked somehow inside the black hole.

Another striking new feature is the horizontal line at the bottom of the diagram, which is also a singularity. Nothing ever falls into this singularity, and anything inside region 4 that lives long enough must eventually cross one of the horizons and enter regions 1 or 3. Anyone in ‘universes’ 1 or 3 could therefore encounter stuff that emerges across the horizons from region 4. This is the reverse of a black hole. It is called a white hole. The black hole lies in the future for astronauts in the two infinite universes, and they may or may not choose to fall into it. Conversely, the white hole lies in the past for these astronauts. They may receive signals from it, but they can never visit it. This is a rather dramatic turn of events.

We’ve referred to this diagram as the maximally extended Schwarzschild spacetime. The term ‘maximal’ has a technical meaning which, unlike many technical terms, is quite illuminating. The astronauts we’ve followed on their journeys around and into the black hole are immortal, which means that their worldlines should be infinitely long unless they hit a singularity. They live forever unless they cross the horizon of the black hole. This means that their worldlines must begin and end at infinity or on a singularity. A spacetime is maximal if it has that property. The spacetime representing the eternal Schwarzschild black hole in Figure 6.1 does not have that property because we can draw a worldline that enters the diagram on the left-hand edge. The maximally extended Schwarzschild spacetime is different. Every edge of the diagram lies either at infinity or on a singularity. For the Schwarzschild spacetime, this diagram is all there can be.

BOX 6.1. Kruskal–Szekeres coordinates

The grid we’ve drawn on Figure 6.2 is different to the Schwarzschild grid we’ve been using so far. Remember that we can choose any grid we like. Nature has no grid. These grid lines are marked out using Kruskal–Szekeres coordinates, discovered by Martin Kruskal and independently by George Szekeres in 1960.† The Kruskal–Szekeres grid lines correspond to spacelike (roughly horizontal) and timelike (roughly vertical) slices of the spacetime. Notice that the bunching up of the Schwarzschild grid lines at the horizons is avoided in Kruskal–Szekeres coordinates, which is more in line with the experience of time for astronauts who fall through the horizon without noticing anything strange. That said, it’s worth remembering that the behaviour of Schwarzschild time at the horizon does tell us something important – that distant observers outside of the black hole see in-falling objects freeze on the horizon. To emphasise again, the coordinate grid we choose to locate events is a free choice, and different coordinate grids are more or less useful to different observers. The Schwarzschild grid is useful in describing the experience of observers outside the black hole because the Schwarzschild time coordinate corresponds directly to something measurable – it is the time as measured on clocks far away from the black hole. Kruskal–Szekeres time does not have that interpretation, but if we want to think about travelling across the horizons, the Kruskal–Szekeres grid is better.

Into the wormhole

Let’s now explore the connection between the two universes in the maximally extended Schwarzschild spacetime. So far, we’ve mostly visualised spacetime using Penrose diagrams that represent a single dimension of space. These diagrams are a great way of visualising the relationship between events in different regions of spacetime – who can influence what and when – but they are not so good for visualising the curvature of spacetime. We can construct a more intuitive visual picture by using what are known as ‘embedding diagrams’.

A good way to understand embedding diagrams is to return to the surface of the Earth. We are going to erase from our minds the idea that the Earth is a sphere in three dimensions and think only of its surface, which is two-dimensional. Picture the surface as a kind of flatland, like the one we encountered in Chapter 2. Flat Albert and his flat companions are now busy doing all the things that geometers do to make themselves happy. They draw circles on the surface and calculate the value of π, which they will discover to be different to the value in Euclidean geometry. They will discover that if they travel far enough over this flatland, they’ll arrive back where they started. If they have drawn a Mercator projection map, they will associate the points along the left and right edges with each other. They may also come to understand that they have introduced a great deal of distortion at the top and bottom of the map by their coordinate choice and will no doubt be motivated to find some new, complementary coordinates to better understand the polar regions. The important point is that all these observations and properties could be a feature of a two-dimensional universe with no third dimension. We are three-dimensional beings, and we can visualise a third dimension. As a consequence, we notice that there is an elegant way of representing this geometry by ‘embedding’ it in three dimensions. Flatland would then be represented as the curled-up surface of a sphere. The important thing is that the third dimension is not necessary and need not even exist. The three-dimensional space into which we imagine flatland to be embedded could be a hypothetical space (sometimes referred to as a hyperspace). Flat philosophers may enquire as to whether a third dimension really exists or not, but flat navigators will not care one way or the other. We 3D beings can use this third (hypothetical) dimension in our imaginations to visualise the curvature of the two-dimensional surface of flatland and gain a new perspective on the geometry. Before some flat-earther misconstrues this analogy, let us make it very clear that the Earth is actually a sphere in three-dimensional space. This is merely an analogy to help our understanding, and hopefully theirs. The point we wish to emphasise is that the ‘curvature’ observed by Flat Albert and his flat companions could be an intrinsic property of their two-dimensional space and does not require the existence of a third dimension. In our real Universe, the ‘curvature’ of our four-dimensional spacetime (which we experience as gravity) is not, as far as we can tell, a result of us living on a surface that is curved into a real fifth dimension. We don’t think we are like the flatlanders, oblivious to some higher-dimensional universe into which our spacetime is curved.

In this sense, the word ‘curvature’ is a little misleading in general relativity because it encourages us to imagine a surface curving into an extra dimension. But curvature is a quantity that can be calculated directly from the metric with no reference to ‘extra’ dimensions at all. John Wheeler managed to say everything we’ve just said in the three-word title of a section in his book with Edwin F. Taylor:20 ‘Distances Determine Geometry’. The authors ask us to imagine a ‘fantastically sculpted iceberg’ floating on a ‘heaving ocean’. To map its curving shape, we can imagine driving thousands of steel pitons into the ice and stretching strings between them. Then we note the positions of the pitons‡ and the lengths of the strings down in a book. This book contains all the information necessary to reconstruct the geometry of the iceberg, including the curvature of the surface. In spacetime, the pitons are the analogue of events – ‘the steel surveying stakes of spacetime’. The distances between nearby events are the intervals. The book is the metric. Nowhere is there any reference to an extra dimension into which the iceberg is curved.

Given that we are three-dimensional beings, we can use our imaginations to picture the curvature of two-dimensional spatial slices of spacetime, just as we could imagine the geometry of flatland as the surface of a sphere. This is the beauty of embedding diagrams.

Figure 6.3. Representing a spacelike slice through spacetime.

Before we head into the black hole, let’s warm up by looking at spacetime in the vicinity of the Earth. Outside the planet, this will be described by the Schwarzschild metric which has three dimensions of space and one dimension of time. Imagine taking a slice of space through Earth’s equator at a moment in time. In the language of relativity, this will be a two-dimensional spacelike surface. On the left of Figure 6.3 we’ve drawn a Penrose diagram with such a slice through it. Earth sits at point O, and the slice runs from the Earth to X. If Earth wasn’t there, this would be the Penrose diagram for flat spacetime. We’ve represented the slice OX through flat spacetime by a straight line at the top of the diagram. If we spin this line around O, we’ll generate a sheet of two-dimensional space (the slice through the equator) centred on O. This is our embedding diagram, and for two-dimensional flat (Euclidean) space it looks like a sheet of graph paper.

Figure 6.4. Five different slices through maximal Schwarzschild spacetime. Each slice can be regarded as all of space at a moment in time.

If we now place the Earth at O, spacetime will be curved and the curvature outside the planet will be described by the Schwarzschild metric. To an astronaut in space close to the Earth, the curvature could be detected by making measurements of the distance between neighbouring events using a ruler, just as the surface of Wheeler’s iceberg can be described using the lengths of pieces of string stretched between steel pitons. If you recall, the distance measured by the astronaut’s ruler between two events, one slightly closer to Earth than the other, would be larger than expected had the space been flat. As for the case of ‘curved’ flatland, we could interpret this distortion with no reference at all to an imaginary extra dimension. Or, we could ask what shape the slice of space would have to be to produce the measured distortion if it were curved into an extra dimension. This is the grid we’ve sketched on Figure 6.3. From this perspective, the Earth makes a dimple in the fabric of space. Now let’s construct some embedding diagrams to explore the geometry of the black hole.

Figure 6.5. An embedding diagram of the spacelike slice YJIHX through the eternal Schwarzschild black hole, as described in the text. We can see the wormhole.

Figure 6.6. A wormhole in flatland.

In Figure 6.4 we’ve drawn five spacelike slices through maximally extended Schwarzschild spacetime. They all span the diagram from X to Y (the two spacelike infinities). These slices are snapshots of the geometry at different moments in time,§ with earlier times towards the bottom of the diagram and later times towards the top. Let’s focus first on the slice labelled (from right to left) YJIHX. We’ve drawn this slice as a line in Figure 6.5, just as we did for the spacetime around the Earth in Figure 6.3. The circles are encouraging you to think about the surface generated when we sweep the line around, but hold that thought for a moment and concentrate on the line. The line is flat towards Y because space is flat far away from the black hole. As we move inwards from Y to the event horizon at J, space starts to curve. So far so normal. On crossing the horizon, however, the line continues bending around until it crosses the second horizon at H. It then flattens out again as it approaches X. Now we can spin this line around as we did in Figure 6.3, and then we see what this interesting geometry corresponds to. Remarkably, we have two flat regions of space joined by what John Wheeler called the throat of a wormhole and what Einstein and Rosen called a bridge. The flat regions can be thought of as two separate universes linked by a wormhole.

A more artistic rendering of the wormhole is shown in Figure 6.6. Thinking in only two dimensions of space, we can imagine Flat Albert and his friend sliding around the black hole. There is an infinite space inside. We humans can see how this works because we can picture the space curving in a third dimension, but for the flatlanders the idea would seem very strange. Similarly, you could imagine wrapping your hands around a low-mass maximally extended Schwarzschild black hole. Its horizon would be a tiny perfect sphere, but inside your cupped hands would reside an infinite other universe.

Figure 6.7. Embedding diagrams of three other spacelike slices from Figure 6.4. Time increases from bottom to top. We can see how the wormhole stretches open and snaps, leaving two disconnected universes. Nothing can travel through the wormhole before it snaps off.

In Figure 6.7 we’ve drawn three more of the embedding diagrams representing the spacelike slices shown in Figure 6.4. We can see how the horizons move apart as the wormhole lengthens and eventually breaks to expose the singularity. The slice at the bottom occurs at earlier time than the slice at the top, which means that the wormhole evolves in time.¶ It is this evolution that prevents anything from travelling through the wormhole (see Box 6.2 for more detail). We don’t need to draw wormholes to appreciate that nobody can travel from region 1 to region 3, and vice versa, which is what a journey through the wormhole would entail. That much is evident from the Penrose diagram since there is no line you can draw at an angle less than 45 degrees to the vertical that connects these two regions. However, the embedding diagrams of the wormhole provide a lovely picture of how the evolution of the wormhole renders travel through it impossible.

Figure 6.8. An astronaut (the dot) falling into a Schwarzschild black hole. Time advances from the top left to bottom right. Notice how the wormhole grows and pinches off before the astronaut can reach the other side. (Illustrations by Jack Jewell)

Figure 6.8 is a visualisation of the geometry of the maximally extended Schwarzschild spacetime together with an astronaut as he falls into the black hole. The wormholes are constant (Kruskal) time embedding diagrams. In the top left picture, the astronaut is approaching the horizon and the wormhole is open, connecting the two universes together. On the top right, the astronaut is about to pass through the horizon. He is still in region 1, but the wormhole has already passed its maximum diameter and is beginning to close. In the next image, he has crossed the black hole’s horizon and the wormhole is pinching closed, which it has done by the time of the second image on the bottom row. We see that the astronaut cannot traverse the wormhole because it pinches shut before he can pass through it. None of this is peculiar to any particular astronaut or how they manoeuvre on their journey into the black hole. The slamming of the door between universes has nothing to do with the details of the journey, and there is nothing anyone can do to change it. This story is written entirely within the Schwarzschild metric, the unique spherically symmetric solution of Einstein’s equations. How wonderful.

BOX 6.2. Evolving wormholes

Let’s start by writing down the Schwarzschild metric again:

The (1 – RS/R) terms in front of the time and space coordinates tell us about the geometry of the spacetime – how it deviates from flat. These terms do not depend on time outside the horizon, which means the geometry does not change as t changes. The words outside the horizon are important. Inside the horizon, the space and time coordinates flip around such that the Schwarzschild R coordinate takes the role of time. If you recall, a key feature of life inside the horizon is that everything is compelled to move to smaller and smaller R, just as outside everything is compelled to move forwards in time. Why? Because the interval must always be positive along the worldline of anything with non-zero mass. This means that dt2 cannot be zero outside the horizon and dR2 cannot be zero inside the horizon. The ticking of time drives us forwards in t outside the horizon and forwards in R inside the horizon. This is why R is the time coordinate inside the horizon. But the (1 – RS/R) terms depend on R, which means that inside the horizon the geometry is compelled to change, just as inexorably as we are compelled to journey towards tomorrow. This is why the spacetime geometry is dynamic inside the horizon. It changes, and in such a way that not even light can make it through the wormhole.

Although travel from one universe to the other is impossible because the wormhole pinches shut, we have noted that it is possible for someone to jump into the black hole from region 1 and meet up with someone jumping in from region 3 before they both end up at the singularity. That is easy to see using the Penrose diagram. Moreover, someone inside the black hole in region 2 can see things in both regions 1 and 3 because they can receive signals from them. That is also evident from the Penrose diagram. This means that, in the moments between jumping into the black hole and hitting the singularity, our intrepid astronauts would be able to peer through the wormhole and see the universe on the other side.

We must of course ask whether any of this might play out in our Universe. Sadly, the answer appears to be ‘probably not’, at least for the case of astronauts attempting to travel between universes. That is because the maximally extended Schwarzschild spacetime does not correspond to the geometry of spacetime created by the gravitational collapse of a star. Rather, the Schwarzschild solution is only valid in the region of empty space outside of the star. The maximally extended Schwarzschild spacetime in Figure 6.2, replete with wormhole and black and white holes, would be the correct description of a non-spinning, eternal black hole. We are not aware that such things exist.

Why ‘probably not’? Because wormhole geometries are valid solutions of Einstein’s equations. In 1988, Michael Morris, Kip Thorne and Ulvi Yurtsever explored the possibility of keeping the wormhole open.21 ‘We begin by asking whether the laws of physics permit an arbitrarily advanced civilisation to construct and maintain wormholes for interstellar travel?’ These wormholes would not be constructed by collapsing stars, but could conceivably be ‘pulled out of the quantum foam and enlarged to classical size and stabilised, potentially, by quantum fields with a negative energy density’. This is great fun but very speculative. As the authors state, such a wormhole would be a time machine, and the consequences of such things existing are disturbing. ‘Can an advanced being measure Schrödinger’s cat to be alive at an event P (thereby collapsing its wave function into a live state), then go backwards in time via the wormhole and kill the cat (collapse its wave function into a “dead” state) before it reaches P?’ Putting the fun aside, these ideas have resurfaced in the search for a resolution to the black hole information paradox. In particular, the idea that microscopic wormholes could be part of the structure of spacetime is part of the ER = EPR hypothesis we’ll meet in the final chapters. So maybe there are time machines in our Universe after all. In any event, the maximal Schwarzschild extension is a solution to the equations of general relativity, and it is a very interesting and beautiful one at that. It alerts us to the outrageous possibilities that spacetime may have in store for us and, as we shall see next, wormholes aren’t even the half of it.

* As early as 1916, in his paper, ‘Contributions to Einstein’s Theory of Gravitation’, Ludwig Flamm had anticipated much of the Einstein–Rosen solution, but he does not appear to have identified the ‘bridge’.

† Kruskal told John Wheeler about his coordinates that covered the entirety of the maximally extended Schwarzschild spacetime smoothly but didn’t bother to publish the idea. Wheeler wrote up a short paper on the matter and sent it for publication with Kruskal as sole author, originally without Kruskal’s knowledge. George Szekeres discovered the same coordinate system, also in 1960.

‡ We might arrange them in a rectangular grid.

§ These slices do not correspond to constant time slices using an array of identical clocks all at rest with respect to each other. It is a feature of flat spacetime that such a network of clocks can be conceived of, but the warping of spacetime makes it impossible to arrange in general. Rather they are slices of constant ‘Kruskal’ time. Nevertheless, the slices are spacelike in the sense that they have the property that no object can travel along any of the five curves (they are everywhere tilted at less than 45 degrees to the horizontal). These slices through spacetime are the best we can do to define the notion of a snapshot of space at some moment in time.

¶ We should really say ‘Kruskal time’ here, as described in Box 6.1.