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.