Figure 2.5. Schematic picture of the distortion of spacetime in the vicinity of the Earth.
An entirely reasonable first response to that last paragraph is to ask what on earth are we talking about. What does it mean to talk about distorting space and time? How should we picture distorted spacetime? Thus far, we have been drawing diagrams like Figure 2.1, with time pointing upwards and one or two directions in space represented horizontally. But our world isn’t like that. We live in three spatial dimensions: forwards and backwards, left and right, up and down. Adding a fourth dimension – time – is very hard to picture.
To help us to comprehend the idea of spacetime, let’s take a step back and imagine a two-dimensional world called flatland, populated by flat creatures.*** The flatlanders can wander around, forwards and backwards, left and right, but never up or down. Their flat eyes can only see flat things on the flat surface and their flat brains can only comprehend flat things. Imagine the reception that renowned flat physicist Flat Albert would receive if he dared to say that space is really three-dimensional. ‘There is another dimension, another direction we cannot point in,’ he claims, and with his mathematics he would have no difficulty describing this three-dimensional world.
Suppose that Flat Albert is correct, and the flatlanders actually do live on the surface of a large messy table in an office, as shown in Figure 2.6. The third dimension is real; it’s the direction upwards from the surface of the table, but the flatlanders can’t see it. Their explorations haven’t yet revealed that space comes to an end at the edge of the table, but they have discovered that there are impenetrable regions where they cannot go. They have to walk around the coffee cup and the lamp and books, and they are left to wonder at why the forbidden regions are sometimes circular, sometimes rectangular and occasionally some other less regular shape. Moreover, the land is covered in regions of light and dark that shift and change in shape and size.
Figure 2.6. Flatland.
How did Flat Albert deduce that there is an extra dimension in the world, based solely on observations made on the flat tabletop? ‘It’s all to do with those changing light and dark regions,’ he says. ‘I know what they are. They are shadows.’
Albert used mathematics to figure out that the shadows are two-dimensional projections of objects that live in three dimensions (coffee cups and books) and to deduce the three-dimensional shapes of the objects that cast them. It helps that the shadows change shape occasionally, which Albert correctly interprets as the higher dimensional source of brightness changing. From our three-dimensional viewpoint, we immediately see that this is due to someone moving the table lamp.
Perhaps you can see the analogy. The interval – the thing that doesn’t vary with point of view – lives in the four dimensions of spacetime. Distances in space and differences in time are mere shadows; they vary as we adopt different points of view in the three-dimensional world of our everyday experience. We can’t picture something that lives in four dimensions, just as Flat Albert couldn’t picture a coffee cup or a lamp or a book. But that didn’t stop him lifting his gaze from the two-dimensional world of his experience to marvel at the true reality of three-dimensional space and the unchanging objects that live on the tabletop.
By pushing the flatland analogy a little further, we can also get a feel for how general relativity fits into this picture. The flatlanders may be inclined to assume that their tabletop is flat. If this were the case, parallel lines across flatland would never meet and the interior angles of triangles would add up to 180 degrees. We call such a flat geometry ‘Euclidean’.
If the table is slightly warped, however, the flatlanders will discover small deviations from Euclid. Using precise measuring devices, they will be shocked to find triangles whose angles do not add up to exactly 180 degrees and parallel lines that converge or diverge from each other. It is in precisely this sense that we speak of space being curved or warped in Einstein’s theory of gravity and it is what we are aiming to illustrate in Figure 2.5.
Flatland helps us to picture how space could have more dimensions than we directly experience, and it helps us to picture how space can be warped. By dropping down a dimension we can see the bigger picture of a two-dimensional space (the tabletop) embedded in a three-dimensional space (the room). We can’t step outside of ourselves to see the bigger picture of four-dimensional spacetime because our imaginations are limited to picturing things in three dimensions or less. In that sense we are very much like flatlanders, fated to view the world in too few dimensions.
It’s not easy to become comfortable with the idea of higher-dimensional spaces, but if it offers some comfort, professional physicists are no better at picturing four-dimensional spacetime than you are. When it comes to spacetime, we are all Flat Albert, peering at shadows. Fortunately, it is not necessary to try to visualise spacetime in all its four-dimensional grandeur. Often, we can drop a couple of dimensions in our mental picture and not lose anything important. We’ve already seen this in the spacetime diagrams we have used to explore the basics of special relativity, where (apart from Figure 2.1) we depicted only a single space dimension and the time dimension. Our understanding wouldn’t have been enhanced by trying to draw two-dimensional space, although it might have made our diagrams look prettier. If we’d attempted to draw all three space dimensions plus the time dimension, we’d have run into problems.
We will rarely need to keep track of more than one space dimension in our study of black holes, because our focus will be on the distance from the black hole. We will be most interested in how the warping of the geometry influences the way cause and effect play out and that means keeping track of the light cones. Physicists have had a century to come up with a nice visual scheme for doing this, and the most widely used is named after Sir Roger Penrose. In the following two chapters, we’ll introduce ‘Penrose diagrams’. Armed with these beautiful maps of spacetime, we’ll be ready to navigate beyond the horizon.
* A nanosecond is one billionth of a second.
† We will use imperial units when discussing cricket.
‡ In more technical language, we are assuming that the cricket ground is an inertial reference frame. We might imagine it detached from Earth and floating freely between the stars. We are also neglecting air resistance.
§ He’ll need to make a correction for the time it takes for the light to travel from the ball to his eyes in order to work out when the flash was actually emitted.
¶ This comes as a shock to a certain type of individual.
** This means that the person travelling along the worldline is not accelerating during the segment. Any curving path, through space or through spacetime, can be thought of as being made up of lots of tiny straight-line paths.
†† To see this, notice that the watch never moves relative to itself, so Δτ = Δt because Δx = 0.
‡‡ Light cones are cones in spacetime if space is two-dimensional because a flash of light will spread out in a circle of increasing radius. In three dimensions, a flash of light spreads out in a spherical shell making a kind of hyper-cone in spacetime. That is not possible to visualise, so we’ll stick to two dimensions of space to make drawing diagrams easier.