§§ Both bars are on, and there is Vim under the sink.
¶¶ Notice that, for pretty much all events we deal with in our everyday lives, (Δτ)2 = (Δt)2 is approximately true. That’s because the distances in space we are usually interested in range over a few metres or kilometres or even a few thousands of kilometres, and all of these are tiny when measured in light seconds. In everyday life Δx is very much smaller than 1 light second, and this is the reason why it feels to us as if time is universal.
*** We are inspired by Edwin Abbott’s 1884 novel Flatland: A Romance of Many Dimensions. And possibly Mr Oizo’s ‘Flat Beat’ featuring Flat Eric.
3
Bringing Infinity to a Finite Place
Physicists often describe general relativity in aesthetic terms; it is the theory to which the word ‘beautiful’ is most often attached. ‘Beautiful’ implies an elegance and economy not easily visible in the mathematics, which is notoriously arcane. There is a well-known anecdote about Arthur Eddington who, when it was put to him that he was one of only three people in the world who understood Einstein’s theory, paused for a moment and replied, ‘I’m trying to think who the third person is.’ Rather the adjective applies to the elegance and economy of the ideas that underlie the theory and to the beautiful idea that gravity is geometry. John Archibald Wheeler expressed this central dogma of general relativity in a single sentence: ‘Spacetime tells matter how to move; matter tells spacetime how to curve.’ The hard part of general relativity is to calculate how spacetime is curved, and for anything other than very simple arrangements of matter and energy, exact solutions to Einstein’s equations are not easy to find. Black holes are one of the few cases in Nature for which we can precisely calculate the spacetime geometry, and once we have the geometry, we can represent it pictorially. The challenge is to find the most useful way of drawing the spacetime around a black hole on a flat sheet of paper. Flat paper is two-dimensional and spacetime is four-dimensional, which makes it difficult to draw (to say the least). If the spacetime is curved, that introduces an additional headache and distortion is inevitable. The trick is to draw the minimum number of dimensions necessary, which as we saw in the previous chapter is often a single dimension of space plus the time dimension, and to choose the distortion such that the features we are interested in are rendered in a way that enhances our understanding.
Figure 3.1. Hand with Reflecting Sphere by M. C. Escher, 1935. (M.C. Escher’s ‘Hand with Reflecting Sphere’ © 2022 The M.C. Escher Company-The Netherlands. All rights reserved. www.mcescher.com)
There is an image with which we are all familiar that introduces distortion in a well-chosen way to represent a curved surface on a flat piece of paper – a map of the surface of the spherical Earth. Many ways of representing the surface have been devised but the one of particular interest is shown in Figure 3.2. The Mercator projection, introduced in 1569 by Gerard de Kremer,* is designed specifically for navigation. Sailors care about compass bearings, and presumably other stuff that’s not relevant to this book, so the Mercator projection is defined such that angles on the map at any point are equal to compass bearings on the Earth’s surface at that point. This means that a navigator can draw a straight line on the map between two places and the angle between the line and the vertical will be the bearing from North that the ship must sail to travel between those places. The price to pay is that distances on the map are distorted, and the distortion increases with latitude. Greenland appears to be the same size as Africa on the Mercator projection, when in fact in area it is over 14 times smaller. The distortion becomes infinite at the poles, which cannot be represented on the map. The Mercator projection is an example of a ‘conformal’ projection, which means that angles and shapes are preserved at the expense of distances and areas.
Figure 3.2. Mercator projection of the Earth’s surface between 85° South and 85° North.
The spacetime diagrams we drew in the last chapter extend infinitely in every direction: time carries on forever up and down, and space extends forever left and right. That’s not necessarily a problem unless we are interested in depicting the physics of forever, but that’s precisely what we would like to do if we want to visualise the spacetime in the vicinity of a black hole. If we are to develop an intuitive picture of a black hole, therefore, we’d like a way of bringing infinity to a finite place on the page. Roger Penrose found an elegant way of doing so.†
Penrose diagrams
Figure 3.3 shows the Penrose diagram for ‘flat’ spacetime. By flat spacetime, we mean a universe without gravity, which is the spacetime of special relativity we discussed in the last chapter. Flat spacetime is often referred to as Minkowski spacetime, after Hermann Minkowski who first introduced the idea of spacetime: ‘The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.’‡ As we’ve discussed, matter and energy distort spacetime, and so in the presence of a planet, star or black hole, the geometry will change. But to warm up, we will focus first on simple, common or garden flat spacetime; a universe with nothing in it to distort the fabric.
Figure 3.3. The Penrose diagram for flat spacetime in the simplified case of one space dimension.
The Penrose diagram for flat spacetime is a thing of beauty: all of space and all of time has been squeezed into a finite-sized diamond-shaped region. Every event in the history and future of an infinite, eternal universe is located somewhere on this diagram. The distortion is extreme, as it must be (we have captured infinity on a single page) but the distortion has been carefully chosen.§ Just as for the Mercator projection, the Penrose diagram is a conformal projection; angles are preserved at the expense of distances on the page. This means that light rays always travel along 45-degree lines, and all the light cones are oriented vertically, just as they were in the spacetime diagrams of the last chapter. At any point, therefore, we can think of time as pointing vertically upwards. Since the light cones define the notion of past and future and tell us whether a particular event can influence another event, they are of utmost importance. If we are interested in how events in spacetime are related to each other in the vicinity of a black hole, a simple and intuitive rule for light cones is what we’re after. The other beautiful feature of the Penrose diagram is that, as advertised, it brings infinity to a finite place on the page; not just one place, but five.
The diamond in Figure 3.3, representing an infinite, flat universe, is centred on a particular event, which we’ve labelled ‘O’. There is nothing special about this central point. We could choose any event in spacetime around which to centre the diagram, but of course we’ll usually choose an event that we are interested in. It would be an eccentric choice to depict things we want to study in the most distorted regions out towards the corners of the diamond, but if we really want to then there’s nothing stopping us. Remember that we are capturing all of space and all of time in this diagram, so every event that ever happened and ever will happen is depicted somewhere. ‘All of space’ is limited to one dimension – left and right – just to make things easier to draw. We could have drawn another spatial dimension, but it wouldn’t add much to our understanding and would complicate the diagram. We’ve shown what adding another spatial dimension looks like in Box 3.2. The freedom to choose where to centre the diagram is also available for the Mercator projection. If we were interested in navigating around the polar regions, for example, we could choose Singapore as the ‘pole’. The distortion would then be extreme around Singapore, but the map would be good for Greenland.