Since we can choose any event in spacetime around which to centre the diagram, let’s go with an important one: your birth. For the purposes of this discussion, and only this discussion, the entire Universe – or at least this representation of it – revolves around you. The diagonal wavy lines that pass through O mark out the future and past light cones of O. We’ve labelled these two regions inside the light cones as ‘O’s causal future’ and ‘O’s causal past’. Any event inside the future light cone can be reached from O and any event inside the past light cone can communicate with O. Since O is your birth, your worldline must snake around inside the future light cone.
The light cones can be thought of as the paths over spacetime of two pulses of light that started their journey in the distant past and happen to pass each other at event O before heading off into the far future. One light beam heads in from the left and another from the right. Remember that there is only a single dimension of space represented on this diagram. We’ve sketched this on a ‘space-only’ diagram at the bottom left of Figure 3.3. Two light beams travel towards and then past O from opposite directions, crossing at O. The flashes, marked on both diagrams, tell us the location of the light beams one day before they reach O and one day after they pass O.
We said that the light beams began their journey in the distant past and headed off into the far future. As we’ve drawn them, each beam originates from one of the bottom diagonal edges of the diamond and ends on the opposite top diagonal edge. The top edges are where all light beams end up if they fly through the Universe forever. You can see that any light beam emitted from any event on the diagram will end up there, because all light beams travel at an angle of 45 degrees. For this reason, the top edges are known as future lightlike infinity. Likewise, any light beam that was emitted in the infinite past will have begun its journey on one of the bottom edges. These edges are known as past lightlike infinity.¶
Now let’s focus on the grid lines on the figure. In the vicinity of O the grid looks like a sheet of graph paper, but closer to the edges the grid is increasingly distorted. This is the ultimate fish-eye lens effect: an infinite amount of space and time are squeezed into the diamond by shrinking spacetime by different amounts from place to place. The further away from the centre we go, the more shrinking is applied. On the Mercator projection of the Earth’s surface, the stretching increases as we move away from the equator and an infinite stretch is applied at the poles, which is why we can’t draw the polar regions. On the Penrose diagram, the shrinking increases as we head away from the centre of the diamond and an infinite shrink is applied at infinity, which is why we can represent an entire infinite universe within a diamond.
The distorted grid is a way to measure the coordinates of events in spacetime, just as the grid of latitude and longitude allows us to measure the coordinates of places on the Earth’s surface. Looking at Figure 3.2, you can see how the distortion of the lines of latitude on the Mercator projection becomes more pronounced towards the poles, which affects the visual representation of the distance between points on the Earth’s surface on the map. It is important to appreciate that the grid itself and the corresponding choice of coordinates is completely arbitrary. On Earth the choice is partly driven by the Earth’s spin and the location of the geographical poles, but it’s also historical. There is nothing about the geometry of a sphere that forces us to measure longitude relative to the meridian that passes through Greenwich Observatory in London.
Likewise, in spacetime any grid can be used although, just as for the Earth, some grids will be more useful than others. For example, non-rotating black holes are spherical and so when we are describing spacetime in the vicinity of a black hole we’ll choose coordinate grids suited to a spherical geometry. It’s worth emphasising, though, that the grids we choose don’t even need to correspond to anybody’s idea of space or time. It’s just a grid, laid over spacetime such that we can label events. All that matters is that when we calculate the interval along a particular path using the grid, that distance will be an invariant quantity (which means it is independent of the grid choice). Analogously on Earth, the distance between London and New York doesn’t depend on how we choose to define latitude and longitude.
That said, the grid on the Penrose diagram in Figure 3.3 does correspond to somebody’s idea of space and time: yours. Let’s focus on the moment of your birth, which we’ve identified as event O in the centre of the diamond. The horizontal dashed line passing though O represents all of space – ‘now’ – according to you. Every event on the ‘now’ line happened at the same time from your point of view at the moment you were born.
We now need to be clearer on what a ‘point of view’ is. Imagine the ‘now’ line is populated by a series of clocks which are all synchronised with the clock present at your birth. They are spaced at uniform intervals along the line; we might imagine them being connected by little rulers. Each clock is at rest relative to the clock at O. As time ticks, this line of clocks will march towards the top of the Penrose diagram into the future. The worldlines of these clocks are represented by the curving vertical lines on the diagram. The clock at O – the clock present at your birth and at rest relative to you when you were born – goes straight up the vertical line into the future. If you don’t move, then the vertical line will also be your worldline. The other clocks also travel along what we will refer to as straight worldlines, even though they are curved on the Penrose diagram.
Let’s say ‘tomorrow’ is exactly 24 hours after your birth. The whole line of clocks will have advanced up the diagram into the future and will now lie along the dashed line labelled ‘O’s tomorrow’.** Likewise, if ‘yesterday’ is exactly 24 hours before you were born, then all the clocks would be located along the line labelled ‘O’s yesterday’. All the curving horizontal lines on the diagram are therefore slices of space that are different moments in time from your point of view.†† Only once we have been this careful can we say precisely what we mean by ‘O’s tomorrow’ (the day after your birth). As this book unfolds, you will come to appreciate the significance of this apparent pedantry. If you stay still for your whole life relative to this ensemble of clocks, the horizontal lines are all your tomorrows, stacked up one after the other. Your tomorrows will end when your worldline ends at the event of your death, but the tomorrows on the Penrose diagram extend into the infinite future. Tomorrow, and tomorrow, and tomorrow, Creeps in this petty pace from day to day, To the last syllable of recorded time.‡‡
We will now identify the remaining three of the five infinities on the diagram. Assuming that our imaginary clocks have always existed and always will exist, the worldline of every clock begins at the bottom vertex of the diamond and ends at the top vertex. If you recall from the previous chapter, anything other than light must move along a timelike worldline and could therefore carry a clock along with it.§§ Any (immortal) object that follows a timelike worldline will therefore begin at the bottom vertex of the diamond and end at the top vertex. The bottom vertex is known as past timelike infinity and the top vertex is known as future timelike infinity: ‘the last syllable of recorded time’.