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